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You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org. 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"pages": [ "020", "026" ], "concepts": [ "concept/circular-measure", "concept/great-circle", "concept/isosceles-spherical-triangle", "concept/polar-triangle", "concept/polyhedral-angle", "concept/primitive-triangle", "concept/spherical-angle", "concept/spherical-triangle", "concept/supplementary-angles", "theorem/angles-opposite-equal-sides-of-a-spherical-triangle-are-equal", "theorem/greater-angle-lies-opposite-greater-side-in-a-spherical-triangle", "theorem/greater-side-lies-opposite-greater-angle-in-a-spherical-triangle", "theorem/polar-triangle-relation-is-symmetric", "theorem/sides-opposite-equal-angles-of-a-spherical-triangle-are-equal", "theorem/sum-of-angles-of-a-spherical-triangle-lies-between-two-and-six-right-angles", "theorem/sum-of-sides-of-a-spherical-triangle-is-less-than-a-great-circle", "theorem/sum-of-two-sides-of-a-spherical-triangle-exceeds-the-third" ], "excerpts": [ "todhunter-spherical-trigonometry-1886/x-7266e99e1f", 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of Oblique-Angled Triangles", "name": "Todhunter 1886, Solution of Oblique-Angled Triangles", "pages": [ "056", "069" ], "concepts": [ "concept/ambiguous-case", "concept/existence-conditions-of-a-triangle", "concept/isosceles-spherical-triangle", "concept/polar-triangle", "concept/quadrant", "concept/right-spherical-triangle", "concept/spherical-triangle", "method/solving-a-right-angled-triangle", "method/solving-a-triangle", "method/solving-a-triangle-from-its-three-angles", "method/solving-a-triangle-from-its-three-sides", "method/solving-a-triangle-from-two-angles-and-the-included-side", "method/solving-a-triangle-from-two-angles-and-the-side-opposite-one-of-them", "method/solving-a-triangle-from-two-sides-and-the-angle-opposite-one-of-them", "method/solving-a-triangle-from-two-sides-and-the-included-angle", "theorem/cosine-formula-for-a-side-of-a-spherical-triangle", "theorem/cosine-formula-for-an-angle-of-a-spherical-triangle", 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"todhunter-spherical-trigonometry-1886/eq-f02f3d8708", "todhunter-spherical-trigonometry-1886/eq-0d82b9577d", "todhunter-spherical-trigonometry-1886/eq-07bf8dd39e", "todhunter-spherical-trigonometry-1886/eq-5d2551fa89", "todhunter-spherical-trigonometry-1886/eq-a0542d9004", "todhunter-spherical-trigonometry-1886/eq-005a749ed8", "todhunter-spherical-trigonometry-1886/eq-0d2af6669e", "todhunter-spherical-trigonometry-1886/eq-9c59fb0e7c", "todhunter-spherical-trigonometry-1886/eq-6042a75488", "todhunter-spherical-trigonometry-1886/eq-fd242a7122", "todhunter-spherical-trigonometry-1886/eq-fda1c375c3", "todhunter-spherical-trigonometry-1886/eq-7d137a01ab", "todhunter-spherical-trigonometry-1886/eq-fe7db1381d", "todhunter-spherical-trigonometry-1886/eq-7f5190e61a", "todhunter-spherical-trigonometry-1886/eq-138053619c", "todhunter-spherical-trigonometry-1886/eq-d098ea636b", "todhunter-spherical-trigonometry-1886/eq-3350ce60f9", "todhunter-spherical-trigonometry-1886/eq-b5bdefe289", 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Spherical Excess", "title": "Area of a Spherical Triangle. Spherical Excess", "name": "Todhunter 1886, Area of a Spherical Triangle. Spherical Excess", "pages": [ "077", "085" ], "concepts": [ "concept/circular-measure", "concept/lune", "concept/spherical-polygon", "concept/spherical-triangle", "concept/symmetrical-spherical-triangles", "quantity/spherical-excess", "theorem/area-of-a-lune", "theorem/area-of-a-sphere", "theorem/area-of-a-spherical-polygon", "theorem/area-of-a-spherical-triangle", "theorem/cagnoli-s-theorem", "theorem/lhuilier-s-theorem" ], "excerpts": [ "todhunter-spherical-trigonometry-1886/x-04622b30bd", "todhunter-spherical-trigonometry-1886/x-668e457509", "todhunter-spherical-trigonometry-1886/x-d8d1b13964", "todhunter-spherical-trigonometry-1886/x-b87ec29ae0", "todhunter-spherical-trigonometry-1886/x-41016c057f", "todhunter-spherical-trigonometry-1886/x-e1388788ed" ], "equations": [ "todhunter-spherical-trigonometry-1886/eq-eb8673a8f4", "todhunter-spherical-trigonometry-1886/eq-304fee8d60", "todhunter-spherical-trigonometry-1886/eq-28c67e571a", "todhunter-spherical-trigonometry-1886/eq-906f30a7ce", "todhunter-spherical-trigonometry-1886/eq-e3c9cf0aee", "todhunter-spherical-trigonometry-1886/eq-2207369818", "todhunter-spherical-trigonometry-1886/eq-ba76121da9", "todhunter-spherical-trigonometry-1886/eq-298ba1333f", "todhunter-spherical-trigonometry-1886/eq-e24709451a", "todhunter-spherical-trigonometry-1886/eq-4d203e9088", "todhunter-spherical-trigonometry-1886/eq-826316e317", "todhunter-spherical-trigonometry-1886/eq-abaf048ff8", "todhunter-spherical-trigonometry-1886/eq-6da0507ea5" ], "exercise_sets": [ "todhunter-spherical-trigonometry-1886/ex-viii" ] }, { "id": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "number": "On certain approximate Formul\\ae", "title": "On certain approximate Formul\\ae", "name": "Todhunter 1886, On certain approximate Formul\\ae", "pages": [ "085", "095" ], "concepts": [ "concept/approximation", "concept/area", "concept/chord", "concept/chordal-triangle", "concept/circular-measure", "concept/cosine", "concept/cotangent", "concept/diameter", "concept/equilateral-triangle", "concept/great-circle", "concept/plane-triangle", "concept/quadrant", "concept/sine", "concept/small-circle", "concept/spherical-triangle", "concept/spherical-trigonometry", "method/approximate-solution-of-a-spherical-triangle", "person/adrien-marie-legendre", "quantity/spherical-excess", "theorem/legendre-s-theorem" ], "excerpts": [ "todhunter-spherical-trigonometry-1886/x-281c33fdb8", "todhunter-spherical-trigonometry-1886/x-8b5df53b01", "todhunter-spherical-trigonometry-1886/x-adc430ec0a", "todhunter-spherical-trigonometry-1886/x-a3dd0ae03f", "todhunter-spherical-trigonometry-1886/x-fbcf2881ce", "todhunter-spherical-trigonometry-1886/x-9670b9ee3a", "todhunter-spherical-trigonometry-1886/x-16fe05fe8a" ], "equations": [ "todhunter-spherical-trigonometry-1886/eq-7ff2e03c9e", "todhunter-spherical-trigonometry-1886/eq-3061b14a0f", "todhunter-spherical-trigonometry-1886/eq-6330ffbec6", "todhunter-spherical-trigonometry-1886/eq-2fe41806f3", "todhunter-spherical-trigonometry-1886/eq-aae4e1aee6", "todhunter-spherical-trigonometry-1886/eq-cfb1c6ec64", "todhunter-spherical-trigonometry-1886/eq-d4ab6f42aa", "todhunter-spherical-trigonometry-1886/eq-9f08f7c802", "todhunter-spherical-trigonometry-1886/eq-8fae053762", "todhunter-spherical-trigonometry-1886/eq-25e34dbffd", "todhunter-spherical-trigonometry-1886/eq-f8dde2dd0a", "todhunter-spherical-trigonometry-1886/eq-bbe33dc6e2", "todhunter-spherical-trigonometry-1886/eq-611426c391", "todhunter-spherical-trigonometry-1886/eq-047d92b0ae", "todhunter-spherical-trigonometry-1886/eq-ff1920fea4", "todhunter-spherical-trigonometry-1886/eq-7b05326c6c", "todhunter-spherical-trigonometry-1886/eq-8c133698d2", "todhunter-spherical-trigonometry-1886/eq-b07d086024", "todhunter-spherical-trigonometry-1886/eq-c9d4bebbcf", "todhunter-spherical-trigonometry-1886/eq-ce2a1367b1", "todhunter-spherical-trigonometry-1886/eq-70dc21c690", "todhunter-spherical-trigonometry-1886/eq-f26e7db759", "todhunter-spherical-trigonometry-1886/eq-fcf71dbcfb", "todhunter-spherical-trigonometry-1886/eq-a9b000eac4" ], "exercise_sets": [ "todhunter-spherical-trigonometry-1886/ex-ix" ] }, { "id": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "number": "Geodetical Operations", "title": "Geodetical Operations", "name": "Todhunter 1886, Geodetical Operations", "pages": [ "095", "103" ], "concepts": [ "concept/approximation", "concept/base-line", "concept/chordal-triangle", "concept/circular-measure", "concept/figure-of-the-earth", "concept/geodesy", "concept/horizontal-angle", "concept/observational-error", "concept/plane-triangle", "concept/plane-trigonometry", "concept/spherical-triangle", "concept/spherical-trigonometry", "concept/spheroid", "concept/station", "concept/temperature", "concept/triangle", "method/approximate-solution-of-a-spherical-triangle", "method/general-roy-s-rule", "method/reduction-to-the-horizon", "method/triangulation", "person/delambre", "person/george-everest", "quantity/radius", "quantity/spherical-excess", "theorem/area-of-a-spherical-triangle", "theorem/legendre-s-theorem", "theorem/sum-of-the-angles-of-a-triangle" ], "excerpts": [ "todhunter-spherical-trigonometry-1886/x-3e33e11a3c", "todhunter-spherical-trigonometry-1886/x-4c5122419c", "todhunter-spherical-trigonometry-1886/x-591ba3ff2f", "todhunter-spherical-trigonometry-1886/x-f6c0143a9c", "todhunter-spherical-trigonometry-1886/x-5600c330c3", "todhunter-spherical-trigonometry-1886/x-7c4fa9c288", "todhunter-spherical-trigonometry-1886/x-2b0ef355a6", "todhunter-spherical-trigonometry-1886/x-290670c8fc", "todhunter-spherical-trigonometry-1886/x-e24a68a1ab", "todhunter-spherical-trigonometry-1886/x-235471c74c", "todhunter-spherical-trigonometry-1886/x-8c55cfa267" ], "equations": [ "todhunter-spherical-trigonometry-1886/eq-a74d6af74a", "todhunter-spherical-trigonometry-1886/eq-9463a7c87c", "todhunter-spherical-trigonometry-1886/eq-67dfd61a4c", "todhunter-spherical-trigonometry-1886/eq-fa88a49409", "todhunter-spherical-trigonometry-1886/eq-26ec02c88b", "todhunter-spherical-trigonometry-1886/eq-3cb795b6e7", "todhunter-spherical-trigonometry-1886/eq-5bd06f71ac", "todhunter-spherical-trigonometry-1886/eq-c49c01a15d", "todhunter-spherical-trigonometry-1886/eq-72fb5d1823", "todhunter-spherical-trigonometry-1886/eq-3fe6422f17", "todhunter-spherical-trigonometry-1886/eq-57583349f0", "todhunter-spherical-trigonometry-1886/eq-0a019d6e23", "todhunter-spherical-trigonometry-1886/eq-bba8f1585a", "todhunter-spherical-trigonometry-1886/eq-0b4b0082ab", "todhunter-spherical-trigonometry-1886/eq-cd94f9a08e" ], "exercise_sets": [] }, { "id": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "number": "On small variations in the parts of a Spherical Triangle", "title": "On small variations in the parts of a Spherical Triangle", "name": "Todhunter 1886, On small variations in the parts of a Spherical Triangle", "pages": [ "103", "106" ], "concepts": [ "concept/observational-error", "concept/polar-triangle", "concept/quadrant", "concept/side", "concept/small-variation", "concept/spherical-angle", "concept/spherical-triangle", "method/neglecting-higher-order-small-quantities", "theorem/cosine-formula-for-a-side-of-a-spherical-triangle" ], "excerpts": [ "todhunter-spherical-trigonometry-1886/x-7f35ebec37", "todhunter-spherical-trigonometry-1886/x-35033ae7a2", "todhunter-spherical-trigonometry-1886/x-ab9a5994ca", "todhunter-spherical-trigonometry-1886/x-a63246acaf", "todhunter-spherical-trigonometry-1886/x-ad2197bc85" ], "equations": [ "todhunter-spherical-trigonometry-1886/eq-6a7b9d112f", "todhunter-spherical-trigonometry-1886/eq-31ee1ad3c2", "todhunter-spherical-trigonometry-1886/eq-768a831f1c", "todhunter-spherical-trigonometry-1886/eq-16b4d8c2b3", "todhunter-spherical-trigonometry-1886/eq-61e2903576", "todhunter-spherical-trigonometry-1886/eq-0b08793202", "todhunter-spherical-trigonometry-1886/eq-bc495b38a7", "todhunter-spherical-trigonometry-1886/eq-ee63d785e7", "todhunter-spherical-trigonometry-1886/eq-260df06816", "todhunter-spherical-trigonometry-1886/eq-df708001e4", "todhunter-spherical-trigonometry-1886/eq-33c75a520f", "todhunter-spherical-trigonometry-1886/eq-22071d4562", "todhunter-spherical-trigonometry-1886/eq-66bd77ebfc", "todhunter-spherical-trigonometry-1886/eq-2e09b66a59", "todhunter-spherical-trigonometry-1886/eq-c60b71bab7", "todhunter-spherical-trigonometry-1886/eq-f54a77482f", "todhunter-spherical-trigonometry-1886/eq-6e45612880", "todhunter-spherical-trigonometry-1886/eq-2697c6b93d", "todhunter-spherical-trigonometry-1886/eq-c8ef95b0b4", "todhunter-spherical-trigonometry-1886/eq-6e3a6a4ea2", "todhunter-spherical-trigonometry-1886/eq-5c59a9ea1e", "todhunter-spherical-trigonometry-1886/eq-b620e7d2fe", "todhunter-spherical-trigonometry-1886/eq-dd3b1e5e9a", "todhunter-spherical-trigonometry-1886/eq-3bab42ac05", "todhunter-spherical-trigonometry-1886/eq-8772bfd74d", "todhunter-spherical-trigonometry-1886/eq-97c62aeed0" ], "exercise_sets": [ "todhunter-spherical-trigonometry-1886/ex-xi" ] }, { "id": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "number": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "title": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "name": "Todhunter 1886, On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "pages": [ "106", "124" ], "concepts": [ "concept/angular-coordinates-on-a-sphere", "concept/angular-radius", "concept/area", "concept/circular-measure", "concept/cosine", "concept/excircle", "concept/great-circle", "concept/inscribed-circle", "concept/nine-points-circle", "concept/perpendicular", "concept/plane-triangle", "concept/plane-trigonometry", "concept/pole-of-a-circle", "concept/side", "concept/sine", "concept/small-circle", "concept/spherical-angle", "concept/spherical-triangle", "concept/spherical-trigonometry", "method/deducing-plane-trigonometry-from-spherical-trigonometry", "method/expansion-in-powers", "method/limiting-case", "person/euclid", "theorem/concurrence-of-altitudes", "theorem/concurrence-of-altitudes-of-a-spherical-triangle", "theorem/delambre-s-analogies", "theorem/napier-s-analogies", "theorem/power-of-a-point" ], "excerpts": [ "todhunter-spherical-trigonometry-1886/x-81728fb2e0", "todhunter-spherical-trigonometry-1886/x-980f4bcb43", "todhunter-spherical-trigonometry-1886/x-046fe9749f", "todhunter-spherical-trigonometry-1886/x-7239031000", "todhunter-spherical-trigonometry-1886/x-87e0a8fafa", "todhunter-spherical-trigonometry-1886/x-16d2bf68f0", "todhunter-spherical-trigonometry-1886/x-31d70cbf5a", "todhunter-spherical-trigonometry-1886/x-ee53284ef9", "todhunter-spherical-trigonometry-1886/x-a372706cfa", "todhunter-spherical-trigonometry-1886/x-55d8335dcd", "todhunter-spherical-trigonometry-1886/x-a3088fdec3", "todhunter-spherical-trigonometry-1886/x-77339bfbed", "todhunter-spherical-trigonometry-1886/x-b1fdcc90aa", "todhunter-spherical-trigonometry-1886/x-52fdf6c1fe", "todhunter-spherical-trigonometry-1886/x-c5ae896b0c" ], "equations": [ "todhunter-spherical-trigonometry-1886/eq-8f42d22be2", "todhunter-spherical-trigonometry-1886/eq-90526d7351", "todhunter-spherical-trigonometry-1886/eq-85a23ecd78", "todhunter-spherical-trigonometry-1886/eq-69c081ae40", "todhunter-spherical-trigonometry-1886/eq-76fb85a5cc", "todhunter-spherical-trigonometry-1886/eq-be7f1476db", "todhunter-spherical-trigonometry-1886/eq-8d706c5694", "todhunter-spherical-trigonometry-1886/eq-e593165e50", "todhunter-spherical-trigonometry-1886/eq-ae226de395", "todhunter-spherical-trigonometry-1886/eq-cfe0db7585", "todhunter-spherical-trigonometry-1886/eq-349dd6820f", "todhunter-spherical-trigonometry-1886/eq-4b0def777d", "todhunter-spherical-trigonometry-1886/eq-fb1c5f1a1d", "todhunter-spherical-trigonometry-1886/eq-661c96c233", "todhunter-spherical-trigonometry-1886/eq-ae89c15e6c", "todhunter-spherical-trigonometry-1886/eq-6da907218a", "todhunter-spherical-trigonometry-1886/eq-5216e40cfd", "todhunter-spherical-trigonometry-1886/eq-ca9cd2c245", "todhunter-spherical-trigonometry-1886/eq-4a37097765", "todhunter-spherical-trigonometry-1886/eq-39ddfa0aba", "todhunter-spherical-trigonometry-1886/eq-ec40b21ace", "todhunter-spherical-trigonometry-1886/eq-f514b47875", "todhunter-spherical-trigonometry-1886/eq-c78ecbf863", "todhunter-spherical-trigonometry-1886/eq-6bf6267c57", "todhunter-spherical-trigonometry-1886/eq-963e982eec", "todhunter-spherical-trigonometry-1886/eq-1476ffaf7f", "todhunter-spherical-trigonometry-1886/eq-2df09c5e30", "todhunter-spherical-trigonometry-1886/eq-e95ecb044c", "todhunter-spherical-trigonometry-1886/eq-770ab45613", "todhunter-spherical-trigonometry-1886/eq-09d63ce8ae", "todhunter-spherical-trigonometry-1886/eq-e72d80c6db" ], "exercise_sets": [ "todhunter-spherical-trigonometry-1886/ex-xii" ] }, { "id": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "number": "Polyhedrons", "title": "Polyhedrons", "name": "Todhunter 1886, Polyhedrons", "pages": [ "124", "135" ], "concepts": [ "concept/area", "concept/diagonal", "concept/dodecahedron", "concept/edge", "concept/face", "concept/great-circle", "concept/hexahedron", "concept/icosahedron", "concept/inclination-of-two-lines", "concept/octahedron", "concept/parallelepiped", "concept/plane-angle", "concept/polygon", "concept/polyhedral-angle", "concept/polyhedron", "concept/polyhedron-inscribed-in-a-sphere", "concept/regular-polyhedron", "concept/sphere-inscribed-in-a-polyhedron", "concept/spherical-triangle", "concept/surface-of-a-polyhedron", "concept/tetrahedron", "concept/vertex", "method/carnot-s-method", "quantity/volume", "theorem/five-regular-polyhedra", "theorem/polyhedral-formula", "theorem/sum-of-plane-angles-of-a-polyhedron", "theorem/volume-of-a-parallelepiped", "theorem/volume-of-a-tetrahedron" ], "excerpts": [ "todhunter-spherical-trigonometry-1886/x-322bf3d4de", "todhunter-spherical-trigonometry-1886/x-38cd10975e", "todhunter-spherical-trigonometry-1886/x-c0c5bad25c", "todhunter-spherical-trigonometry-1886/x-32f9d5e38f", "todhunter-spherical-trigonometry-1886/x-def1a53745", "todhunter-spherical-trigonometry-1886/x-adaa97ce20", "todhunter-spherical-trigonometry-1886/x-fd27761983", "todhunter-spherical-trigonometry-1886/x-9f65b68cb8" ], "equations": [ "todhunter-spherical-trigonometry-1886/eq-79e7ded3c6", "todhunter-spherical-trigonometry-1886/eq-0fd343d679", "todhunter-spherical-trigonometry-1886/eq-52f362833c", "todhunter-spherical-trigonometry-1886/eq-5eac6c4aae", "todhunter-spherical-trigonometry-1886/eq-189d427919", "todhunter-spherical-trigonometry-1886/eq-20359ce215", "todhunter-spherical-trigonometry-1886/eq-b9a3b34cea", "todhunter-spherical-trigonometry-1886/eq-fabebb2c1f", "todhunter-spherical-trigonometry-1886/eq-125ecfb5bb", "todhunter-spherical-trigonometry-1886/eq-048dde3aef", "todhunter-spherical-trigonometry-1886/eq-1b8bbfe0ac", "todhunter-spherical-trigonometry-1886/eq-ac9b3748cf", "todhunter-spherical-trigonometry-1886/eq-fd85cd3a6f", "todhunter-spherical-trigonometry-1886/eq-121ebd5f42", "todhunter-spherical-trigonometry-1886/eq-732d278f02", "todhunter-spherical-trigonometry-1886/eq-4f8a7f8e26", "todhunter-spherical-trigonometry-1886/eq-e9ce095f9f", "todhunter-spherical-trigonometry-1886/eq-68cce3b6e6", "todhunter-spherical-trigonometry-1886/eq-64f572916f", "todhunter-spherical-trigonometry-1886/eq-a8bb94e492", "todhunter-spherical-trigonometry-1886/eq-b05425a14d", "todhunter-spherical-trigonometry-1886/eq-89ffab1455", "todhunter-spherical-trigonometry-1886/eq-df1d298afc", "todhunter-spherical-trigonometry-1886/eq-778dfb0254", "todhunter-spherical-trigonometry-1886/eq-22c8b9899a", "todhunter-spherical-trigonometry-1886/eq-bccf076a5d", "todhunter-spherical-trigonometry-1886/eq-bf65169ac5", "todhunter-spherical-trigonometry-1886/eq-be81949a9f", "todhunter-spherical-trigonometry-1886/eq-1e1de4d61b" ], "exercise_sets": [ "todhunter-spherical-trigonometry-1886/ex-xiii" ] }, { "id": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "number": "Arcs drawn to fixed points on the Surface of a Sphere", "title": "Arcs drawn to fixed points on the Surface of a Sphere", "name": "Todhunter 1886, Arcs drawn to fixed points on the Surface of a Sphere", "pages": [ "135", "145" ], "concepts": [ "concept/quadrantal-triangle", "theorem/sum-of-cosines-to-fixed-points-varies-as-the-cosine-of-one-arc", "theorem/sum-of-cosines-to-the-solid-angles-of-a-regular-polyhedron-is-zero", "theorem/sum-of-squared-cosines-to-the-solid-angles-of-a-regular-polyhedron-is-one-third-of-their-number", "theorem/sum-of-squared-cosines-to-the-vertices-of-a-quadrantal-triangle-is-one" ], "excerpts": [ "todhunter-spherical-trigonometry-1886/x-0ce78283f5", "todhunter-spherical-trigonometry-1886/x-e210bfbeed", "todhunter-spherical-trigonometry-1886/x-773969ff87", "todhunter-spherical-trigonometry-1886/x-c39907c4c1" ], "equations": [ "todhunter-spherical-trigonometry-1886/eq-1fc7892b52", "todhunter-spherical-trigonometry-1886/eq-2366db96d7", "todhunter-spherical-trigonometry-1886/eq-20392bb642", "todhunter-spherical-trigonometry-1886/eq-6e52058936", "todhunter-spherical-trigonometry-1886/eq-d26aa9aeb5", "todhunter-spherical-trigonometry-1886/eq-4cc7277b9c", "todhunter-spherical-trigonometry-1886/eq-cf7fe690be", "todhunter-spherical-trigonometry-1886/eq-049a870f1f", "todhunter-spherical-trigonometry-1886/eq-c2dc6f3362", "todhunter-spherical-trigonometry-1886/eq-7596a277f8", "todhunter-spherical-trigonometry-1886/eq-0feaf39c63", "todhunter-spherical-trigonometry-1886/eq-30146f55e2", "todhunter-spherical-trigonometry-1886/eq-5f37530b5a", "todhunter-spherical-trigonometry-1886/eq-ecec46ec0a", "todhunter-spherical-trigonometry-1886/eq-308ea681d3", "todhunter-spherical-trigonometry-1886/eq-1d85f9938a" ], "exercise_sets": [] }, { "id": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "number": "Miscellaneous Propositions", "title": "Miscellaneous Propositions", "name": "Todhunter 1886, Miscellaneous Propositions", "pages": [ "145", "159" ], "concepts": [ "concept/angular-coordinates-on-a-sphere", "concept/circle", "concept/circumscribed-circle", "concept/edge", "concept/escribed-circle", "concept/excircle", "concept/face", "concept/great-circle", "concept/inscribed-circle", "concept/locus", "concept/mechanical-equilibrium", "concept/polar-triangle", "concept/pole-of-a-circle", "concept/polyhedral-angle", "concept/polyhedron", "concept/quadrant", "concept/re-entrant-angle", "concept/regular-polyhedron", "concept/rotation", "concept/sine-of-a-solid-angle", "concept/small-circle", "concept/spherical-distance", "concept/spherical-triangle", "concept/symmetry", "concept/vertex", "person/augustin-louis-cauchy", "person/leonhard-euler", "quantity/force", "quantity/spherical-excess", "theorem/arc-through-midpoints-of-two-sides-of-a-spherical-triangle", "theorem/cauchy-s-network-theorem", "theorem/concurrence-of-altitudes-of-a-spherical-triangle", "theorem/concurrence-of-arcs-through-a-point-of-a-spherical-triangle", "theorem/euler-s-theorem", "theorem/volume-of-a-tetrahedron" ], "excerpts": [ "todhunter-spherical-trigonometry-1886/x-24696586dc", "todhunter-spherical-trigonometry-1886/x-165d0ffa96", "todhunter-spherical-trigonometry-1886/x-87f70aac01", "todhunter-spherical-trigonometry-1886/x-b1ec984451", "todhunter-spherical-trigonometry-1886/x-675c2a3867", "todhunter-spherical-trigonometry-1886/x-d3abf4f9d0", "todhunter-spherical-trigonometry-1886/x-3ce611b5bc", "todhunter-spherical-trigonometry-1886/x-8b0b466615", "todhunter-spherical-trigonometry-1886/x-b595b4f71e", "todhunter-spherical-trigonometry-1886/x-65746fd40b" ], "equations": [ "todhunter-spherical-trigonometry-1886/eq-52db6629ab", "todhunter-spherical-trigonometry-1886/eq-5c0a940a72", "todhunter-spherical-trigonometry-1886/eq-7b4d659cac", "todhunter-spherical-trigonometry-1886/eq-d4633f403d", "todhunter-spherical-trigonometry-1886/eq-c2afe48dd1", "todhunter-spherical-trigonometry-1886/eq-cb9be38ba7", "todhunter-spherical-trigonometry-1886/eq-df24f3de26", "todhunter-spherical-trigonometry-1886/eq-fff9b4163b", "todhunter-spherical-trigonometry-1886/eq-fcbe7e2519", "todhunter-spherical-trigonometry-1886/eq-57fd2af42d", "todhunter-spherical-trigonometry-1886/eq-f4a8b7299f", "todhunter-spherical-trigonometry-1886/eq-cd0a423266", "todhunter-spherical-trigonometry-1886/eq-3ff02703cc", "todhunter-spherical-trigonometry-1886/eq-88a959a7cc", "todhunter-spherical-trigonometry-1886/eq-8d9436e32a", "todhunter-spherical-trigonometry-1886/eq-d6651eee52", "todhunter-spherical-trigonometry-1886/eq-dfc4828fcb", "todhunter-spherical-trigonometry-1886/eq-bab61a2e0c", "todhunter-spherical-trigonometry-1886/eq-37de993ee3", "todhunter-spherical-trigonometry-1886/eq-6086171e27", "todhunter-spherical-trigonometry-1886/eq-cd8bdfda20", "todhunter-spherical-trigonometry-1886/eq-a791c31890", "todhunter-spherical-trigonometry-1886/eq-83d4ebb4d8", "todhunter-spherical-trigonometry-1886/eq-160d7d9c36", "todhunter-spherical-trigonometry-1886/eq-26519385f6", "todhunter-spherical-trigonometry-1886/eq-982541c041", "todhunter-spherical-trigonometry-1886/eq-ab005abd91", "todhunter-spherical-trigonometry-1886/eq-615696e1eb", "todhunter-spherical-trigonometry-1886/eq-f3337c6b61", "todhunter-spherical-trigonometry-1886/eq-42242cebb6", "todhunter-spherical-trigonometry-1886/eq-9753ddcc87", "todhunter-spherical-trigonometry-1886/eq-74caa3a88b", "todhunter-spherical-trigonometry-1886/eq-c0969412fd", "todhunter-spherical-trigonometry-1886/eq-bfde24d6fd", "todhunter-spherical-trigonometry-1886/eq-697a3bf215", "todhunter-spherical-trigonometry-1886/eq-8b99ce8883", "todhunter-spherical-trigonometry-1886/eq-38d65d06bc", "todhunter-spherical-trigonometry-1886/eq-652302df59", "todhunter-spherical-trigonometry-1886/eq-6dc53ac76f", "todhunter-spherical-trigonometry-1886/eq-5bde5974a3", "todhunter-spherical-trigonometry-1886/eq-7db4fb0f5a", "todhunter-spherical-trigonometry-1886/eq-01380bf85c", "todhunter-spherical-trigonometry-1886/eq-e0d5206458", "todhunter-spherical-trigonometry-1886/eq-e0f5c63ad2", "todhunter-spherical-trigonometry-1886/eq-c5ede1faa2", "todhunter-spherical-trigonometry-1886/eq-1eb32181fb", "todhunter-spherical-trigonometry-1886/eq-f585cb1c96", "todhunter-spherical-trigonometry-1886/eq-3fabae2229", "todhunter-spherical-trigonometry-1886/eq-5fed82367d", "todhunter-spherical-trigonometry-1886/eq-874e33a367", "todhunter-spherical-trigonometry-1886/eq-0a1b0e48a5" ], "exercise_sets": [ "todhunter-spherical-trigonometry-1886/ex-xv" ] }, { "id": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "number": "Numerical Solution of Spherical Triangles", "title": "Numerical Solution of Spherical Triangles", "name": "Todhunter 1886, Numerical Solution of Spherical Triangles", "pages": [ "159", "168" ], "concepts": [ "concept/ambiguous-case", "concept/cosine", "concept/cotangent", "concept/logarithm", "concept/polar-triangle", "concept/right-spherical-triangle", "concept/sine", "concept/tangent-function", "method/numerical-solution-of-a-spherical-triangle", "method/solving-a-right-spherical-triangle", "method/solving-a-spherical-triangle-from-its-three-sides", "method/solving-an-oblique-spherical-triangle", "method/verification", "theorem/half-angle-formulae-for-a-spherical-triangle", "theorem/napier-s-analogies", "theorem/right-spherical-triangle-relations" ], "excerpts": [ "todhunter-spherical-trigonometry-1886/x-e7eb828cda", "todhunter-spherical-trigonometry-1886/x-c0d4e3b27c", "todhunter-spherical-trigonometry-1886/x-bc35865994", "todhunter-spherical-trigonometry-1886/x-4bc5a22075", "todhunter-spherical-trigonometry-1886/x-ebdb84046b", "todhunter-spherical-trigonometry-1886/x-37e4e98cdc", "todhunter-spherical-trigonometry-1886/x-ecb7519ed7", "todhunter-spherical-trigonometry-1886/x-d8317ddd78" ], "equations": [ "todhunter-spherical-trigonometry-1886/eq-3f37ec6aa9", "todhunter-spherical-trigonometry-1886/eq-36de107c6c", "todhunter-spherical-trigonometry-1886/eq-f06e05b729", "todhunter-spherical-trigonometry-1886/eq-082cf7ae05", "todhunter-spherical-trigonometry-1886/eq-2eb4c207e8", "todhunter-spherical-trigonometry-1886/eq-9c35c7f2e6", "todhunter-spherical-trigonometry-1886/eq-7b665ead8b", "todhunter-spherical-trigonometry-1886/eq-1965fa6c3f", "todhunter-spherical-trigonometry-1886/eq-18b885383b", "todhunter-spherical-trigonometry-1886/eq-affc03b8b6", "todhunter-spherical-trigonometry-1886/eq-0439ff4fe9", "todhunter-spherical-trigonometry-1886/eq-92630403c2", "todhunter-spherical-trigonometry-1886/eq-a744f1059a", "todhunter-spherical-trigonometry-1886/eq-6804b95e2b", "todhunter-spherical-trigonometry-1886/eq-fe2bfe5ccd", "todhunter-spherical-trigonometry-1886/eq-ec148c206b", "todhunter-spherical-trigonometry-1886/eq-89542f733c", "todhunter-spherical-trigonometry-1886/eq-432e2e689d", "todhunter-spherical-trigonometry-1886/eq-69c3063014" ], "exercise_sets": [ "todhunter-spherical-trigonometry-1886/ex-xvi" ] } ], "excerpts": [ { "id": "todhunter-spherical-trigonometry-1886/x-15efe6643b", "chapter": "todhunter-spherical-trigonometry-1886/ch-great-and-small-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 11", "location": "Great and Small Circles", "latex": "\\textsc{A sphere} is a solid bounded by a surface every point of which is equally distant from a fixed point which is called the \\textit{centre} of the sphere.", "markdown": "A sphere is a solid bounded by a surface every point of which is equally distant from a fixed point which is called the *centre* of the sphere.", "why": "Gives the learner the defining property of a sphere as a set of equidistant points before any circles appear.", "use": [ "lesson" ], "concepts": [ "concept/sphere" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-970cc3b75f", "chapter": "todhunter-spherical-trigonometry-1886/ch-great-and-small-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 12", "location": "Great and Small Circles", "latex": "The section of the surface of a sphere by a plane is called a \\textit{great circle} if the plane passes through the centre of the sphere, and a \\textit{small circle} if the plane does not pass through the centre of the sphere.", "markdown": "The section of the surface of a sphere by a plane is called a *great circle* if the plane passes through the centre of the sphere, and a *small circle* if the plane does not pass through the centre of the sphere.", "why": "States the great/small circle distinction in one sentence, which is the central classification of the chapter.", "use": [ "lesson" ], "concepts": [ "concept/great-circle", "concept/small-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-8f6fab5596", "chapter": "todhunter-spherical-trigonometry-1886/ch-great-and-small-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 12", "location": "Great and Small Circles", "latex": "When only one great circle can be drawn through two given points, the great circle is unequally divided at the two points; we shall for brevity speak of the shorter of the two arcs as \\textit{the} arc of a great circle joining the two points.", "markdown": "When only one great circle can be drawn through two given points, the great circle is unequally divided at the two points; we shall for brevity speak of the shorter of the two arcs as *the* arc of a great circle joining the two points.", "why": "Explains why the shorter arc is taken as the distance between two points on a sphere, a point learners often miss.", "use": [ "lesson" ], "concepts": [ "concept/great-circle", "concept/spherical-distance" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-a793b4f83a", "chapter": "todhunter-spherical-trigonometry-1886/ch-great-and-small-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 13", "location": "Great and Small Circles", "latex": "Then $PO$ is at right angles to the plane $ABC$, because $P$ is the pole of $ABC$, therefore $POA$ is a right angle, and the arc $PA$ is a quadrant.", "markdown": "Then $PO$ is at right angles to the plane $ABC$, because $P$ is the pole of $ABC$, therefore $POA$ is a right angle, and the arc $PA$ is a quadrant.", "why": "A short proof showing how the pole of a great circle produces a quarter-circle arc, tied to the right angle at the centre.", "use": [ "lesson" ], "concepts": [ "concept/pole-of-a-circle", "concept/quadrant" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-7ecdd48a80", "chapter": "todhunter-spherical-trigonometry-1886/ch-great-and-small-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 13", "location": "Great and Small Circles", "latex": "Thus the distance of a pole of a circle from every point of the circumference of the circle is constant, whether that distance be measured by the straight line joining the points, or by the arc of a great circle intercepted between the points.", "markdown": "Thus the distance of a pole of a circle from every point of the circumference of the circle is constant, whether that distance be measured by the straight line joining the points, or by the arc of a great circle intercepted between the points.", "why": "Shows that the pole is a single point whose distance to the rim is the same measured two ways, which gives learners a concrete picture of the pole.", "use": [ "lesson", "website" ], "concepts": [ "concept/pole-of-a-circle", "concept/spherical-distance" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-7031fc74b2", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 16", "location": "Spherical Triangles", "latex": "Spherical Trigonometry investigates the relations which subsist between the angles of the plane faces which form a solid angle and the angles at which the plane faces are inclined to each other.", "markdown": "Spherical Trigonometry investigates the relations which subsist between the angles of the plane faces which form a solid angle and the angles at which the plane faces are inclined to each other.", "why": "It states in one sentence what the subject is about, so a learner knows what the later theorems are for.", "use": [ "lesson" ], "concepts": [ "concept/polyhedral-angle", "concept/spherical-trigonometry" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-9ba9bf0470", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 17", "location": "Spherical Triangles", "latex": "Thus a figure will be formed on the surface of the sphere which is called a \\textit{spherical triangle} if it is bounded by \\textit{three} arcs of great circles; this will be the case when the solid angle is formed by the meeting of \\textit{three} plane angles.", "markdown": "Thus a figure will be formed on the surface of the sphere which is called a *spherical triangle* if it is bounded by *three* arcs of great circles; this will be the case when the solid angle is formed by the meeting of *three* plane angles.", "why": "It shows how a spherical triangle arises geometrically from a solid angle at the centre of a sphere.", "use": [ "lesson" ], "concepts": [ "concept/great-circle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-d2580e6a15", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 18", "location": "Spherical Triangles", "latex": "It will be seen that what are called \\textit{sides} of a spherical triangle are really \\textit{arcs} of great circles, and these arcs are proportional to the three plane angles which form the solid angle corresponding to the spherical triangle.", "markdown": "It will be seen that what are called *sides* of a spherical triangle are really *arcs* of great circles, and these arcs are proportional to the three plane angles which form the solid angle corresponding to the spherical triangle.", "why": "It corrects the natural idea that a side is a straight line, showing it is an arc whose length tracks a plane angle at the centre.", "use": [ "lesson" ], "concepts": [ "concept/side", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-5a086e42fa", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 18", "location": "Spherical Triangles", "latex": "As in the case of plane triangles, $A$, $B$, and $C$ may be used to denote the numerical values of the angles expressed in \\textit{terms of any unit}, provided we understand distinctly what the unit is.", "markdown": "As in the case of plane triangles, $A$, $B$, and $C$ may be used to denote the numerical values of the angles expressed in *terms of any unit*, provided we understand distinctly what the unit is.", "why": "It warns the learner that an angle's numerical value depends on the chosen unit, so the unit must always be stated.", "use": [ "lesson" ], "concepts": [ "concept/circular-measure", "unit/degree-of-angle", "unit/radian" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-db04fdbe3d", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 19", "location": "Spherical Triangles", "latex": "In spherical triangles each side is restricted to be less than a semicircle; this is of course a \\textit{convention}, and it is adopted because it is found convenient.", "markdown": "In spherical triangles each side is restricted to be less than a semicircle; this is of course a *convention*, and it is adopted because it is found convenient.", "why": "It makes plain that the restriction on side length is a choice of convention, not a fact of nature, which helps learners read the later theorems correctly.", "use": [ "lesson" ], "concepts": [ "concept/side", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-3d97267e4a", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 19", "location": "Spherical Triangles", "latex": "From the restriction of the preceding Article it will follow that \\textit{any angle of a spherical triangle is less than two right angles}.", "markdown": "From the restriction of the preceding Article it will follow that *any angle of a spherical triangle is less than two right angles*.", "why": "It states a consequence of the convention that a learner would otherwise have to rediscover, and marks where it applies.", "use": [ "lesson" ], "concepts": [ "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-7266e99e1f", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 20", "location": "Spherical Geometry", "latex": "Since there are two poles for each side of a spherical triangle, \\textit{eight} triangles can be formed having for their angular points poles of the sides of the given triangle; but there is only one triangle in which these poles $A'$, $B'$, $C'$ lie towards the same parts with the corresponding angles $A$, $B$, $C$; and this is the triangle which is known under the name of the \\textit{polar triangle}.", "markdown": "Since there are two poles for each side of a spherical triangle, *eight* triangles can be formed having for their angular points poles of the sides of the given triangle; but there is only one triangle in which these poles $A'$, $B'$, $C'$ lie towards the same parts with the corresponding angles $A$, $B$, $C$; and this is the triangle which is known under the name of the *polar triangle*.", "why": "It shows why the polar triangle is unique among the eight triangles that could be formed from the poles of the sides, which is the key to picking the right one.", "use": [ "lesson" ], "concepts": [ "concept/polar-triangle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-213a6a4986", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 21", "location": "Spherical Geometry", "latex": "The sides and angles of the polar triangle are respectively the supplements of the angles and sides of the primitive triangle.", "markdown": "The sides and angles of the polar triangle are respectively the supplements of the angles and sides of the primitive triangle.", "why": "It gives the central duality in one sentence, so a learner can see that every fact about one triangle has a counterpart in its polar.", "use": [ "lesson", "website" ], "concepts": [ "concept/polar-triangle", "concept/primitive-triangle", "concept/supplementary-angles" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-a087f483bf", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 22", "location": "Spherical Geometry", "latex": "Thus any such theorem will remain true when the angles are changed into the supplements of the corresponding sides and the sides into the supplements of the corresponding angles.", "markdown": "Thus any such theorem will remain true when the angles are changed into the supplements of the corresponding sides and the sides into the supplements of the corresponding angles.", "why": "It states the principle that lets one proof of a theorem on a spherical triangle give a second theorem for free.", "use": [ "lesson" ], "concepts": [ "concept/polar-triangle", "concept/supplementary-angles" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-3908216c5c", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 22", "location": "Spherical Geometry", "latex": "Any two sides of a spherical triangle are together greater than the third side.", "markdown": "Any two sides of a spherical triangle are together greater than the third side.", "why": "It is the spherical form of the triangle inequality, and a learner who knows the plane version will recognise it at once.", "use": [ "lesson", "website" ], "concepts": [ "concept/spherical-triangle", "theorem/sum-of-two-sides-of-a-spherical-triangle-exceeds-the-third" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-c0dcf14836", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 23", "location": "Spherical Geometry", "latex": "The three angles of a spherical triangle are together greater than two right angles and less than six right angles.", "markdown": "The three angles of a spherical triangle are together greater than two right angles and less than six right angles.", "why": "It is a surprising contrast with the plane case, where the angles always sum to exactly two right angles, and it is quotable for that reason.", "use": [ "website", "lesson" ], "concepts": [ "concept/spherical-triangle", "theorem/sum-of-angles-of-a-spherical-triangle-lies-between-two-and-six-right-angles" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-97d3e6c824", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 25", "location": "Spherical Geometry", "latex": "This Chapter might be extended; but it is unnecessary to do so because the Trigonometrical formul\\ae\\ of the next Chapter supply an easy method of investigating the theorems of Spherical Geometry.", "markdown": "This Chapter might be extended; but it is unnecessary to do so because the Trigonometrical formul of the next Chapter supply an easy method of investigating the theorems of Spherical Geometry.", "why": "It tells the learner that this chapter is a deliberate stepping stone and that the trigonometric formulae are the faster route, which places the chapter in the wider course.", "use": [ "history" ], "concepts": [ "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-ffe7d5c760", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 29", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "The sines of the angles of a spherical triangle are proportional to the sines of the opposite sides.", "markdown": "The sines of the angles of a spherical triangle are proportional to the sines of the opposite sides.", "why": "States the law of sines plainly, which a learner can test on any triangle once the chapter's proof is understood.", "use": [ "lesson" ], "concepts": [ "concept/spherical-triangle", "theorem/law-of-sines" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-24ff309928", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 29", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "The radical on the right-hand side must be taken with the positive sign, because $\\sin b$, $\\sin c$, and $\\sin A$ are all positive.", "markdown": "The radical on the right-hand side must be taken with the positive sign, because $\\sin b$, $\\sin c$, and $\\sin A$ are all positive.", "why": "Shows a learner why a square root in a spherical formula carries only the positive sign, a common place to slip.", "use": [ "lesson" ], "concepts": [ "concept/sine", "theorem/law-of-sines" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-aee12a2ff4", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 38", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "It should be observed that the two triangles in this case are \\textit{not} necessarily such that one may be made to \\textit{coincide with the other by superposition}.", "markdown": "It should be observed that the two triangles in this case are *not* necessarily such that one may be made to *coincide with the other by superposition*.", "why": "Warns that two spherical triangles with the same parts can be mirror images rather than superposable, which is the usual mistake with congruence on a sphere.", "use": [ "lesson", "website" ], "concepts": [ "concept/spherical-triangle", "concept/symmetrical-spherical-triangles" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-3a3e44fecb", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 35", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "The formul\\ae\\ (4), (5), (6), (7) may be put in the form of proportions or analogies, and are called from their discoverer \\textit{Napier's Analogies:} the last two may be demonstrated without recurring to the polar triangle by starting with the formul\\ae\\ in Art.~39.", "markdown": "The formul (4), (5), (6), (7) may be put in the form of proportions or analogies, and are called from their discoverer *Napier’s Analogies:* the last two may be demonstrated without recurring to the polar triangle by starting with the formul in Art. 39.", "why": "Names the four analogies after Napier and shows that two of them can be reached without the polar triangle, which is a useful historical and structural remark.", "use": [ "history" ], "concepts": [ "concept/polar-triangle", "person/john-napier", "theorem/napier-s-analogies" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-b21c40ba20", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 37", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "The last four formul\\ae\\ are commonly, but improperly, called \\textit{Gauss's Theorems}; they were first given by Delambre in the \\textit{Connaissance des Tems} for 1809, page~445.", "markdown": "The last four formul are commonly, but improperly, called *Gauss’s Theorems*; they were first given by Delambre in the *Connaissance des Tems* for 1809, page 445.", "why": "Credits the formulae to their true first author and records a misattribution, in the book's own careful style.", "use": [ "history" ], "concepts": [ "person/delambre" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-bf9c3ef821", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 42", "location": "Solution of Right-angled Triangles", "latex": "The solution of spherical triangles is the process by which, when the values of a sufficient number of the six elements are given, we calculate the values of the remaining elements.", "markdown": "The solution of spherical triangles is the process by which, when the values of a sufficient number of the six elements are given, we calculate the values of the remaining elements.", "why": "It states plainly what solving a spherical triangle means before any formula is used.", "use": [ "lesson" ], "concepts": [ "concept/spherical-triangle", "method/solving-a-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-fe3829ed82", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 44", "location": "Solution of Right-angled Triangles", "latex": "These six formul\\ae\\ comprise ten equations; and thus we can solve every case of right-angled triangles.", "markdown": "These six formul comprise ten equations; and thus we can solve every case of right-angled triangles.", "why": "It shows the learner that a small set of relations covers every right-triangle case.", "use": [ "lesson" ], "concepts": [ "method/solving-a-right-angled-triangle", "theorem/right-spherical-triangle-relations" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-2d5f3913eb", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 45", "location": "Solution of Right-angled Triangles", "latex": "the side opposite the right angle is called the \\textit{hypotenuse:}", "markdown": "the side opposite the right angle is called the *hypotenuse:*", "why": "It gives the name and position of the hypotenuse, the anchor for the right-triangle relations.", "use": [ "lesson" ], "concepts": [ "concept/hypotenuse", "concept/right-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-3134c57182", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 45", "location": "Solution of Right-angled Triangles", "latex": "Napier was also the inventor of Logarithms, and the Rules of Circular Parts were first published by him in a work entitled \\textit{Mirifici Logarithmorum Canonis Descriptio}\\dots\\dots Edinburgh, 1614.", "markdown": "Napier was also the inventor of Logarithms, and the Rules of Circular Parts were first published by him in a work entitled *Mirifici Logarithmorum Canonis Descriptio*……Edinburgh, 1614.", "why": "It places Napier's Rules in their historical source, with the publication date.", "use": [ "history" ], "concepts": [ "person/john-napier", "theorem/napier-s-rules-of-circular-parts" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-9b98fafd49", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 50", "location": "Solution of Right-angled Triangles", "latex": "We do not give them, because we are convinced that they only create confusion instead of assisting the memory.", "markdown": "We do not give them, because we are convinced that they only create confusion instead of assisting the memory.", "why": "It records a dissenting view on the rules, so a learner sees that teachers disagree about their value.", "use": [ "history" ], "concepts": [ "theorem/napier-s-rules-of-circular-parts" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-e73455694d", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 52", "location": "Solution of Right-angled Triangles", "latex": "There are limitations of the data in order to insure a possible triangle.", "markdown": "There are limitations of the data in order to insure a possible triangle.", "why": "It warns the learner that not every set of given parts yields a triangle.", "use": [ "lesson" ], "concepts": [ "concept/existence-conditions-of-a-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-69ea8cfea0", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 52", "location": "Solution of Right-angled Triangles", "latex": "Thus if one triangle exists with the given parts, there will be \\textit{in general} two, and only two, triangles with the given parts.", "markdown": "Thus if one triangle exists with the given parts, there will be *in general* two, and only two, triangles with the given parts.", "why": "It explains the ambiguous case, where a side is found from its sine and two triangles can satisfy the data.", "use": [ "lesson" ], "concepts": [ "concept/ambiguous-case" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-0928c089f4", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 45", "location": "Solution of Right-angled Triangles", "latex": "From (4) it follows that $\\tan a$ has the same sign as $\\tan A$.", "markdown": "From (4) it follows that $\\tan a$ has the same sign as $\\tan A$.", "why": "It shows how a side and its opposite angle are tied together by sign, which decides the solution.", "use": [ "lesson" ], "concepts": [ "concept/same-affection" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-052b8e5459", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 56", "location": "Solution of Oblique-Angled Triangles", "latex": "The solution of oblique-angled triangles may be made in some cases to depend immediately on the solution of right-angled triangles; we will indicate these cases before considering the subject generally.", "markdown": "The solution of oblique-angled triangles may be made in some cases to depend immediately on the solution of right-angled triangles; we will indicate these cases before considering the subject generally.", "why": "It tells the learner that many oblique triangles reduce to right-angled ones, which is the reason the right-triangle rules come first.", "use": [ "lesson" ], "concepts": [ "method/solving-a-right-angled-triangle", "method/solving-a-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-0797e7599a", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 57", "location": "Solution of Oblique-Angled Triangles", "latex": "these determine $\\tfrac{1}{2}(A + B)$ and $\\tfrac{1}{2}(A - B)$, and thence $A$ and $B$.", "markdown": "these determine $\\tfrac{1}{2}(A + B)$ and $\\tfrac{1}{2}(A - B)$, and thence $A$ and $B$.", "why": "It shows the two-step pattern of Napier's analogies, first the half-sum and half-difference and then the angles themselves.", "use": [ "lesson" ], "concepts": [ "method/solving-a-triangle-from-two-sides-and-the-included-angle", "theorem/napier-s-analogies" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-6f3e9017bd", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 58", "location": "Solution of Oblique-Angled Triangles", "latex": "Thus, in the present case, there is no real ambiguity, and the triangle is always possible.", "markdown": "Thus, in the present case, there is no real ambiguity, and the triangle is always possible.", "why": "It explains why the case of two sides and the included angle never needs a check for existence, which contrasts with the cases that do.", "use": [ "lesson" ], "concepts": [ "concept/existence-conditions-of-a-triangle", "method/solving-a-triangle-from-two-sides-and-the-included-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-b0bb9a841e", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 60", "location": "Solution of Oblique-Angled Triangles", "latex": "In this case, since $B$ is found from its sine, there will sometimes be two solutions; and sometimes there will be no solution at all, namely, when the value found for $\\sin B$ is greater than unity.", "markdown": "In this case, since $B$ is found from its sine, there will sometimes be two solutions; and sometimes there will be no solution at all, namely, when the value found for $\\sin B$ is greater than unity.", "why": "It warns the learner that a sine can give two angles or none, a common source of error in the opposite-angle case.", "use": [ "lesson" ], "concepts": [ "concept/ambiguous-case", "method/solving-a-triangle-from-two-sides-and-the-angle-opposite-one-of-them" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-8fc5a5d324", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 63", "location": "Solution of Oblique-Angled Triangles", "latex": "Hence, when $a = b$, there will be no solution at all, unless $A$ and $a$ are of the same affection, and then there will be only one solution; except when $A$ and $a$ are both right angles, and then $\\cot \\tfrac{1}{2}C$ and $\\tan\\tfrac{1}{2}c$ are indeterminate, and there is an infinite number of solutions.", "markdown": "Hence, when $a = b$, there will be no solution at all, unless $A$ and $a$ are of the same affection, and then there will be only one solution; except when $A$ and $a$ are both right angles, and then $\\cot \\tfrac{1}{2}C$ and $\\tan\\tfrac{1}{2}c$ are indeterminate, and there is an infinite number of solutions.", "why": "It gives a worked special case that shows how the sign conditions decide whether a triangle exists, which helps with the exercises on ambiguity.", "use": [ "lesson" ], "concepts": [ "concept/ambiguous-case", "concept/existence-conditions-of-a-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-d2baa13134", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 63", "location": "Solution of Oblique-Angled Triangles", "latex": "If $\\sin b \\sin A$ be greater than $\\sin a$, there is no triangle which satisfies the given conditions;", "markdown": "If $\\sin b \\sin A$ be greater than $\\sin a$, there is no triangle which satisfies the given conditions;", "why": "It states the simple test that rules out a triangle before any other work is done.", "use": [ "lesson" ], "concepts": [ "concept/ambiguous-case", "concept/existence-conditions-of-a-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-9944470e3f", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 69", "location": "Circumscribed and Inscribed Circles", "latex": "Let $ABC$ be the triangle; bisect the angles $A$ and $B$ by arcs meeting at $P$; from $P$ draw $PD$, $PE$, $PF$ perpendicular to the sides. Then it may be shewn that $PD$, $PE$, $PF$ are all equal; also that $AE = AF$, $BF = BD$, $CD = CE$.", "markdown": "Let $ABC$ be the triangle; bisect the angles $A$ and $B$ by arcs meeting at $P$; from $P$ draw $PD$, $PE$, $PF$ perpendicular to the sides. Then it may be shewn that $PD$, $PE$, $PF$ are all equal; also that $AE = AF$, $BF = BD$, $CD = CE$.", "why": "It shows a learner how the pole of the inscribed small circle is found by bisecting two angles and why the tangent lengths are equal.", "use": [ "lesson" ], "concepts": [ "concept/inscribed-circle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-288619f49d", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 72", "location": "Circumscribed and Inscribed Circles", "latex": "A circle which touches one side of a triangle and the other sides produced is called an \\textit{escribed circle;} thus there are three escribed circles belonging to a given triangle.", "markdown": "A circle which touches one side of a triangle and the other sides produced is called an *escribed circle;* thus there are three escribed circles belonging to a given triangle.", "why": "It gives the defining picture of an excircle, which students often confuse with the inscribed circle.", "use": [ "lesson", "website" ], "concepts": [ "concept/excircle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-695650b82d", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 73", "location": "Circumscribed and Inscribed Circles", "latex": "Then $P$ will be the pole of the small circle described about $ABC$.", "markdown": "Then $P$ will be the pole of the small circle described about $ABC$.", "why": "It states the key fact that the perpendicular bisectors of two sides meet at the pole of the circumscribed small circle.", "use": [ "lesson" ], "concepts": [ "concept/circumscribed-circle", "concept/small-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-68f65f62d8", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 74", "location": "Circumscribed and Inscribed Circles", "latex": "Many examples may be proposed involving properties of the circles inscribed in and described about the associated triangles.", "markdown": "Many examples may be proposed involving properties of the circles inscribed in and described about the associated triangles.", "why": "It invites the reader to see that the inscribed and circumscribed circles belong to a whole family of linked triangles.", "use": [ "history", "lesson" ], "concepts": [ "concept/associated-triangles", "concept/circumscribed-circle", "concept/inscribed-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-2b7dd60977", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 75", "location": "Circumscribed and Inscribed Circles", "latex": "Thus $P$ is the pole of the small circle \\textit{described round} the polar triangle, and the angular radius of the small circle described round the polar triangle is the complement of the angular radius of the small circle inscribed in the primitive triangle.", "markdown": "Thus $P$ is the pole of the small circle *described round* the polar triangle, and the angular radius of the small circle described round the polar triangle is the complement of the angular radius of the small circle inscribed in the primitive triangle.", "why": "It shows how the inscribed circle of one triangle and the circumscribed circle of its polar triangle have complementary angular radii.", "use": [ "lesson" ], "concepts": [ "concept/angular-radius", "concept/circumscribed-circle", "concept/inscribed-circle", "concept/polar-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-7905c724cb", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 76", "location": "Circumscribed and Inscribed Circles", "latex": "Shew that in an equilateral triangle $\\tan R = 2\\tan r$.", "markdown": "Shew that in an equilateral triangle $\\tan R = 2\\tan r$.", "why": "It is a short exercise that tests whether the learner can carry the circumscribed and inscribed radii of one triangle into a single relation.", "use": [ "lesson" ], "concepts": [ "concept/circumscribed-circle", "concept/inscribed-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-04622b30bd", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 77", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "A \\textit{Lune} is that portion of the surface of a sphere which is comprised between two great semicircles.", "markdown": "A *Lune* is that portion of the surface of a sphere which is comprised between two great semicircles.", "why": "Gives the learner the exact picture of a lune before any area is computed.", "use": [ "lesson", "website" ], "concepts": [ "concept/lune" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-668e457509", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 78", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "The expression $A+B+C-\\pi$ is called the \\textit{spherical excess} of the triangle;", "markdown": "The expression $A+B+C-\\pi$ is called the *spherical excess* of the triangle;", "why": "Names the quantity that the whole chapter builds on, in one sentence.", "use": [ "lesson" ], "concepts": [ "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-d8d1b13964", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 79", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "\\textit{the area of a spherical triangle is the same fraction of half the surface of the sphere as the spherical excess is of four right angles.}", "markdown": "*the area of a spherical triangle is the same fraction of half the surface of the sphere as the spherical excess is of four right angles.*", "why": "States the area rule as a ratio a learner can check against the sphere's total surface.", "use": [ "lesson", "website" ], "concepts": [ "quantity/spherical-excess", "theorem/area-of-a-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-b87ec29ae0", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 77", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "Hence since the whole surface of a sphere may be considered as a lune with an angle equal to four right angles, we have for a lune with an angle of which the circular measure is $A$,", "markdown": "Hence since the whole surface of a sphere may be considered as a lune with an angle equal to four right angles, we have for a lune with an angle of which the circular measure is $A$,", "why": "Shows how the lune formula follows from treating the whole sphere as a lune of angle four right angles.", "use": [ "lesson" ], "concepts": [ "concept/circular-measure", "concept/lune", "theorem/area-of-a-lune" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-41016c057f", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 79", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "The triangles are, however, not absolutely equal, but \\textit{symmetrically} equal (Art.\\ 57), so that one cannot be made to coincide with the other by superposition.", "markdown": "The triangles are, however, not absolutely equal, but *symmetrically* equal (Art. 57), so that one cannot be made to coincide with the other by superposition.", "why": "Warns that symmetrically equal triangles can be equal in area without being superposable, a common slip.", "use": [ "lesson" ], "concepts": [ "concept/symmetrical-spherical-triangles" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-e1388788ed", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 79", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "This expression is true even when the polygon has some of its angles greater than two right angles, provided it can be decomposed into triangles, of which each of the angles is less than two right angles.", "markdown": "This expression is true even when the polygon has some of its angles greater than two right angles, provided it can be decomposed into triangles, of which each of the angles is less than two right angles.", "why": "Tells the learner exactly when the polygon area formula applies, which prevents misuse on non-convex shapes.", "use": [ "lesson" ], "concepts": [ "concept/spherical-polygon", "theorem/area-of-a-spherical-polygon" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-281c33fdb8", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 85", "location": "On certain approximate Formul\\ae", "latex": "If the sides of the triangle are small compared with the radius of the sphere, $EF$ will not differ much from $A$;", "markdown": "If the sides of the triangle are small compared with the radius of the sphere, $EF$ will not differ much from $A$;", "why": "It explains why a small-sides expansion is reasonable, since the chord angle stays close to the spherical angle when the sides are short.", "use": [ "lesson" ], "concepts": [ "concept/approximation", "theorem/legendre-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-8b5df53b01", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 86", "location": "On certain approximate Formul\\ae", "latex": "This gives the \\textit{circular measure} of $\\theta$", "markdown": "This gives the *circular measure* of $\\theta$", "why": "It reminds the learner that the derived angle is in radians and must be converted to seconds of arc.", "use": [ "lesson" ], "concepts": [ "concept/circular-measure" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-adc430ec0a", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 89", "location": "On certain approximate Formul\\ae", "latex": "The importance of Legendre's Theorem in the application\nof Spherical Trigonometry to the measurement of the Earth's\nsurface has given rise to various developments of it which enable\nus to test the degree of exactness of the approximation.", "markdown": "The importance of Legendre’s Theorem in the application of Spherical Trigonometry to the measurement of the Earth’s surface has given rise to various developments of it which enable us to test the degree of exactness of the approximation.", "why": "It shows the learner why an approximate theorem matters: it underlies measuring the Earth's surface.", "use": [ "history", "website" ], "concepts": [ "concept/approximation", "concept/spherical-trigonometry", "theorem/legendre-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-a3dd0ae03f", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 86", "location": "On certain approximate Formul\\ae", "latex": "\\textit{If the sides of a spherical triangle\nbe small compared with the radius of the sphere, then each angle\nof the spherical triangle exceeds by one third of the spherical excess\nthe corresponding angle of the plane triangle, the sides\nof which are of the same length as the arcs of the spherical triangle.}", "markdown": "*If the sides of a spherical triangle be small compared with the radius of the sphere, then each angle of the spherical triangle exceeds by one third of the spherical excess the corresponding angle of the plane triangle, the sides of which are of the same length as the arcs of the spherical triangle.*", "why": "It states Legendre's theorem in full, so the learner sees exactly which plane triangle is compared with which spherical one.", "use": [ "lesson" ], "concepts": [ "concept/plane-triangle", "concept/spherical-triangle", "quantity/spherical-excess", "theorem/legendre-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-fbcf2881ce", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 88", "location": "On certain approximate Formul\\ae", "latex": "It will be seen that in the above approximation the area of\nthe spherical triangle is considered equal to the area of the plane\ntriangle which can be formed with sides of the same length.", "markdown": "It will be seen that in the above approximation the area of the spherical triangle is considered equal to the area of the plane triangle which can be formed with sides of the same length.", "why": "It warns the learner that the area comparison is itself an approximation, not an exact identity.", "use": [ "lesson" ], "concepts": [ "concept/approximation", "concept/area", "concept/plane-triangle", "concept/spherical-triangle", "theorem/legendre-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-9670b9ee3a", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 86", "location": "On certain approximate Formul\\ae", "latex": "Thus when the sides of\nthe spherical triangle and the radius of the sphere are known, we\ncan calculate the angles and sides of the chordal triangle.", "markdown": "Thus when the sides of the spherical triangle and the radius of the sphere are known, we can calculate the angles and sides of the chordal triangle.", "why": "It explains the link between a spherical triangle and its chordal triangle, which makes the chord approximation concrete.", "use": [ "lesson" ], "concepts": [ "concept/chord", "concept/chordal-triangle", "concept/circular-measure", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-16fe05fe8a", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 88", "location": "On certain approximate Formul\\ae", "latex": "therefore $\\dfrac{S}{r^2}$ is approximately equal to the spherical excess of the\nspherical triangle, and thus the theorem is established.", "markdown": "therefore $\\dfrac{S}{r^2}$ is approximately equal to the spherical excess of the spherical triangle, and thus the theorem is established.", "why": "It shows the step where the plane area divided by the squared radius becomes the spherical excess, which is the heart of the proof.", "use": [ "lesson" ], "concepts": [ "concept/area", "quantity/spherical-excess", "theorem/legendre-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-3e33e11a3c", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 97", "location": "Geodetical Operations", "latex": "If the three angles of a plane triangle be observed, the fact that their sum ought to be equal to two right angles affords a test of the accuracy with which the observations are made.", "markdown": "If the three angles of a plane triangle be observed, the fact that their sum ought to be equal to two right angles affords a test of the accuracy with which the observations are made.", "why": "It shows a familiar plane fact being used as a check on real measurements.", "use": [ "lesson" ], "concepts": [ "concept/observational-error", "theorem/sum-of-the-angles-of-a-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-4c5122419c", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 99", "location": "Geodetical Operations", "latex": "Now in modern observations $h$ will not exceed the circular measure of a few seconds, so that, if $C$ be not very small, $h\\cot C$ is practically insensible.", "markdown": "Now in modern observations $h$ will not exceed the circular measure of a few seconds, so that, if $C$ be not very small, $h\\cot C$ is practically insensible.", "why": "It teaches that a small angular error can matter little when the angle is not small, a sense of scale a learner needs.", "use": [ "lesson" ], "concepts": [ "concept/circular-measure", "concept/observational-error" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-591ba3ff2f", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 96", "location": "Geodetical Operations", "latex": "The degree of closeness with which the measured length agrees with the calculated length is a test of the accuracy of the survey.", "markdown": "The degree of closeness with which the measured length agrees with the calculated length is a test of the accuracy of the survey.", "why": "It states the survey's self-check in one sentence: a measured base checked against a calculated one.", "use": [ "lesson", "website" ], "concepts": [ "concept/base-line", "concept/geodesy" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-f6c0143a9c", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 95", "location": "Geodetical Operations", "latex": "One of the most important applications of Trigonometry, both Plane and Spherical, is to the determination of the figure and dimensions of the Earth itself, and of any portion of its surface.", "markdown": "One of the most important applications of Trigonometry, both Plane and Spherical, is to the determination of the figure and dimensions of the Earth itself, and of any portion of its surface.", "why": "It states plainly why spherical and plane trigonometry matter to surveying, which gives a learner the purpose of the whole chapter.", "use": [ "lesson", "history" ], "concepts": [ "concept/figure-of-the-earth", "concept/geodesy", "concept/plane-trigonometry", "concept/spherical-trigonometry" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-5600c330c3", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 95", "location": "Geodetical Operations", "latex": "An important part of any survey consists in the measurement of a horizontal line, which is called a \\textit{base}.", "markdown": "An important part of any survey consists in the measurement of a horizontal line, which is called a *base*.", "why": "It defines the base line in one sentence, the starting point for every triangulation.", "use": [ "lesson" ], "concepts": [ "concept/base-line" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-7c4fa9c288", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 95", "location": "Geodetical Operations", "latex": "At various points of the country suitable stations are selected and signals erected; then by supposing lines to be drawn connecting the signals, the country is divided into a series of triangles.", "markdown": "At various points of the country suitable stations are selected and signals erected; then by supposing lines to be drawn connecting the signals, the country is divided into a series of triangles.", "why": "It gives a clear picture of how triangulation turns a country into a network of triangles that can be calculated.", "use": [ "lesson", "website" ], "concepts": [ "concept/geodesy", "concept/station", "concept/triangle", "method/triangulation" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-2b0ef355a6", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 98", "location": "Geodetical Operations", "latex": "This formula is called General Roy's rule, as it was used by him in the Trigonometrical survey of Great Britain and Ireland.", "markdown": "This formula is called General Roy’s rule, as it was used by him in the Trigonometrical survey of Great Britain and Ireland.", "why": "It names the rule and places it in a real national survey, which gives the learner a concrete historical anchor.", "use": [ "history" ], "concepts": [ "method/general-roy-s-rule" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-290670c8fc", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 98", "location": "Geodetical Operations", "latex": "Now the area is not known \\textit{exactly} unless the elements of the spherical triangle are known \\textit{exactly}; but it is found that in such cases as occur in practice an approximate value of the area is sufficient.", "markdown": "Now the area is not known *exactly* unless the elements of the spherical triangle are known *exactly*; but it is found that in such cases as occur in practice an approximate value of the area is sufficient.", "why": "It warns that exact inputs are rarely available and explains why an approximate area still gives a useful answer.", "use": [ "lesson", "website" ], "concepts": [ "concept/approximation", "theorem/area-of-a-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-e24a68a1ab", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 101", "location": "Geodetical Operations", "latex": "thus in observing three angles, we suppose that in one observation a certain error is made, in a second observation the same numerical error is made but with an opposite sign, and in the remaining observation no error is made.", "markdown": "thus in observing three angles, we suppose that in one observation a certain error is made, in a second observation the same numerical error is made but with an opposite sign, and in the remaining observation no error is made.", "why": "It shows a learner a concrete case of how an assumption about observational error can mislead, which is a useful caution about the method.", "use": [ "lesson" ], "concepts": [ "concept/observational-error", "theorem/legendre-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-235471c74c", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 97", "location": "Geodetical Operations", "latex": "The three methods which we have indicated were all used by Delambre in calculating the triangles in the French survey (\\textit{Base du Syst\\`eme M\\'etrique}, Tome~\\textsc{iii}.\\ page~7).", "markdown": "The three methods which we have indicated were all used by Delambre in calculating the triangles in the French survey (*Base du Systeme Metrique*, Tome iii. page 7).", "why": "It ties the three approximation methods to a historical survey and gives the source for readers who want to look further.", "use": [ "history" ], "concepts": [ "person/delambre" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-8c55cfa267", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 103", "location": "Geodetical Operations", "latex": "This process, by which we find the angle $COD$ from the angle $AOB$, is called \\textit{reducing an angle to the horizon}.", "markdown": "This process, by which we find the angle $COD$ from the angle $AOB$, is called *reducing an angle to the horizon*.", "why": "It names the reduction process in one sentence, so a learner can recognise it when it appears in a survey problem.", "use": [ "lesson", "website" ], "concepts": [ "concept/horizontal-angle", "method/reduction-to-the-horizon" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-7f35ebec37", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 103", "location": "On small variations in the parts of a Spherical Triangle", "latex": "It is sometimes important to know what amount of error will be introduced into one of the calculated parts of a triangle by reason of any small error which may exist in the given parts.", "markdown": "It is sometimes important to know what amount of error will be introduced into one of the calculated parts of a triangle by reason of any small error which may exist in the given parts.", "why": "It tells the learner why a small change in the given data matters for the accuracy of the answer.", "use": [ "lesson", "website" ], "concepts": [ "concept/observational-error", "concept/small-variation", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-35033ae7a2", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 103", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\textit{A side and the opposite angle of a spherical triangle remain constant: determine the connexion between the small variations of any other pair of elements}.", "markdown": "*A side and the opposite angle of a spherical triangle remain constant: determine the connexion between the small variations of any other pair of elements*.", "why": "It states the central problem: hold one side and its opposite angle fixed, then find how the other parts change together.", "use": [ "lesson" ], "concepts": [ "concept/side", "concept/small-variation", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-ab9a5994ca", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 103", "location": "On small variations in the parts of a Spherical Triangle", "latex": "Suppose $C$ and $c$ to remain constant.", "markdown": "Suppose $C$ and $c$ to remain constant.", "why": "It shows the method of fixing a pair of opposite parts and varying the rest.", "use": [ "lesson" ], "concepts": [ "concept/side", "concept/small-variation", "concept/spherical-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-a63246acaf", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 103", "location": "On small variations in the parts of a Spherical Triangle", "latex": "then we require the ratio of $\\delta a$ to $\\delta b$ when both are extremely small.", "markdown": "then we require the ratio of $\\delta a$ to $\\delta b$ when both are extremely small.", "why": "It names the target of the calculation: a ratio between two small variations, not their absolute sizes.", "use": [ "lesson" ], "concepts": [ "concept/small-variation" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-ad2197bc85", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 106", "location": "On small variations in the parts of a Spherical Triangle", "latex": "If $A$ and $C$ are constant, and $b$ be increased by a small quantity, shew that $a$ will be increased or diminished according as $c$ is less or greater than a quadrant.", "markdown": "If $A$ and $C$ are constant, and $b$ be increased by a small quantity, shew that $a$ will be increased or diminished according as $c$ is less or greater than a quadrant.", "why": "It asks the learner to predict the direction of a change from the size of a side relative to a quadrant, which builds geometric intuition.", "use": [ "lesson" ], "concepts": [ "concept/quadrant", "concept/side", "concept/small-variation", "concept/spherical-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-81728fb2e0", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 106", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "then if we suppose $r$ to become indefinitely great, the limiting form of the proposed formula will be a relation in Plane Trigonometry.", "markdown": "then if we suppose $r$ to become indefinitely great, the limiting form of the proposed formula will be a relation in Plane Trigonometry.", "why": "It states the whole idea behind deriving plane results from spherical ones: let the radius grow without bound.", "use": [ "lesson" ], "concepts": [ "concept/plane-trigonometry", "method/deducing-plane-trigonometry-from-spherical-trigonometry" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-980f4bcb43", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 107", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "that is, in a plane triangle the sides are as the sines of the opposite angles.", "markdown": "that is, in a plane triangle the sides are as the sines of the opposite angles.", "why": "It turns the spherical law of sines into the familiar plane version, showing a learner what the limit looks like.", "use": [ "lesson" ], "concepts": [ "method/deducing-plane-trigonometry-from-spherical-trigonometry", "theorem/law-of-sines" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-046fe9749f", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 107", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "Let $OS = \\alpha$, $OSX = \\beta$; then the position of $O$ is determined by means of these angular co-ordinates $\\alpha$ and $\\beta$.", "markdown": "Let $OS = \\alpha$, $OSX = \\beta$; then the position of $O$ is determined by means of these angular co-ordinates $\\alpha$ and $\\beta$.", "why": "It shows how two angles fix a point on a sphere, which a learner needs before the equation of a small circle makes sense.", "use": [ "lesson" ], "concepts": [ "concept/angular-coordinates-on-a-sphere", "concept/pole-of-a-circle", "concept/small-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-7239031000", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 108", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "It will be observed that the angular co-ordinates here used are analogous to the \\textit{latitude} and \\textit{longitude} which serve to determine the positions of places on the Earth's surface;", "markdown": "It will be observed that the angular co-ordinates here used are analogous to the *latitude* and *longitude* which serve to determine the positions of places on the Earth’s surface;", "why": "It ties an abstract coordinate pair to latitude and longitude, a picture a learner already knows from maps.", "use": [ "lesson", "website" ], "concepts": [ "concept/angular-coordinates-on-a-sphere" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-87e0a8fafa", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 111", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "It is known in Plane Geometry that a certain circle touches the inscribed and escribed circles of any triangle; this circle is called the \\textit{Nine points circle}:", "markdown": "It is known in Plane Geometry that a certain circle touches the inscribed and escribed circles of any triangle; this circle is called the *Nine points circle*:", "why": "It names a famous plane result and sets up the spherical analogue the chapter goes on to prove.", "use": [ "history", "website" ], "concepts": [ "concept/inscribed-circle", "concept/nine-points-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-16d2bf68f0", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 110", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "The principal use of Art.\\ 137 is to determine whether three given points are on the same great circle; an illustration will be given in Art.\\ 146.", "markdown": "The principal use of Art. 137 is to determine whether three given points are on the same great circle; an illustration will be given in Art. 146.", "why": "It tells a learner what the preceding perpendicular-sines result is for, before its application appears.", "use": [ "lesson" ], "concepts": [ "concept/great-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-31d70cbf5a", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 110", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "The arcs drawn from the angles of a spherical triangle perpendicular to the opposite sides respectively meet at a point.", "markdown": "The arcs drawn from the angles of a spherical triangle perpendicular to the opposite sides respectively meet at a point.", "why": "It states a clear geometric result that learners can picture and then check with a diagram.", "use": [ "lesson", "website" ], "concepts": [ "concept/spherical-triangle", "theorem/concurrence-of-altitudes-of-a-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-ee53284ef9", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 106", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "The student must have perceived that many of the results obtained in \\textit{Spherical} Trigonometry resemble others with which he is familiar in \\textit{Plane} Trigonometry.", "markdown": "The student must have perceived that many of the results obtained in *Spherical* Trigonometry resemble others with which he is familiar in *Plane* Trigonometry.", "why": "It tells the learner why the two trigonometries are being compared before any formula appears.", "use": [ "lesson" ], "concepts": [ "concept/plane-trigonometry", "concept/spherical-trigonometry" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-a372706cfa", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 106", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "if we suppose $r$ to become indefinitely great, the limiting form of the proposed formula will be a relation in Plane Trigonometry.", "markdown": "if we suppose $r$ to become indefinitely great, the limiting form of the proposed formula will be a relation in Plane Trigonometry.", "why": "It states the method by which a spherical formula becomes a plane one: let the sphere's radius grow without bound.", "use": [ "lesson" ], "concepts": [ "concept/plane-trigonometry", "concept/spherical-trigonometry", "method/limiting-case" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-55d8335dcd", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 107", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "Let $O$ be the pole of a small circle, $S$ a fixed point on the sphere, $SX$ a fixed great circle of the sphere.", "markdown": "Let $O$ be the pole of a small circle, $S$ a fixed point on the sphere, $SX$ a fixed great circle of the sphere.", "why": "It sets up the diagram and the reference points needed to write the equation of a small circle.", "use": [ "lesson" ], "concepts": [ "concept/angular-coordinates-on-a-sphere", "concept/great-circle", "concept/pole-of-a-circle", "concept/small-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-a3088fdec3", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 108", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "this gives a relation between the angular co-ordinates of any point on the circumference of the circle.", "markdown": "this gives a relation between the angular co-ordinates of any point on the circumference of the circle.", "why": "It explains that one equation links the two angular coordinates of every point on the circle.", "use": [ "lesson" ], "concepts": [ "concept/angular-coordinates-on-a-sphere", "concept/small-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-77339bfbed", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 108", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "this result corresponds to the well-known property of a circle in Plane Geometry which is demonstrated in Euclid \\textsc{iii.}\\ 36 \\textit{Corollary}.", "markdown": "this result corresponds to the well-known property of a circle in Plane Geometry which is demonstrated in Euclid iii. 36 *Corollary*.", "why": "It links a new spherical result to a familiar plane theorem the learner already knows from Euclid.", "use": [ "history", "lesson" ], "concepts": [ "person/euclid", "theorem/power-of-a-point" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-b1fdcc90aa", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 112", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "We shall now shew that a small circle can always be determined on the sphere to touch the inscribed and escribed circles of any spherical triangle.", "markdown": "We shall now shew that a small circle can always be determined on the sphere to touch the inscribed and escribed circles of any spherical triangle.", "why": "It introduces the spherical analogue of the nine points circle as the main goal of the section.", "use": [ "lesson", "website" ], "concepts": [ "concept/excircle", "concept/inscribed-circle", "concept/nine-points-circle", "concept/small-circle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-52fdf6c1fe", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 110", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "The student should convince himself by examination that the result holds for all relative positions of $P$, $P_1$ and $P_2$, when due regard is paid to algebraical signs.", "markdown": "The student should convince himself by examination that the result holds for all relative positions of $P$, $P_1$ and $P_2$, when due regard is paid to algebraical signs.", "why": "It warns that the sign of each term depends on the relative positions of the points, a common source of error.", "use": [ "lesson" ], "concepts": [ "concept/perpendicular", "concept/sine" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-c5ae896b0c", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 121", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "The results which have been demonstrated with respect to the circle which touches the inscribed and escribed circles of a spherical triangle are mainly due to Dr Hart and Dr Salmon.", "markdown": "The results which have been demonstrated with respect to the circle which touches the inscribed and escribed circles of a spherical triangle are mainly due to Dr Hart and Dr Salmon.", "why": "It credits the authors of the spherical nine-points result, giving the learner a historical thread.", "use": [ "history", "website" ], "concepts": [ "concept/nine-points-circle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-322bf3d4de", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 124", "location": "Polyhedrons", "latex": "A polyhedron is a solid bounded by any number of plane rectilineal figures which are called its faces.", "markdown": "A polyhedron is a solid bounded by any number of plane rectilineal figures which are called its faces.", "why": "Gives the learner the basic picture of a polyhedron as a solid whose boundary is made of flat faces.", "use": [ "lesson" ], "concepts": [ "concept/face", "concept/polyhedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-38cd10975e", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 125", "location": "Polyhedrons", "latex": "It will be seen that the demonstration establishes something more than the enunciation states; for it is not assumed that the faces are equilateral and equiangular and all equal.", "markdown": "It will be seen that the demonstration establishes something more than the enunciation states; for it is not assumed that the faces are equilateral and equiangular and all equal.", "why": "Shows that the proof of the five regular polyhedra needs less than the usual definition of regularity, which is a useful point for a learner to notice.", "use": [ "lesson", "history" ], "concepts": [ "concept/regular-polyhedron", "theorem/five-regular-polyhedra" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-c0c5bad25c", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 124", "location": "Polyhedrons", "latex": "If $\\mathrm{S}$ be the number of solid angles in any polyhedron, $\\mathrm{F}$ the number of its faces, $\\mathrm{E}$ the number of its edges, then $\\mathrm{S+F=E+2}$.", "markdown": "If $\\mathrm{S}$ be the number of solid angles in any polyhedron, $\\mathrm{F}$ the number of its faces, $\\mathrm{E}$ the number of its edges, then $\\mathrm{S+F=E+2}$.", "why": "States the counting relation between vertices, faces and edges that the rest of the chapter builds on.", "use": [ "lesson" ], "concepts": [ "theorem/polyhedral-formula" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-32f9d5e38f", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 124", "location": "Polyhedrons", "latex": "Take any point within the polyhedron as centre, and describe a sphere of radius $r$, and draw straight lines from the centre to each of the angular points of the polyhedron;", "markdown": "Take any point within the polyhedron as centre, and describe a sphere of radius $r$, and draw straight lines from the centre to each of the angular points of the polyhedron;", "why": "Shows a clear geometric device, projecting a polyhedron onto a sphere, that makes the counting proof visible.", "use": [ "lesson", "website" ], "concepts": [ "concept/sphere", "theorem/polyhedral-formula" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-def1a53745", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 125", "location": "Polyhedrons", "latex": "but $n$ cannot be less than 3, so that $\\dfrac{1}{n}$ cannot be greater than $\\dfrac{1}{3}$,", "markdown": "but $n$ cannot be less than 3, so that $\\dfrac{1}{n}$ cannot be greater than $\\dfrac{1}{3}$,", "why": "Models the step-by-step inequality reasoning that reduces the possible face and angle counts to a finite list.", "use": [ "lesson" ], "concepts": [ "concept/regular-polyhedron", "theorem/five-regular-polyhedra" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-adaa97ce20", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 130", "location": "Polyhedrons", "latex": "Thus for a \\textit{regular} tetrahedron we have $144\\hspace{3pt}V^2=2a^6$.", "markdown": "Thus for a *regular* tetrahedron we have $144\\hspace{3pt}V^2=2a^6$.", "why": "Gives a concrete closed-form result a learner can check against the general formula for a tetrahedron's volume.", "use": [ "lesson" ], "concepts": [ "concept/tetrahedron", "theorem/volume-of-a-tetrahedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-fd27761983", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 134", "location": "Polyhedrons", "latex": "A regular octahedron is inscribed in a cube so that the corners of the octahedron are at the centres of the faces of the cube: shew that the volume of the cube is six times that of the octahedron.", "markdown": "A regular octahedron is inscribed in a cube so that the corners of the octahedron are at the centres of the faces of the cube: shew that the volume of the cube is six times that of the octahedron.", "why": "A clean exercise linking two regular solids through their inscribed relation, suitable for practice.", "use": [ "lesson" ], "concepts": [ "concept/hexahedron", "concept/octahedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-9f65b68cb8", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 130", "location": "Polyhedrons", "latex": "Or we may adopt Carnot's method, in which this relation is established independently, and the expression for the volume of a tetrahedron is deduced from it;", "markdown": "Or we may adopt Carnot’s method, in which this relation is established independently, and the expression for the volume of a tetrahedron is deduced from it;", "why": "Records that the chapter offers a second route to the tetrahedron volume, which gives historians a named method to trace.", "use": [ "history" ], "concepts": [ "method/carnot-s-method", "theorem/volume-of-a-tetrahedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-0ce78283f5", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 138", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "Thus, whatever may be the position of $T$, the sum of the cosines\nof the arcs which join $T$ to the fixed points varies as the cosine\nof the single arc which joins $T$ to a certain fixed point $U$.", "markdown": "Thus, whatever may be the position of $T$, the sum of the cosines of the arcs which join $T$ to the fixed points varies as the cosine of the single arc which joins $T$ to a certain fixed point $U$.", "why": "It shows a learner that a sum of many cosines collapses to one cosine, which is the central idea of the chapter.", "use": [ "lesson" ], "concepts": [ "concept/cosine", "concept/point", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-e210bfbeed", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 138", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "A sphere is described about a regular polyhedron; from any point on the surface of the sphere arcs are drawn to the solid angles of the polyhedron: to shew that the sum of the cosines of these arcs is zero.", "markdown": "A sphere is described about a regular polyhedron; from any point on the surface of the sphere arcs are drawn to the solid angles of the polyhedron: to shew that the sum of the cosines of these arcs is zero.", "why": "It poses a concrete problem whose answer is zero, giving a learner a clear target to test by symmetry and computation.", "use": [ "lesson" ], "concepts": [ "concept/cosine", "concept/polyhedral-angle", "concept/regular-polyhedron", "concept/sum" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-773969ff87", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 143", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "\\textit{Thus the sum of the squares of the cosines of the arcs which\njoin any point on the surface of the sphere to the solid angles of\nthe regular polyhedron is one third of the number of the solid\nangles.}", "markdown": "*Thus the sum of the squares of the cosines of the arcs which join any point on the surface of the sphere to the solid angles of the regular polyhedron is one third of the number of the solid angles.*", "why": "It gives a memorable result that links a trigonometric sum to a simple count of vertices.", "use": [ "website" ], "concepts": [ "concept/cosine", "concept/polyhedral-angle", "concept/regular-polyhedron", "concept/square" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-c39907c4c1", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 136", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "We leave to the student the exercise of shewing that\nthe formul\\ae\\ of the two preceding Articles are perfectly general for\nall positions of $T$ and $U$, outside or inside the triangle $ABC$: the\ndemonstrations will remain essentially the same for all modifications\nof the diagrams.", "markdown": "We leave to the student the exercise of shewing that the formul of the two preceding Articles are perfectly general for all positions of $T$ and $U$, outside or inside the triangle $ABC$: the demonstrations will remain essentially the same for all modifications of the diagrams.", "why": "It shows how the author expected a student to extend a proof to all diagram positions, a habit worth teaching; the old spelling is for the history use only.", "use": [ "history" ], "concepts": [ "concept/quadrantal-triangle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-24696586dc", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 148", "location": "Miscellaneous Propositions", "latex": "\\textit{If three arcs be drawn from the angles of a spherical triangle through any point to meet the opposite sides, the products of the sines of the alternate segments of the sides are equal.}", "markdown": "*If three arcs be drawn from the angles of a spherical triangle through any point to meet the opposite sides, the products of the sines of the alternate segments of the sides are equal.*", "why": "It states a clear, checkable sine-product theorem that a learner can test on a diagram.", "use": [ "lesson", "website" ], "concepts": [ "concept/spherical-triangle", "theorem/concurrence-of-arcs-through-a-point-of-a-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-165d0ffa96", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 147", "location": "Miscellaneous Propositions", "latex": "\\textit{The arc which passes through the middle points of the sides of any triangle upon a given base will meet the base produced at a fixed point, the distance of which from the middle point of the base is a quadrant.}", "markdown": "*The arc which passes through the middle points of the sides of any triangle upon a given base will meet the base produced at a fixed point, the distance of which from the middle point of the base is a quadrant.*", "why": "It states a surprising fixed-point result on the sphere that invites the learner to test it on a sketch.", "use": [ "website", "lesson" ], "concepts": [ "concept/quadrant", "theorem/arc-through-midpoints-of-two-sides-of-a-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-87f70aac01", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 145", "location": "Miscellaneous Propositions", "latex": "It may be presumed from symmetry that the pole of this circle is in the great circle which bisects $AB$ at right angles; and this presumption is easily verified.", "markdown": "It may be presumed from symmetry that the pole of this circle is in the great circle which bisects $AB$ at right angles; and this presumption is easily verified.", "why": "It shows a learner how a likely answer is first guessed from symmetry and then checked by substitution.", "use": [ "lesson" ], "concepts": [ "concept/great-circle", "concept/locus", "concept/pole-of-a-circle", "concept/symmetry" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-b1ec984451", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 146", "location": "Miscellaneous Propositions", "latex": "Let $P$ denote the pole of the inscribed circle, and $Q$ the pole of the circumscribed circle of a triangle $ABC$;", "markdown": "Let $P$ denote the pole of the inscribed circle, and $Q$ the pole of the circumscribed circle of a triangle $ABC$;", "why": "It names the two special circles of a triangle and their poles, which the following result compares.", "use": [ "lesson" ], "concepts": [ "concept/circumscribed-circle", "concept/inscribed-circle", "concept/pole-of-a-circle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-675c2a3867", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 150", "location": "Miscellaneous Propositions", "latex": "then the spherical triangle which corresponds to the three planes $LPM$, $MPN$, $NPL$ is the \\textit{polar triangle} of the spherical triangle which corresponds to the solid angle at $O$.", "markdown": "then the spherical triangle which corresponds to the three planes $LPM$, $MPN$, $NPL$ is the *polar triangle* of the spherical triangle which corresponds to the solid angle at $O$.", "why": "It gives a concrete geometric picture of how a polar triangle arises from perpendiculars inside a solid angle.", "use": [ "lesson" ], "concepts": [ "concept/polar-triangle", "concept/polyhedral-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-d3abf4f9d0", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 150", "location": "Miscellaneous Propositions", "latex": "Again, we know in mechanics that if three forces acting at a point are in equilibrium, each force is as the sine of the angle between the directions of the other two: the following proposition is analogous; if four forces acting at a point are in equilibrium each force is as the sine of the solid angle formed by the directions of the other three.", "markdown": "Again, we know in mechanics that if three forces acting at a point are in equilibrium, each force is as the sine of the angle between the directions of the other two: the following proposition is analogous; if four forces acting at a point are in equilibrium each force is as the sine of the solid angle formed by the directions of the other three.", "why": "It links the new idea of a solid angle's sine to the mechanics of forces the learner already knows.", "use": [ "lesson" ], "concepts": [ "concept/mechanical-equilibrium", "concept/sine-of-a-solid-angle", "quantity/force" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-3ce611b5bc", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 150", "location": "Miscellaneous Propositions", "latex": "the volume of a tetrahedron is one sixth of the product of three edges into the sine of the solid angle which they form.", "markdown": "the volume of a tetrahedron is one sixth of the product of three edges into the sine of the solid angle which they form.", "why": "It states a clear three-dimensional analogue of the area formula for a plane triangle.", "use": [ "lesson" ], "concepts": [ "concept/edge", "concept/sine-of-a-solid-angle", "theorem/volume-of-a-tetrahedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-8b0b466615", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 151", "location": "Miscellaneous Propositions", "latex": "The result, however, is generally true, even in cases in which the condition required by the demonstration of Art.~150 is not satisfied.", "markdown": "The result, however, is generally true, even in cases in which the condition required by the demonstration of Art. 150 is not satisfied.", "why": "It warns that a proof's hypotheses can be weaker than the truth of the result it proves.", "use": [ "history" ], "concepts": [ "concept/re-entrant-angle", "theorem/euler-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-b595b4f71e", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 151", "location": "Miscellaneous Propositions", "latex": "We begin with a theorem which is due to Cauchy.", "markdown": "We begin with a theorem which is due to Cauchy.", "why": "It credits the network theorem to its author and sets up the proof that follows.", "use": [ "history" ], "concepts": [ "person/augustin-louis-cauchy", "theorem/cauchy-s-network-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-65746fd40b", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 150", "location": "Miscellaneous Propositions", "latex": "Hence it may be inferred that any change of position in a rigid body, of which one point is fixed, may be effected by rotation round some axis through the fixed point.", "markdown": "Hence it may be inferred that any change of position in a rigid body, of which one point is fixed, may be effected by rotation round some axis through the fixed point.", "why": "It shows a surprising consequence of a plane-geometry fact, applied to the motion of a spinning body.", "use": [ "website" ], "concepts": [ "concept/rotation", "concept/spherical-distance" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-e7eb828cda", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 159", "location": "Numerical Solution of Spherical Triangles", "latex": "We shall give in this Chapter examples of the numerical solution of Spherical Triangles.", "markdown": "We shall give in this Chapter examples of the numerical solution of Spherical Triangles.", "why": "It states what the chapter sets out to do, so a learner knows the worked examples are a numerical method rather than new theory.", "use": [ "lesson", "history" ], "concepts": [ "method/numerical-solution-of-a-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-c0d4e3b27c", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 159", "location": "Numerical Solution of Spherical Triangles", "latex": "We shall first take right-angled triangles, and then oblique-angled triangles.", "markdown": "We shall first take right-angled triangles, and then oblique-angled triangles.", "why": "It gives the plan of the chapter: right-angled cases first, then oblique ones, which is the order a learner should follow.", "use": [ "lesson" ], "concepts": [ "concept/right-spherical-triangle", "method/numerical-solution-of-a-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-bc35865994", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 161", "location": "Numerical Solution of Spherical Triangles", "latex": "Here $\\tan c$ is \\textit{negative}; and therefore $\\tan b$ will be negative and $b$ greater than a quadrant.", "markdown": "Here $\\tan c$ is *negative*; and therefore $\\tan b$ will be negative and $b$ greater than a quadrant.", "why": "It shows the same sign reasoning for the tangent relation, so the learner sees that the sign of each ratio controls which quadrant the answer falls in.", "use": [ "lesson" ], "concepts": [ "concept/right-spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-4bc5a22075", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 160", "location": "Numerical Solution of Spherical Triangles", "latex": "The numerical value of $\\cos c$ is the same as that of $\\cos 81^\\circ\\, 45'\\, 36''$.", "markdown": "The numerical value of $\\cos c$ is the same as that of $\\cos 81^\\circ\\, 45'\\, 36''$.", "why": "It shows a useful device: a negative cosine is handled by taking the supplementary angle's value and correcting the final answer.", "use": [ "lesson", "website" ], "concepts": [ "concept/cosine", "concept/right-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-ebdb84046b", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 160", "location": "Numerical Solution of Spherical Triangles", "latex": "Here $\\cos c$ \\textit{is negative}; and therefore $\\cot B$ will be negative, and $B$ greater than a right angle.", "markdown": "Here $\\cos c$ *is negative*; and therefore $\\cot B$ will be negative, and $B$ greater than a right angle.", "why": "It shows a learner how the sign of a cosine decides whether an angle is acute or obtuse before any table work begins.", "use": [ "lesson" ], "concepts": [ "concept/cosine", "concept/cotangent", "concept/right-spherical-triangle", "quantity/right-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-37e4e98cdc", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 164", "location": "Numerical Solution of Spherical Triangles", "latex": "since $\\sin C$ is greater than $\\sin A$ we shall obtain two values for $c$ both greater than $a$, and we shall not know which is the value to be taken.", "markdown": "since $\\sin C$ is greater than $\\sin A$ we shall obtain two values for $c$ both greater than $a$, and we shall not know which is the value to be taken.", "why": "It explains plainly why a sine-rule calculation can leave two answers and why a further formula is needed to decide between them.", "use": [ "lesson" ], "concepts": [ "concept/ambiguous-case", "concept/sine" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-ecb7519ed7", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 166", "location": "Numerical Solution of Spherical Triangles", "latex": "Thus by taking only the nearest number of seconds in the tables the two methods give values of $c$ which differ by $1''$; if, however, we estimate fractions of a second both methods will agree in giving about $43\\tfrac12$ as the number of seconds.", "markdown": "Thus by taking only the nearest number of seconds in the tables the two methods give values of $c$ which differ by $1''$; if, however, we estimate fractions of a second both methods will agree in giving about $43\\tfrac12$ as the number of seconds.", "why": "It shows a learner how two independent routes to the same answer serve as a check, and how table precision affects the last digit.", "use": [ "lesson", "website" ], "concepts": [ "method/verification", "unit/second-of-arc" ] }, { "id": "todhunter-spherical-trigonometry-1886/x-d8317ddd78", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 168", "location": "Numerical Solution of Spherical Triangles", "latex": "The student can obtain more examples, which can be easily verified, from those here worked out, by interchanging the given and required quantities, or by making use of the polar triangle.", "markdown": "The student can obtain more examples, which can be easily verified, from those here worked out, by interchanging the given and required quantities, or by making use of the polar triangle.", "why": "It invites a learner to extend the worked problems on their own by swapping given and required parts, and to check each new result.", "use": [ "lesson" ], "concepts": [ "concept/polar-triangle", "method/verification" ] } ], "equations": [ { "id": "todhunter-spherical-trigonometry-1886/eq-e1147ea536", "chapter": "todhunter-spherical-trigonometry-1886/ch-great-and-small-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 11", "location": "Great and Small Circles", "latex": "CD=\\surd(OD^2-OC^2)", "name": null, "statement": "The radius CD of the plane section of a sphere is the square root of the sphere's radius squared minus the squared distance from the centre of the sphere to the plane.", "kind": "result", "symbols": [ { "unit": null, "symbol": "CD", "meaning": "radius of the plane section (distance from the centre of the section C to a point D on it)" }, { "unit": null, "symbol": "OD", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "OC", "meaning": "perpendicular distance from the centre of the sphere O to the cutting plane" } ], "sympy": "Eq(CD, sqrt(OD**2 - OC**2))", "physics": false, "states": [], "concepts": [ "concept/centre-of-a-circle", "concept/circle", "concept/perpendicular", "concept/radius-of-a-regular-polygon", "concept/small-circle", "concept/sphere", "quantity/right-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f86b7866b7", "chapter": "todhunter-spherical-trigonometry-1886/ch-great-and-small-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 12", "location": "Great and Small Circles", "latex": "PD=\\surd(PC^2+CD^2)", "name": null, "statement": "The straight-line distance from a pole P of a circle to a point D on its circumference is the square root of the sum of the squares of PC (pole to centre of the circle) and CD (radius of the circle).", "kind": "result", "symbols": [ { "unit": null, "symbol": "PD", "meaning": "straight-line distance (chord) from the pole P to a point D on the circumference" }, { "unit": null, "symbol": "PC", "meaning": "distance from the pole P to the centre C of the circle" }, { "unit": null, "symbol": "CD", "meaning": "radius of the circle" } ], "sympy": "Eq(PD, sqrt(PC**2 + CD**2))", "physics": false, "states": [], "concepts": [ "concept/centre-of-a-circle", "concept/chord", "concept/circle", "concept/perpendicular", "concept/pole-of-a-circle", "concept/radius-of-a-regular-polygon", "quantity/right-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-543ae7f36f", "chapter": "todhunter-spherical-trigonometry-1886/ch-great-and-small-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 14", "location": "Great and Small Circles", "latex": "AOB = AOM - BOM = BON - BOM = MON", "name": null, "statement": "The angle AOB subtended at the centre of the sphere by the arc joining two poles equals the inclination MON of the two great circles, since AOB is found by subtracting equal parts.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AOB", "meaning": "angle at the centre O subtended by the arc joining the poles A and B" }, { "unit": null, "symbol": "AOM", "meaning": "angle at O between the pole A and the point M" }, { "unit": null, "symbol": "BOM", "meaning": "angle at O between the pole B and the point M" }, { "unit": null, "symbol": "BON", "meaning": "angle at O between the pole B and the point N" }, { "unit": null, "symbol": "MON", "meaning": "angle of inclination of the planes of the two great circles" } ], "sympy": "Eq(AOB, AOM - BOM, BON - BOM, MON)", "physics": false, "states": [], "concepts": [ "concept/angle-at-the-centre-of-a-regular-polygon", "concept/great-circle", "concept/inclination-of-two-lines", "concept/perpendicular", "concept/plane-angle", "concept/pole-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-46c72377a0", "chapter": "todhunter-spherical-trigonometry-1886/ch-great-and-small-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 16", "location": "Great and Small Circles", "latex": "\\frac{\\operatorname{arc} ab} {\\operatorname{radius} Ca}=\\frac{\\operatorname{arc} AB} {\\operatorname{radius} OA}", "name": null, "statement": "The ratio of arc to radius is the same for the small-circle arc ab and the great-circle arc AB, since both subtend the same angle at their centres.", "kind": "result", "symbols": [ { "unit": null, "symbol": "arc ab", "meaning": "arc of the small circle subtending the angle at its centre C" }, { "unit": null, "symbol": "radius Ca", "meaning": "radius of the small circle" }, { "unit": null, "symbol": "arc AB", "meaning": "arc of the great circle subtending the same angle at its centre O" }, { "unit": null, "symbol": "radius OA", "meaning": "radius of the sphere (radius of the great circle)" } ], "sympy": "Eq(arc_ab/radius_Ca, arc_AB/radius_OA)", "physics": false, "states": [], "concepts": [ "concept/angle-at-the-centre-of-a-regular-polygon", "concept/great-circle", "concept/plane-trigonometry", "concept/radius-of-a-regular-polygon", "concept/small-circle", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-a2eba91f59", "chapter": "todhunter-spherical-trigonometry-1886/ch-great-and-small-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 16", "location": "Great and Small Circles", "latex": "\\frac{\\operatorname{arc} ab}{\\operatorname{arc} AB}=\\frac{Ca}{OA} =\\frac{Ca}{Oa}=\\sin POa", "name": null, "statement": "The ratio of the small-circle arc ab to the great-circle arc AB equals the sine of the angle POa, the ratio of the radii of the two circles.", "kind": "result", "symbols": [ { "unit": null, "symbol": "arc ab", "meaning": "arc of the small circle subtending the angle at its centre C" }, { "unit": null, "symbol": "arc AB", "meaning": "arc of the great circle subtending the same angle at its centre O" }, { "unit": null, "symbol": "Ca", "meaning": "radius of the small circle" }, { "unit": null, "symbol": "OA", "meaning": "radius of the great circle (radius of the sphere)" }, { "unit": null, "symbol": "Oa", "meaning": "radius of the sphere, equal to OA" }, { "unit": null, "symbol": "POa", "meaning": "angle at the centre O between the pole P and the point a" } ], "sympy": "Eq(arc_ab/arc_AB, sin(POa))", "physics": false, "states": [], "concepts": [ "concept/angle-at-the-centre-of-a-regular-polygon", "concept/great-circle", "concept/plane-trigonometry", "concept/pole-of-a-circle", "concept/radius-of-a-regular-polygon", "concept/sine", "concept/small-circle", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-c8e5a5b19f", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 18", "location": "Spherical Triangles", "latex": "\\dfrac{\\operatorname{arc} AB} {\\operatorname{radius} OA}", "name": null, "statement": "The plane angle AOB at the centre O is measured by the arc AB divided by the radius OA, so an arc is proportional to its angle on the same sphere.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "AOB", "meaning": "plane angle at the centre O of the sphere, subtended by the arc AB" }, { "unit": null, "symbol": "arc AB", "meaning": "arc of a great circle between the points A and B" }, { "unit": null, "symbol": "OA", "meaning": "radius of the sphere" } ], "sympy": "Eq(AOB, arc_AB/radius_OA)", "physics": false, "states": [], "concepts": [ "concept/circular-measure", "concept/great-circle", "concept/plane-angle", "concept/radius-of-a-regular-polygon", "concept/sphere", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-14d3380c7e", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 18", "location": "Spherical Triangles", "latex": "C = 90^\\circ", "name": null, "statement": "If the angle C of a spherical triangle is a right angle, its numerical value is 90 when the unit is the degree.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle at the vertex C" }, { "unit": "degree of angle", "symbol": "90^\\circ", "meaning": "a right angle expressed in degrees" } ], "sympy": "Eq(C, 90)", "physics": false, "states": [], "concepts": [ "concept/plane-angle", "concept/spherical-triangle", "quantity/right-angle", "unit/degree-of-angle", "unit/unit" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-b540f47156", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 18", "location": "Spherical Triangles", "latex": "C = \\dfrac{\\pi}{2}", "name": null, "statement": "If the angle C of a spherical triangle is a right angle, its numerical value is pi/2 when the unit is the angle subtended at the centre by an arc equal to the radius.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle at the vertex C" }, { "unit": null, "symbol": "\\pi", "meaning": "the ratio of circumference to diameter of a circle" } ], "sympy": "Eq(C, pi/2)", "physics": false, "states": [], "concepts": [ "concept/circular-measure", "concept/plane-angle", "concept/radius-of-a-regular-polygon", "quantity/pi", "quantity/right-angle", "unit/unit" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-9da182692e", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 22", "location": "Spherical Geometry", "latex": "A' &= \\pi - a", "name": null, "statement": "The angle A' of the polar triangle is the supplement of the side a of the primitive triangle, in circular measure.", "kind": "result", "symbols": [ { "unit": "radian (circular measure)", "symbol": "A'", "meaning": "angle of the polar triangle at the pole of side BC" }, { "unit": "radian (circular measure)", "symbol": "a", "meaning": "side BC of the primitive triangle" } ], "sympy": "Eq(Ap, pi - a)", "physics": false, "states": [], "concepts": [ "concept/circular-measure", "concept/plane-angle", "concept/polar-triangle", "concept/primitive-triangle", "concept/side", "concept/supplementary-angles" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-a3167bef84", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 22", "location": "Spherical Geometry", "latex": "B' &= \\pi - b", "name": null, "statement": "The angle B' of the polar triangle is the supplement of the side b of the primitive triangle, in circular measure.", "kind": "result", "symbols": [ { "unit": "radian (circular measure)", "symbol": "B'", "meaning": "angle of the polar triangle at the pole of side CA" }, { "unit": "radian (circular measure)", "symbol": "b", "meaning": "side CA of the primitive triangle" } ], "sympy": "Eq(Bp, pi - b)", "physics": false, "states": [], "concepts": [ "concept/circular-measure", "concept/plane-angle", "concept/polar-triangle", "concept/primitive-triangle", "concept/side", "concept/supplementary-angles" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-a3ec71e761", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 22", "location": "Spherical Geometry", "latex": "C' &= \\pi - c", "name": null, "statement": "The angle C' of the polar triangle is the supplement of the side c of the primitive triangle, in circular measure.", "kind": "result", "symbols": [ { "unit": "radian (circular measure)", "symbol": "C'", "meaning": "angle of the polar triangle at the pole of side AB" }, { "unit": "radian (circular measure)", "symbol": "c", "meaning": "side AB of the primitive triangle" } ], "sympy": "Eq(Cp, pi - c)", "physics": false, "states": [], "concepts": [ "concept/circular-measure", "concept/plane-angle", "concept/polar-triangle", "concept/primitive-triangle", "concept/side", "concept/supplementary-angles" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ca97c139f1", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 22", "location": "Spherical Geometry", "latex": "a' &= \\pi - A", "name": null, "statement": "The side a' of the polar triangle is the supplement of the angle A of the primitive triangle, in circular measure.", "kind": "result", "symbols": [ { "unit": "radian (circular measure)", "symbol": "a'", "meaning": "side of the polar triangle opposite A'" }, { "unit": "radian (circular measure)", "symbol": "A", "meaning": "angle A of the primitive triangle" } ], "sympy": "Eq(ap, pi - A)", "physics": false, "states": [], "concepts": [ "concept/circular-measure", "concept/plane-angle", "concept/polar-triangle", "concept/primitive-triangle", "concept/side", "concept/supplementary-angles" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-8912284cb1", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 22", "location": "Spherical Geometry", "latex": "b' &= \\pi - B", "name": null, "statement": "The side b' of the polar triangle is the supplement of the angle B of the primitive triangle, in circular measure.", "kind": "result", "symbols": [ { "unit": "radian (circular measure)", "symbol": "b'", "meaning": "side of the polar triangle opposite B'" }, { "unit": "radian (circular measure)", "symbol": "B", "meaning": "angle B of the primitive triangle" } ], "sympy": "Eq(bp, pi - B)", "physics": false, "states": [], "concepts": [ "concept/circular-measure", "concept/plane-angle", "concept/polar-triangle", "concept/primitive-triangle", "concept/side", "concept/supplementary-angles" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-537af0cb96", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 22", "location": "Spherical Geometry", "latex": "c' &= \\pi - C", "name": null, "statement": "The side c' of the polar triangle is the supplement of the angle C of the primitive triangle, in circular measure.", "kind": "result", "symbols": [ { "unit": "radian (circular measure)", "symbol": "c'", "meaning": "side of the polar triangle opposite C'" }, { "unit": "radian (circular measure)", "symbol": "C", "meaning": "angle C of the primitive triangle" } ], "sympy": "Eq(cp, pi - C)", "physics": false, "states": [], "concepts": [ "concept/circular-measure", "concept/plane-angle", "concept/polar-triangle", "concept/primitive-triangle", "concept/side", "concept/supplementary-angles" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-758185b4d1", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 23", "location": "Spherical Geometry", "latex": "\\dfrac{AB}{OA} + \\dfrac{BC}{OA} + \\dfrac{CA}{OA} \\text{ is less than } 2\\pi", "name": null, "statement": "Dividing the three sides of a spherical triangle by the sphere's radius, their sum is less than 2π, because the three plane angles at the centre O sum to less than four right angles.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "arc AB, a side of the spherical triangle" }, { "unit": null, "symbol": "BC", "meaning": "arc BC, a side of the spherical triangle" }, { "unit": null, "symbol": "CA", "meaning": "arc CA, a side of the spherical triangle" }, { "unit": null, "symbol": "OA", "meaning": "radius of the sphere (O is the centre)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/centre-of-a-circle", "concept/circular-measure", "concept/great-circle", "concept/inequality", "concept/side", "concept/sphere", "concept/spherical-triangle", "concept/sum", "quantity/circumference", "quantity/pi" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-5d7dea565b", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 23", "location": "Spherical Geometry", "latex": "AB+BC+CD \\text{ is less than } 2\\pi\\times OA", "name": null, "statement": "The sum of the arcs forming the sides of the spherical figure is less than the circumference of a great circle, 2π times the radius OA. The book writes CD where the sides of the figure are CA; this looks like a typo for CA, flagged for checking.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "arc AB" }, { "unit": null, "symbol": "BC", "meaning": "arc BC" }, { "unit": null, "symbol": "CD", "meaning": "arc CD as printed (likely CA in the figure)" }, { "unit": null, "symbol": "OA", "meaning": "radius of the sphere" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/great-circle", "concept/inequality", "concept/side", "concept/sphere", "concept/sum", "quantity/circumference", "quantity/pi" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-fa9d499657", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 23", "location": "Spherical Geometry", "latex": "AD+BC\\text{ is greater than }AC", "name": null, "statement": "In a four-sided spherical polygon with each angle less than two right angles, the sum of two opposite-ended sides AD and BC exceeds the diagonal AC; obtained by repeated use of the triangle inequality on the sphere.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AD", "meaning": "arc AD, a side of the polygon" }, { "unit": null, "symbol": "BC", "meaning": "arc BC, a side of the polygon" }, { "unit": null, "symbol": "AC", "meaning": "arc AC, a diagonal of the polygon" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/polygon", "concept/side", "concept/spherical-triangle", "concept/sum" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-38db1db2bc", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 23", "location": "Spherical Geometry", "latex": "AB+BC+CD\\text{ is greater than }AC+CD", "name": null, "statement": "For the four-sided spherical polygon, the sum of the sides AB, BC, CD exceeds the sum AC+CD, and so exceeds AD.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "arc AB, a side of the polygon" }, { "unit": null, "symbol": "BC", "meaning": "arc BC, a side of the polygon" }, { "unit": null, "symbol": "CD", "meaning": "arc CD, a side of the polygon" }, { "unit": null, "symbol": "AC", "meaning": "arc AC, a diagonal" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/polygon", "concept/side", "concept/sum" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-c6e76fa328", "chapter": "todhunter-spherical-trigonometry-1886/ch-spherical-geometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 23", "location": "Spherical Geometry", "latex": "A+B+C\\text{ is greater than }\\pi", "name": null, "statement": "The three angles of a spherical triangle sum to more than π (two right angles), from the polar triangle's side-sum being less than 2π.", "kind": "result", "symbols": [ { "unit": "radian (circular measure)", "symbol": "A", "meaning": "angle A of the spherical triangle" }, { "unit": "radian (circular measure)", "symbol": "B", "meaning": "angle B of the spherical triangle" }, { "unit": "radian (circular measure)", "symbol": "C", "meaning": "angle C of the spherical triangle" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/plane-angle", "concept/polar-triangle", "concept/spherical-triangle", "concept/sum", "quantity/pi", "quantity/right-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-a37e877cc1", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 26", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\cos a = \\cos b \\cos c + \\sin b \\sin c \\cos A", "name": null, "statement": "The cosine of a side equals the product of the cosines of the other two sides plus the product of their sines times the cosine of the included angle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle opposite the angle A" }, { "unit": null, "symbol": "b", "meaning": "side of the spherical triangle opposite the angle B" }, { "unit": null, "symbol": "c", "meaning": "side of the spherical triangle opposite the angle C" }, { "unit": null, "symbol": "A", "meaning": "spherical angle of the triangle at the vertex A" } ], "sympy": "Eq(cos(a), cos(b)*cos(c) + sin(b)*sin(c)*cos(A))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle", "quantity/angle" ], "pages": [ "scan 26", "scan 61" ], "chapters": [ "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-6332f4dec3", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 26", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\cos A = \\dfrac{\\cos a - \\cos b \\cos c}{\\sin b \\sin c}", "name": null, "statement": "The cosine of an angle of a spherical triangle is expressed in terms of the cosines and sines of the sides.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "spherical angle of the triangle at the vertex A" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": "Eq(cos(A), (cos(a) - cos(b)*cos(c))/(sin(b)*sin(c)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/spherical-angle", "concept/spherical-triangle", "method/solving-an-oblique-spherical-triangle", "quantity/angle" ], "pages": [ "scan 26", "scan 56" ], "chapters": [ "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-9b25003591", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 28", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\cos b = \\cos c \\cos a + \\sin c \\sin a \\cos B", "name": null, "statement": "The cosine of side b equals the cosine of c times the cosine of a plus the sines of c and a times the cosine of the angle B.", "kind": "result", "symbols": [ { "unit": null, "symbol": "B", "meaning": "spherical angle of the triangle at the vertex B" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" } ], "sympy": "Eq(cos(b), cos(c)*cos(a) + sin(c)*sin(a)*cos(B))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-bd9923af51", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 28", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\cos c = \\cos a \\cos b + \\sin a \\sin b \\cos C", "name": null, "statement": "The cosine of side c equals the cosine of a times the cosine of b plus the sines of a and b times the cosine of the angle C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "C", "meaning": "spherical angle of the triangle at the vertex C" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": "Eq(cos(c), cos(a)*cos(b) + sin(a)*sin(b)*cos(C))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-64857c7cd8", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 28", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\cos a = \\sin b \\cos A", "name": null, "statement": "In the case where one side containing the angle A is a quadrant, the cosine of the opposite side equals the sine of b times the cosine of A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side containing A, here taken as a side with sine b" } ], "sympy": "Eq(cos(a), sin(b)*cos(A))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/quadrant", "concept/sine", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ce6db00535", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 29", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\sin A=\\dfrac{\\surd(1-\\cos^2 a-\\cos^2 b-\\cos^2 c+2\\cos a\\cos b\\cos c)}{\\sin b \\sin c}", "name": null, "statement": "The sine of an angle of a spherical triangle is given in terms of the trigonometrical functions of the sides, with the positive root taken.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": "Eq(sin(A), sqrt(1 - cos(a)**2 - cos(b)**2 - cos(c)**2 + 2*cos(a)*cos(b)*cos(c))/(sin(b)*sin(c)))", "physics": false, "states": [], "concepts": [ "concept/root", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-b391a8c265", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 29", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\dfrac{\\sin A}{\\sin a}=\\dfrac{\\sin B}{\\sin b}=\\dfrac{\\sin C}{\\sin c}", "name": "sine rule for spherical triangles", "statement": "The sines of the angles of a spherical triangle are proportional to the sines of the opposite sides.", "kind": "law", "symbols": [ { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" } ], "sympy": "Eq(sin(A)/sin(a), sin(B)/sin(b))", "physics": false, "states": [ "law/sine-rule-for-spherical-triangles" ], "concepts": [ "concept/proportion", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-e97756a3ef", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 30", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\dfrac{\\sin B}{\\sin C}=\\dfrac{\\sin b}{\\sin c}", "name": null, "statement": "The sines of two angles of a spherical triangle are in the same ratio as the sines of the opposite sides.", "kind": "law", "symbols": [ { "unit": null, "symbol": "B", "meaning": "spherical angle at the vertex B" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" } ], "sympy": "Eq(sin(B)/sin(C), sin(b)/sin(c))", "physics": false, "states": [], "concepts": [ "concept/proportion", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-a6e7f009e2", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 30", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\cot a \\sin b = \\cot A \\sin C + \\cos b \\cos C", "name": null, "statement": "A four-part relation between two sides and two angles of a spherical triangle, involving cotangents and sines of one side and cosines of the other.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" }, { "unit": null, "symbol": "C", "meaning": "spherical angle at the vertex C" } ], "sympy": "Eq(cot(a)*sin(b), cot(A)*sin(C) + cos(b)*cos(C))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cotangent", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-10946a306d", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 31", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\sin^2 \\dfrac{A}{2} = \\dfrac{\\sin \\tfrac{1}{2}(a+b-c)\\sin\\tfrac{1}{2}(a-b+c)}{\\sin b \\sin c}", "name": null, "statement": "The square of the sine of half an angle of a spherical triangle is expressed as a product of sines of half-sums of the sides over the product of the sines of the two sides that contain the angle.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side containing A" }, { "unit": null, "symbol": "c", "meaning": "side containing A" } ], "sympy": "Eq(sin(A/2)**2, sin((a+b-c)/2)*sin((a-b+c)/2)/(sin(b)*sin(c)))", "physics": false, "states": [], "concepts": [ "concept/half-angle-formula", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ce76df0ac1", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 31", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\sin^2 \\dfrac{A}{2} = \\dfrac{\\sin(s - b)\\sin(s - c)}{\\sin b \\sin c}", "name": null, "statement": "The square of the sine of half an angle equals the product of the sines of s minus each adjacent side over the product of the sines of the two sides containing the angle, where 2s is the sum of the sides.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "s", "meaning": "half the sum of the sides of the triangle, so 2s = a + b + c" }, { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" } ], "sympy": "Eq(sin(A/2)**2, sin(s-b)*sin(s-c)/(sin(b)*sin(c)))", "physics": false, "states": [], "concepts": [ "concept/half-angle-formula", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-cd80df2828", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 32", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\tan\\dfrac{A}{2}=\\Surd{\\left\\{ \\dfrac{\\sin(s-b)\\sin(s-c)}{\\sin s \\sin(s-a)} \\right\\}}", "name": null, "statement": "The tangent of half an angle of a spherical triangle is the square root of a ratio of sines of s minus the sides.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "s", "meaning": "half the sum of the sides of the triangle" }, { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" } ], "sympy": "Eq(tan(A/2), sqrt(sin(s-b)*sin(s-c)/(sin(s)*sin(s-a))))", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-0e6a10feee", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 32", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\sin A = \\dfrac{2}{\\sin b \\sin c}\\{\\sin s \\sin(s-a)\\sin(s-b)\\sin(s-c)\\}^{\\tfrac{1}{2}}", "name": null, "statement": "The sine of an angle of a spherical triangle expressed as twice the square root of the product of four sines of s and s minus the sides, over the product of the sines of the two adjacent sides.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "s", "meaning": "half the sum of the sides of the triangle" }, { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" } ], "sympy": "Eq(sin(A), 2*(sin(s)*sin(s-a)*sin(s-b)*sin(s-c))**(1/2)/(sin(b)*sin(c)))", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-8fc3b3b3f2", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 33", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\cos A =-\\cos B \\cos C + \\sin B \\sin C \\cos a", "name": null, "statement": "The cosine of an angle of a spherical triangle expressed in terms of the cosines and sines of the other two angles and the opposite side (the polar form of the cosine rule).", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" }, { "unit": null, "symbol": "B", "meaning": "spherical angle at the vertex B" }, { "unit": null, "symbol": "C", "meaning": "spherical angle at the vertex C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" } ], "sympy": "Eq(cos(A), -cos(B)*cos(C) + sin(B)*sin(C)*cos(a))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/polar-triangle", "concept/side", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-40ca774a25", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 33", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\sin^{2}\\frac{a}{2}=-\\frac{\\cos S\\cos(S-A)}{\\sin B\\sin C}", "name": null, "statement": "The square of the sine of half a side is expressed in terms of cosines of S and S minus A over the product of the sines of the two angles adjacent to the side, where 2S is the sum of the angles.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "S", "meaning": "half the sum of the angles of the triangle, so 2S = A + B + C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" } ], "sympy": "Eq(sin(a/2)**2, -cos(S)*cos(S-A)/(sin(B)*sin(C)))", "physics": false, "states": [], "concepts": [ "concept/half-angle-formula", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-2ecde172ad", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 34", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\sin a=\\dfrac{2}{\\sin B\\sin C}\\left\\{-\\cos S\\cos (S-A)\\cos(S-B)\\cos (S-C)\\right\\}^{\\tfrac{1}{2}}", "name": null, "statement": "The sine of a side is twice the square root of minus the product of four cosines of S and S minus the angles, over the product of the sines of the two adjacent angles.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "S", "meaning": "half the sum of the angles of the triangle" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" } ], "sympy": "Eq(sin(a), 2*(-cos(S)*cos(S-A)*cos(S-B)*cos(S-C))**(1/2)/(sin(B)*sin(C)))", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-8958eae439", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 35", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\tan\\tfrac{1}{2}(A + B) = \\frac{\\cos\\tfrac{1}{2}(a - b)}{\\cos\\tfrac{1}{2}(a + b)}\\cot\\frac{C}{2}", "name": "Napier's analogy", "statement": "Napier's first analogy: the tangent of half the sum of two angles equals the ratio of cosines of half the difference and half the sum of the opposite sides, times the cotangent of half the third angle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" }, { "unit": null, "symbol": "B", "meaning": "spherical angle at the vertex B" }, { "unit": null, "symbol": "C", "meaning": "spherical angle at the vertex C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" } ], "sympy": "Eq(tan((A+B)/2), cos((a-b)/2)/cos((a+b)/2)*cot(C/2))", "physics": false, "states": [ "theorem/napier-s-analogy" ], "concepts": [ "concept/cosine", "concept/cotangent", "concept/side", "concept/spherical-angle", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-919dc6e313", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 35", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\tan\\frac{1}{2}(A - B) = \\frac{\\sin\\tfrac{1}{2}(a - b)} {\\sin\\tfrac{1}{2}(a + b)} \\cot\\frac{C}{2}", "name": "Napier's analogy", "statement": "Napier's second analogy: the tangent of half the difference of two angles equals the ratio of sines of half the difference and half the sum of the opposite sides, times the cotangent of half the third angle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" }, { "unit": null, "symbol": "B", "meaning": "spherical angle at the vertex B" }, { "unit": null, "symbol": "C", "meaning": "spherical angle at the vertex C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" } ], "sympy": "Eq(tan((A-B)/2), sin((a-b)/2)/sin((a+b)/2)*cot(C/2))", "physics": false, "states": [ "theorem/napier-s-analogy" ], "concepts": [ "concept/cotangent", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-12f4998788", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 35", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\tan\\tfrac{1}{2}(a + b) = \\frac{\\cos\\tfrac{1}{2}(A - B)} {\\cos\\tfrac{1}{2}(A + B)} \\tan\\frac{c}{2}", "name": "Napier's analogy", "statement": "Napier's third analogy, obtained from the first by the supplemental triangle: the tangent of half the sum of two sides equals the ratio of cosines of half the difference and half the sum of the opposite angles, times the tangent of half the third side.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" }, { "unit": null, "symbol": "B", "meaning": "spherical angle at the vertex B" } ], "sympy": "Eq(tan((a+b)/2), cos((A-B)/2)/cos((A+B)/2)*tan(c/2))", "physics": false, "states": [ "theorem/napier-s-analogy" ], "concepts": [ "concept/cosine", "concept/polar-triangle", "concept/side", "concept/spherical-angle", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-81b7d10954", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 35", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\tan\\tfrac{1}{2}(a - b) = \\frac{\\sin\\tfrac{1}{2}(A - B)} {\\sin\\tfrac{1}{2}(A + B)} \\tan\\frac{c}{2}", "name": "Napier's analogy", "statement": "Napier's fourth analogy: the tangent of half the difference of two sides equals the ratio of sines of half the difference and half the sum of the opposite angles, times the tangent of half the third side.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" }, { "unit": null, "symbol": "B", "meaning": "spherical angle at the vertex B" } ], "sympy": "Eq(tan((a-b)/2), sin((A-B)/2)/sin((A+B)/2)*tan(c/2))", "physics": false, "states": [ "theorem/napier-s-analogy" ], "concepts": [ "concept/polar-triangle", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-600fbc0a24", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 36", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\cos^2\\tfrac{1}{2}c = \\cos^2\\tfrac{1}{2}(a - b) \\cos^2\\tfrac{1}{2}C + \\cos^2\\tfrac{1}{2}(a + b) \\sin^2\\tfrac{1}{2}C", "name": "Delambre's analogy", "statement": "The square of the cosine of half the side c is expressed as a weighted sum of the squares of cosines of half the difference and half the sum of a and b, weighted by the squared cosine and sine of half the angle C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "C", "meaning": "spherical angle at the vertex C" } ], "sympy": "Eq(cos(c/2)**2, cos((a-b)/2)**2*cos(C/2)**2 + cos((a+b)/2)**2*sin(C/2)**2)", "physics": false, "states": [ "theorem/delambre-s-analogy" ], "concepts": [ "concept/cosine", "concept/side", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-a7a71fa57d", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 36", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\cos\\tfrac{1}{2}(A + B) \\cos\\tfrac{1}{2}c = \\cos\\tfrac{1}{2}(a + b) \\sin\\tfrac{1}{2}C", "name": "Delambre's analogy", "statement": "Delambre's relation between the cosines of half the sum of two angles and half the third side, and the cosine of half the sum of the sides and the sine of half the third angle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" }, { "unit": null, "symbol": "B", "meaning": "spherical angle at the vertex B" }, { "unit": null, "symbol": "C", "meaning": "spherical angle at the vertex C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": "Eq(cos((A+B)/2)*cos(c/2), cos((a+b)/2)*sin(C/2))", "physics": false, "states": [ "theorem/delambre-s-analogy" ], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-5d7b6aedba", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 36", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\cos\\tfrac{1}{2}(A - B) \\sin\\tfrac{1}{2}c = \\sin\\tfrac{1}{2}(a + b) \\sin\\tfrac{1}{2}C", "name": "Delambre's analogy", "statement": "Delambre's relation between the cosine of half the difference of two angles times the sine of half the third side, and the sine of half the sum of the two sides times the sine of half the third angle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" }, { "unit": null, "symbol": "B", "meaning": "spherical angle at the vertex B" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "C", "meaning": "spherical angle at the vertex C" } ], "sympy": "Eq(cos((A-B)/2)*sin(c/2), sin((a+b)/2)*sin(C/2))", "physics": false, "states": [ "theorem/delambre-s-analogy" ], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ccb91821f8", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 36", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\sin\\tfrac{1}{2}(A + B) \\cos\\tfrac{1}{2}c = \\cos\\tfrac{1}{2}(a - b) \\cos\\tfrac{1}{2}C", "name": "Gauss's theorem (Delambre's)", "statement": "The sine of half the sum of two angles times the cosine of half the third side equals the cosine of half the difference of the sides times the cosine of half the third angle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" }, { "unit": null, "symbol": "B", "meaning": "spherical angle at the vertex B" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "C", "meaning": "spherical angle at the vertex C" } ], "sympy": "Eq(sin((A+B)/2)*cos(c/2), cos((a-b)/2)*cos(C/2))", "physics": false, "states": [ "theorem/gauss-s-theorem-delambre-s" ], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-2b782b6695", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 36", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\sin\\tfrac{1}{2}(A - B) \\sin\\tfrac{1}{2}c = \\sin\\tfrac{1}{2}(a - b) \\cos\\tfrac{1}{2}C", "name": "Gauss's theorem (Delambre's)", "statement": "The sine of half the difference of two angles times the sine of half the third side equals the sine of half the difference of the sides times the cosine of half the third angle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "spherical angle at the vertex A" }, { "unit": null, "symbol": "B", "meaning": "spherical angle at the vertex B" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "C", "meaning": "spherical angle at the vertex C" } ], "sympy": "Eq(sin((A-B)/2)*sin(c/2), sin((a-b)/2)*cos(C/2))", "physics": false, "states": [ "theorem/gauss-s-theorem-delambre-s" ], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f799456ece", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 40", "location": "Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle", "latex": "\\cos a \\cos b + \\sin a \\sin b \\cos C = \\cos c", "name": null, "statement": "The fundamental cosine relation for the sides, derived by the coordinate method with the origin at the centre of the sphere.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { 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{ "unit": null, "symbol": "B", "meaning": "angle at B" }, { "unit": null, "symbol": "c", "meaning": "hypotenuse" } ], "sympy": "Eq(sin(b), sin(B)*sin(c))", "physics": false, "states": [], "concepts": [ "concept/hypotenuse", "concept/plane-angle", "concept/right-spherical-triangle", "concept/side", "concept/sine" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-b2677c517d", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 47", "location": "Solution of Right-angled Triangles", "latex": "\\sin a = \\sin A \\sin c", "name": null, "statement": "In a right-angled spherical triangle the sine of a side equals the sine of its opposite angle times the sine of the hypotenuse.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side opposite angle A" }, { "unit": null, "symbol": "A", "meaning": "angle at A" }, 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false, "states": [], "concepts": [ "concept/cosine", "concept/cotangent", "concept/hypotenuse", "concept/spherical-triangle", "concept/tangent-function", "method/solving-a-right-spherical-triangle" ], "pages": [ "scan 51", "scan 160" ], "chapters": [ "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-082cf7ae05", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 51", "location": "Solution of Right-angled Triangles", "latex": "\\sin a = \\sin c \\sin A", "name": null, "statement": "Given hypotenuse c and angle A, the sine of side a equals sin c times sin A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side opposite angle A" }, { "unit": null, "symbol": "c", "meaning": "hypotenuse" }, { "unit": null, "symbol": "A", "meaning": "angle at A" } ], "sympy": "Eq(sin(a), sin(c)*sin(A))", "physics": false, "states": [], "concepts": [ "concept/hypotenuse", "concept/side", "concept/sine", "concept/spherical-triangle", "method/solving-a-right-spherical-triangle" ], "pages": [ "scan 51", "scan 160" ], "chapters": [ "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f69419a136", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 51", "location": "Solution of Right-angled Triangles", "latex": "\\tan c=\\dfrac{\\tan b}{\\cos A}", "name": null, "statement": "Given side b and adjacent angle A, the tangent of the hypotenuse equals tan b divided by cos A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c", "meaning": "hypotenuse" }, { "unit": null, "symbol": "b", "meaning": "side opposite angle B" }, { "unit": null, "symbol": "A", "meaning": "angle at A" } ], "sympy": "Eq(tan(c), tan(b)/cos(A))", "physics": false, "states": [], "concepts": [ "concept/adjacent-angles", "concept/cosine", "concept/hypotenuse", "concept/tangent-function", "method/solving-a-right-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f4b2adc053", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 51", "location": "Solution of Right-angled Triangles", "latex": "\\tan a = \\tan A \\sin b", "name": null, "statement": "Given side b and adjacent angle A, the tangent of side a equals tan A times sin b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side opposite angle A" }, { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "b", "meaning": "side opposite angle B" } ], "sympy": "Eq(tan(a), tan(A)*sin(b))", "physics": false, "states": [], "concepts": [ "concept/adjacent-angles", "concept/right-spherical-triangle", "concept/side", "concept/sine", "concept/tangent-function", "method/solving-a-right-spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-22c30c146b", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 47", "location": "Solution of Right-angled Triangles", "latex": "\\cos B = \\cos b \\sin A", "name": null, "statement": "Given side b and adjacent angle A, the cosine of angle B equals cos b times sin A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "B", "meaning": "angle at B" }, { "unit": null, "symbol": "b", "meaning": "side opposite angle B" }, { "unit": null, "symbol": "A", "meaning": "angle at A" } ], "sympy": "Eq(cos(B), cos(b)*sin(A))", "physics": false, "states": [], "concepts": [ "concept/adjacent-angles", "concept/cosine", "concept/sine", "method/solving-a-right-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-3f37ec6aa9", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 47", "location": "Solution of Right-angled Triangles", "latex": "\\cos c = \\cos a \\cos b", "name": null, "statement": "Given two sides a and b, the cosine of the hypotenuse equals cos a times cos b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c", "meaning": "hypotenuse" }, { "unit": null, "symbol": "a", "meaning": "side opposite angle A" }, { "unit": null, "symbol": "b", "meaning": "side opposite angle B" } ], "sympy": "Eq(cos(c), cos(a)*cos(b))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/hypotenuse", "concept/right-spherical-triangle", "concept/side", "concept/spherical-triangle", "method/solving-a-right-spherical-triangle", "quantity/right-angle" ], "pages": [ "scan 47", "scan 159" ], "chapters": [ "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-36de107c6c", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 51", "location": "Solution of Right-angled Triangles", "latex": "\\cot A = \\cot a \\sin b", "name": null, "statement": "Given two sides a and b, the cotangent of angle A equals cot a times sin b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "a", "meaning": "side opposite angle A" }, { "unit": null, "symbol": "b", "meaning": "side opposite angle B" } ], "sympy": "Eq(cot(A), cot(a)*sin(b))", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/side", "concept/sine", "concept/spherical-triangle", "method/solving-a-right-spherical-triangle" ], "pages": [ "scan 51", "scan 159" ], "chapters": [ "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f06e05b729", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 51", "location": "Solution of 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"todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 51", "location": "Solution of Right-angled Triangles", "latex": "\\cos b=\\dfrac{\\cos c}{\\cos a}", "name": null, "statement": "Given hypotenuse c and side a, the cosine of side b equals cos c divided by cos a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "b", "meaning": "side opposite angle B" }, { "unit": null, "symbol": "c", "meaning": "hypotenuse" }, { "unit": null, "symbol": "a", "meaning": "side opposite angle A" } ], "sympy": "Eq(cos(b), cos(c)/cos(a))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/hypotenuse", "concept/side", "method/solving-a-right-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-78cf8bd21e", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 51", "location": "Solution of Right-angled Triangles", "latex": "\\cos B=\\dfrac{\\tan a}{\\tan c}", "name": null, "statement": "Given hypotenuse c and side a, the cosine of angle B equals tan a divided by tan c.", "kind": "result", "symbols": [ { "unit": null, "symbol": "B", "meaning": "angle at B" }, { "unit": null, "symbol": "a", "meaning": "side opposite angle A" }, { "unit": null, "symbol": "c", "meaning": "hypotenuse" } ], "sympy": "Eq(cos(B), tan(a)/tan(c))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/hypotenuse", "concept/tangent-function", "method/solving-a-right-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-c87d0e3888", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 51", "location": "Solution of Right-angled Triangles", "latex": "\\sin A=\\dfrac{\\sin a}{\\sin c}", "name": null, "statement": "Given hypotenuse c and side a, the sine of angle A equals sin a divided by sin c.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "a", "meaning": "side opposite angle A" }, { "unit": null, "symbol": "c", "meaning": "hypotenuse" } ], "sympy": "Eq(sin(A), sin(a)/sin(c))", "physics": false, "states": [], "concepts": [ "concept/hypotenuse", "concept/side", "concept/sine", "method/solving-a-right-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-423b109bc0", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 44", "location": "Solution of Right-angled Triangles", "latex": "\\cos c = \\cot A \\cot B", "name": null, "statement": "Given two angles A and B, the cosine of the hypotenuse equals cot A times cot B.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c", "meaning": "hypotenuse" }, { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "B", "meaning": "angle at B" } ], "sympy": "Eq(cos(c), cot(A)*cot(B))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cotangent", "concept/hypotenuse", "concept/right-spherical-triangle", "method/solving-a-right-spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-a1789ce80c", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 52", "location": "Solution of Right-angled Triangles", "latex": "\\cos a = \\frac{\\cos A}{\\sin B}", "name": null, "statement": "Given two angles A and B, the cosine of side a equals cos A divided by sin B.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side opposite angle A" }, { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "B", "meaning": "angle at B" } ], "sympy": "Eq(cos(a), cos(A)/sin(B))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "method/solving-a-right-spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-8812eb9fda", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 52", "location": "Solution of Right-angled Triangles", "latex": "\\cos b = \\frac{\\cos B}{\\sin A}", "name": null, "statement": "Given two angles A and B, the cosine of side b equals cos B divided by sin A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "b", "meaning": "side opposite angle B" }, { "unit": null, "symbol": "B", "meaning": "angle at B" }, { "unit": null, "symbol": "A", "meaning": "angle at A" } ], "sympy": "Eq(cos(b), cos(B)/sin(A))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "method/solving-a-right-spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-efcaf3aa22", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 52", "location": "Solution of Right-angled Triangles", "latex": "\\sin c = \\dfrac{\\sin a}{\\sin A}", "name": null, "statement": "Given side a and its opposite angle A, the sine of the hypotenuse equals sin a divided by sin A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c", "meaning": "hypotenuse" }, { "unit": null, "symbol": "a", "meaning": "side opposite angle A" }, { "unit": null, "symbol": "A", "meaning": "angle at A" } ], "sympy": "Eq(sin(c), sin(a)/sin(A))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/hypotenuse", "concept/side", "concept/sine", "method/solving-a-right-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-fe81930014", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 52", "location": "Solution of Right-angled Triangles", "latex": "\\sin b = \\tan a\\, \\cot A", "name": null, "statement": "Given side a and its opposite angle A, the sine of side b equals tan a times cot A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "b", "meaning": "side opposite angle B" }, { "unit": null, "symbol": "a", "meaning": "side opposite angle A" }, { "unit": null, "symbol": "A", "meaning": "angle at A" } ], "sympy": "Eq(sin(b), tan(a)*cot(A))", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/side", "concept/sine", "concept/tangent-function", "method/solving-a-right-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-18b885383b", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 52", "location": "Solution of Right-angled Triangles", "latex": "\\sin B = \\dfrac{\\cos A}{\\cos a}", "name": null, "statement": "Given side a and its opposite angle A, the sine of angle B equals cos A divided by cos a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "B", "meaning": "angle at B" }, { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "a", "meaning": "side opposite angle A" } ], "sympy": "Eq(sin(B), cos(A)/cos(a))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/cosine", "concept/side", "concept/sine", "concept/spherical-triangle", "method/solving-a-right-spherical-triangle", "quantity/angle" ], "pages": [ "scan 52", "scan 162" ], "chapters": [ "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-0f31c72245", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 49", "location": "Solution of Right-angled Triangles", "latex": "a_1 + p_1 = a_2 + p_2 = a_5 + p_5 = \\dfrac{\\pi}{2}", "name": null, "statement": "Auxiliary quantities p_1, p_2, p_5 are defined so that each pairs with a_1, a_2, a_5 to make a right angle; this characterises the five allied triangles of Napier's Rules.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a_1", "meaning": "first element of triangle BAC taken from the hypotenuse, omitting the right angle" }, { "unit": null, "symbol": "a_2", "meaning": "second element of triangle BAC in order" }, { "unit": null, "symbol": "a_5", "meaning": "fifth element of triangle BAC in order" }, { "unit": null, "symbol": "p_1", "meaning": "auxiliary quantity characterising the triangle" }, { "unit": null, "symbol": "p_2", "meaning": "auxiliary quantity characterising the triangle" }, { "unit": null, "symbol": "p_5", "meaning": "auxiliary quantity characterising the triangle" } ], "sympy": "Eq(a_1 + p_1, pi/2)", "physics": false, "states": [], "concepts": [ "concept/circular-parts", "concept/right-spherical-triangle", "quantity/right-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-bdfe4a7981", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 49", "location": "Solution of Right-angled Triangles", "latex": "p_3 = a_3", "name": null, "statement": "The auxiliary quantity p_3 is set equal to the element a_3.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "p_3", "meaning": "auxiliary quantity characterising the triangle" }, { "unit": null, "symbol": "a_3", "meaning": "third element of triangle BAC in order" } ], "sympy": "Eq(p_3, a_3)", "physics": false, "states": [], "concepts": [ "concept/circular-parts", "concept/equality" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-990d540bee", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 49", "location": "Solution of Right-angled Triangles", "latex": "p_4 = a_4", "name": null, "statement": "The auxiliary quantity p_4 is set equal to the element a_4.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "p_4", "meaning": "auxiliary quantity characterising the triangle" }, { "unit": null, "symbol": "a_4", "meaning": "fourth element of triangle BAC in order" } ], "sympy": "Eq(p_4, a_4)", "physics": false, "states": [], "concepts": [ "concept/circular-parts", "concept/equality" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-4a22df4026", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 57", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cos a = \\dfrac{\\cos A + \\cos B \\cos C}{\\sin B \\sin C}", "name": null, "statement": "Gives the cosine of a side of a spherical triangle from the three angles.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" }, { "unit": null, "symbol": "C", "meaning": "angle at vertex C" } ], "sympy": "Eq(cos(a), (cos(A) + cos(B)*cos(C))/(sin(B)*sin(C)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/spherical-triangle", "method/solving-an-oblique-spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-05abc0038d", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 57", "location": "Solution of Oblique-Angled Triangles", "latex": "\\tan \\tfrac{1}{2}(A + B) = \\dfrac{\\cos\\tfrac{1}{2}(a - b)}{\\cos\\tfrac{1}{2}(a + b)}\\cot\\tfrac{1}{2}C", "name": "Napier's analogies", "statement": "First of Napier's analogies: half the sum of two angles from two sides and the included angle.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" }, { "unit": null, "symbol": "C", "meaning": "included angle at vertex C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" } ], "sympy": "Eq(tan((A + B)/2), cos((a - b)/2)/cos((a + b)/2)*cot(C/2))", "physics": false, "states": [ "theorem/napier-s-analogies" ], "concepts": [ "concept/cotangent", "concept/side", "concept/tangent-function", "method/solving-an-oblique-spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-33b84efeb1", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 57", "location": "Solution of Oblique-Angled Triangles", "latex": "\\tan \\tfrac{1}{2}(A - B) = \\dfrac{\\sin\\tfrac{1}{2}(a - b)}{\\sin\\tfrac{1}{2}(a + b)}\\cot\\tfrac{1}{2}C", "name": "Napier's analogies", "statement": "Second of Napier's analogies: half the difference of two angles from two sides and the included angle.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" }, { "unit": null, "symbol": "C", "meaning": "included angle at vertex C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" } ], "sympy": "Eq(tan((A - B)/2), sin((a - b)/2)/sin((a + b)/2)*cot(C/2))", "physics": false, "states": [ "theorem/napier-s-analogies" ], "concepts": [ "concept/cotangent", "concept/side", "concept/sine", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-3d19927f42", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 57", "location": "Solution of Oblique-Angled Triangles", "latex": "\\sin c = \\dfrac{\\sin a\\, \\sin C}{\\sin A}", "name": null, "statement": "Sine rule: the sine of side c is proportional to the sine of its opposite angle C.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "C", "meaning": "angle at vertex C" } ], "sympy": "Eq(sin(c), sin(a)*sin(C)/sin(A))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/side", "concept/sine", "concept/spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ffe0de22f5", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 57", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cos c = \\cos a\\, \\cos b + \\sin a\\, \\sin b\\, \\cos C", "name": null, "statement": "Cosine rule for a side of a spherical triangle, from two sides and the included angle; free from ambiguity.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "C", "meaning": "included angle at vertex C" } ], "sympy": "Eq(cos(c), cos(a)*cos(b) + sin(a)*sin(b)*cos(C))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/spherical-triangle", "method/solving-an-oblique-spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-d925511c1c", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 57", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cos c = \\cos b\\, (\\cos a + \\sin a \\tan b \\cos C)", "name": null, "statement": "The cosine rule for c rearranged into a form suited to logarithms.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "C", "meaning": "included angle at vertex C" } ], "sympy": "Eq(cos(c), cos(b)*(cos(a) + sin(a)*tan(b)*cos(C)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/logarithm", "concept/side", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-255143d90f", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 58", "location": "Solution of Oblique-Angled Triangles", "latex": "\\tan \\theta = \\tan b\\, \\cos C", "name": null, "statement": "Auxiliary angle theta defined by tan theta equal to tan b times cos C.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "theta", "meaning": "auxiliary angle introduced for the logarithmic form" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "C", "meaning": "included angle at vertex C" } ], "sympy": "Eq(tan(theta), tan(b)*cos(C))", "physics": false, "states": [], "concepts": [ "concept/logarithm", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-4908843390", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 58", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cos c = \\cos b\\, (\\cos a + \\sin a\\, \\tan \\theta) = \\dfrac{\\cos b\\, \\cos (a - \\theta)}{\\cos \\theta}", "name": null, "statement": "With the auxiliary angle theta, the cosine of side c is written as a single quotient adapted to logarithms.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "theta", "meaning": "auxiliary angle with tan theta = tan b cos C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" } ], "sympy": "Eq(cos(c), cos(b)*cos(a - theta)/cos(theta))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/logarithm", "concept/side", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f02f3d8708", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 58", "location": "Solution of Oblique-Angled Triangles", "latex": "\\tan CD = \\tan b \\cos C", "name": null, "statement": "In the right-angled decomposition, the tangent of the segment CD equals tan b times cos C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "CD", "meaning": "arc from C to the foot D of the perpendicular from A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "C", "meaning": "angle at vertex C" } ], "sympy": "Eq(tan(CD), tan(b)*cos(C))", "physics": false, "states": [], "concepts": [ "concept/perpendicular", "concept/right-spherical-triangle", "concept/tangent-function", "method/solving-an-oblique-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-0d82b9577d", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 58", "location": "Solution of Oblique-Angled Triangles", "latex": "\\tan AD = \\tan C \\sin CD", "name": null, "statement": "Right-angled relation giving the tangent of the perpendicular AD from the angle C and the segment CD.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AD", "meaning": "perpendicular arc from A to CB" }, { "unit": null, "symbol": "C", "meaning": "angle at vertex C" }, { "unit": null, "symbol": "CD", "meaning": "arc from C to D" } ], "sympy": "Eq(tan(AD), tan(C)*sin(CD))", "physics": false, "states": [], "concepts": [ "concept/perpendicular", "concept/right-spherical-triangle", "concept/sine", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-07bf8dd39e", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 58", "location": "Solution of Oblique-Angled Triangles", "latex": "\\tan ABD \\sin DB = \\tan C \\sin \\theta", "name": null, "statement": "Relation that finds angle B independently of A, from the right-angled decomposition.", "kind": "result", "symbols": [ { "unit": null, "symbol": "ABD", "meaning": "angle B or its supplement" }, { "unit": null, "symbol": "DB", "meaning": "arc from D to B" }, { "unit": null, "symbol": "C", "meaning": "angle at vertex C" }, { "unit": null, "symbol": "theta", "meaning": "auxiliary angle tan theta = tan b cos C" } ], "sympy": "Eq(tan(ABD)*sin(DB), tan(C)*sin(theta))", "physics": false, "states": [], "concepts": [ "concept/sine", "concept/tangent-function", "method/solving-an-oblique-spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-5d2551fa89", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 59", "location": "Solution of Oblique-Angled Triangles", "latex": "\\tan \\tfrac{1}{2} (a + b) = \\dfrac{\\cos \\tfrac{1}{2} (A - B)}{\\cos \\tfrac{1}{2} (A + B)} \\tan \\tfrac{1}{2} c", "name": "Napier's analogies", "statement": "Napier's analogy giving half the sum of two sides from two angles and the included side.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "included side" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" } ], "sympy": "Eq(tan((a + b)/2), cos((A - B)/2)/cos((A + B)/2)*tan(c/2))", "physics": false, "states": [ "theorem/napier-s-analogies" ], "concepts": [ "concept/side", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-a0542d9004", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 59", "location": "Solution of Oblique-Angled Triangles", "latex": "\\tan \\tfrac{1}{2} (a - b) = \\dfrac{\\sin \\tfrac{1}{2} (A - B)}{\\sin \\tfrac{1}{2} (A + B)} \\tan \\tfrac{1}{2} c", "name": "Napier's analogies", "statement": "Napier's analogy giving half the difference of two sides from two angles and the included side.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "included side" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" } ], "sympy": "Eq(tan((a - b)/2), sin((A - B)/2)/sin((A + B)/2)*tan(c/2))", "physics": false, "states": [ "theorem/napier-s-analogies" ], "concepts": [ "concept/side", "concept/sine", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-005a749ed8", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 59", "location": "Solution of Oblique-Angled Triangles", "latex": "\\sin C = \\dfrac{\\sin A \\sin c}{\\sin a}", "name": null, "statement": "Sine rule giving the angle C from its sine, with the usual ambiguity.", "kind": "law", "symbols": [ { "unit": null, "symbol": "C", "meaning": "angle at vertex C" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" } ], "sympy": "Eq(sin(C), sin(A)*sin(c)/sin(a))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/side", "concept/sine", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-0d2af6669e", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 59", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cos C = -\\cos A \\cos B + \\sin A \\sin B \\cos c", "name": null, "statement": "Cosine rule for an angle of a spherical triangle from two angles and the included side; free from ambiguity.", "kind": "law", "symbols": [ { "unit": null, "symbol": "C", "meaning": "angle at vertex C" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": "Eq(cos(C), -cos(A)*cos(B) + sin(A)*sin(B)*cos(c))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-9c59fb0e7c", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 59", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cot \\phi = \\tan B\\, \\cos c", "name": null, "statement": "Auxiliary angle phi defined by cot phi equal to tan B times cos c.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "auxiliary angle for the logarithmic form" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": "Eq(cot(phi), tan(B)*cos(c))", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/logarithm", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-6042a75488", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 59", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cos C = \\cos B (-\\cos A + \\cot \\phi \\sin A) = \\dfrac{\\cos B \\sin (A-\\phi)}{\\sin \\phi}", "name": null, "statement": "The cosine rule for C written with the auxiliary angle phi, adapted to logarithms.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "C", "meaning": "angle at vertex C" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" }, { "unit": null, "symbol": "phi", "meaning": "auxiliary angle with cot phi = tan B cos c" } ], "sympy": "Eq(cos(C), cos(B)*sin(A - phi)/sin(phi))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cotangent", "concept/logarithm", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-fd242a7122", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 59", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cos c = \\cot B \\cot DAB", "name": null, "statement": "Right-angled relation giving cos c from the angle B and the angle DAB.", "kind": "result", "symbols": [ { "unit": null, "symbol": "DAB", "meaning": "angle at A between AD and AB" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": "Eq(cos(c), cot(B)*cot(DAB))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cotangent", "concept/right-spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-fda1c375c3", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 59", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cos AD \\sin CAD = \\cos C", "name": null, "statement": "Right-angled relation between the perpendicular AD, the angle CAD and the angle C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AD", "meaning": "perpendicular arc from A" }, { "unit": null, "symbol": "CAD", "meaning": "angle at A between AC and AD" }, { "unit": null, "symbol": "C", "meaning": "angle at vertex C" } ], "sympy": "Eq(cos(AD)*sin(CAD), cos(C))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/perpendicular", "concept/right-spherical-triangle", "concept/sine" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-7d137a01ab", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 59", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cos AD \\sin BAD = \\cos B", "name": null, "statement": "Right-angled relation between the perpendicular AD, the angle BAD and the angle B.", "kind": "result", "symbols": [ { "unit": null, "symbol": "AD", "meaning": "perpendicular arc from A" }, { "unit": null, "symbol": "BAD", "meaning": "angle at A between AB and AD" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" } ], "sympy": "Eq(cos(AD)*sin(BAD), cos(B))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/perpendicular", "concept/right-spherical-triangle", "concept/sine" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-fe7db1381d", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 59", "location": "Solution of Oblique-Angled Triangles", "latex": "\\dfrac{\\cos C}{\\sin CAD} = \\dfrac{\\cos B}{\\sin BAD}", "name": null, "statement": "Ratio relation from which the angle C is found in the perpendicular decomposition.", "kind": "result", "symbols": [ { "unit": null, "symbol": "C", "meaning": "angle at vertex C" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" }, { "unit": null, "symbol": "CAD", "meaning": "angle at A between AC and AD" }, { "unit": null, "symbol": "BAD", "meaning": "angle at A between AB and AD" } ], "sympy": "Eq(cos(C)/sin(CAD), cos(B)/sin(BAD))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/sine", "method/solving-an-oblique-spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-7f5190e61a", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 60", "location": "Solution of Oblique-Angled Triangles", "latex": "\\tan b \\cos CAD = \\tan c \\cos \\phi", "name": null, "statement": "Relation that finds side b independently of a, in the perpendicular decomposition.", "kind": "result", "symbols": [ { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "CAD", "meaning": "angle A minus phi" }, { "unit": null, "symbol": "phi", "meaning": "auxiliary angle DAB" } ], "sympy": "Eq(tan(b)*cos(CAD), tan(c)*cos(phi))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/tangent-function", "method/solving-an-oblique-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-138053619c", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 60", "location": "Solution of Oblique-Angled Triangles", "latex": "\\sin B = \\frac{\\sin b}{\\sin a} \\sin A", "name": null, "statement": "Sine rule giving angle B from its sine; it may take two values, so the solution is ambiguous.", "kind": "law", "symbols": [ { "unit": null, "symbol": "B", "meaning": "angle at vertex B" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" } ], "sympy": "Eq(sin(B), sin(b)/sin(a)*sin(A))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/side", "concept/sine", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-d098ea636b", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 60", "location": "Solution of Oblique-Angled Triangles", "latex": "\\tan \\tfrac{1}{2} C = \\dfrac{\\cos \\tfrac{1}{2} (a - b)}{\\cos \\tfrac{1}{2} (a + b)} \\cot \\tfrac{1}{2} (A + B)", "name": "Napier's analogies", "statement": "Napier's analogy giving half of angle C from two sides, one opposite angle, and the sum of the two angles.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "C", "meaning": "angle at vertex C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" } ], "sympy": "Eq(tan(C/2), cos((a - b)/2)/cos((a + b)/2)*cot((A + B)/2))", "physics": false, "states": [ "theorem/napier-s-analogies" ], "concepts": [ "concept/cotangent", "concept/side", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-3350ce60f9", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 60", "location": "Solution of Oblique-Angled Triangles", "latex": "\\tan \\tfrac{1}{2} c = \\dfrac{\\cos \\tfrac{1}{2} (A + B)}{\\cos \\tfrac{1}{2} (A - B)} \\tan \\tfrac{1}{2} (a + b)", "name": "Napier's analogies", "statement": "Napier's analogy giving half of side c from two sides, one opposite angle, and the sum of the two angles.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" } ], "sympy": "Eq(tan(c/2), cos((A + B)/2)/cos((A - B)/2)*tan((a + b)/2))", "physics": false, "states": [ "theorem/napier-s-analogies" ], "concepts": [ "concept/side", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-b5bdefe289", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 60", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cot a\\, \\sin b = \\cos b\\, \\cos C + \\sin C\\, \\cot A", "name": null, "statement": "Four-part (cotangent) relation linking two sides, the included angle C and the angle A.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "C", "meaning": "angle at vertex C" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" } ], "sympy": "Eq(cot(a)*sin(b), cos(b)*cos(C) + sin(C)*cot(A))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cotangent", "concept/side", "concept/sine", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-534695d56e", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 60", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cos (C - \\phi) = \\cos \\phi \\cot a \\tan b", "name": null, "statement": "Equation determining C minus phi, from which C is found; it may have two admissible values.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "C", "meaning": "angle at vertex C" }, { "unit": null, "symbol": "phi", "meaning": "auxiliary angle with tan phi = cot A / cos b" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" } ], "sympy": "Eq(cos(C - phi), cos(phi)*cot(a)*tan(b))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/cosine", "concept/cotangent", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-faf7877fe3", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 61", "location": "Solution of Oblique-Angled Triangles", "latex": "\\tan\\theta = \\tan b \\cos A", "name": null, "statement": "Auxiliary angle theta defined by tan theta equal to tan b times cos A.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "theta", "meaning": "auxiliary angle for the logarithmic form" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" } ], "sympy": "Eq(tan(theta), tan(b)*cos(A))", "physics": false, "states": [], "concepts": [ "concept/logarithm", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-701895c36e", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 61", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cos(c - \\theta) = \\dfrac{\\cos a \\cos\\theta}{\\cos b}", "name": null, "statement": "Equation determining c minus theta, from which side c is found; may be ambiguous.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "theta", "meaning": "auxiliary angle with tan theta = tan b cos A" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" } ], "sympy": "Eq(cos(c - theta), cos(a)*cos(theta)/cos(b))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/cosine", "concept/logarithm", "concept/side" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f07fb02d95", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 61", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cos b = \\cot A \\cot ACD", "name": null, "statement": "Right-angled relation giving cos b from the angle A and the angle ACD.", "kind": "result", "symbols": [ { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "ACD", "meaning": "angle at C between CA and CD" } ], "sympy": "Eq(cos(b), cot(A)*cot(ACD))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cotangent", "concept/right-spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-5f9a147c3f", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 62", "location": "Solution of Oblique-Angled Triangles", "latex": "\\sin b = \\dfrac{\\sin B \\sin a}{\\sin A}", "name": null, "statement": "Sine rule giving side b from its sine; the solution is ambiguous.", "kind": "law", "symbols": [ { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" } ], "sympy": "Eq(sin(b), sin(B)*sin(a)/sin(A))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/side", "concept/sine", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-65524eacc4", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 62", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cos A= -\\cos B\\cos C + \\sin B\\sin C\\cos a", "name": null, "statement": "Cosine rule for angle A from two angles and the included side a.", "kind": "law", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" }, { "unit": null, "symbol": "C", "meaning": "angle at vertex C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" } ], "sympy": "Eq(cos(A), -cos(B)*cos(C) + sin(B)*sin(C)*cos(a))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-d7f2b836fe", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 62", "location": "Solution of Oblique-Angled Triangles", "latex": "\\sin(C - \\phi)=\\dfrac{\\cos A \\sin \\phi}{\\cos B}", "name": null, "statement": "Equation determining C minus phi from its sine; may be ambiguous.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "C", "meaning": "angle at vertex C" }, { "unit": null, "symbol": "phi", "meaning": "auxiliary angle with cot phi = tan B cos a" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" } ], "sympy": "Eq(sin(C - phi), cos(A)*sin(phi)/cos(B))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/cosine", "concept/sine", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-492b807b28", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 63", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cot A \\sin B = \\cot a \\sin c - \\cos c \\cos B", "name": null, "statement": "Relation between the cotangent of A, the sine of B and sides a and c, used to find side c.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": "Eq(cot(A)*sin(B), cot(a)*sin(c) - cos(c)*cos(B))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cotangent", "concept/side", "concept/sine", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-2041131c9f", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 63", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cot \\theta = \\dfrac{\\cot a}{\\cos B}", "name": null, "statement": "Auxiliary angle theta defined by cot theta equal to cot a divided by cos B.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "theta", "meaning": "auxiliary angle for the logarithmic form" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" } ], "sympy": "Eq(cot(theta), cot(a)/cos(B))", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/logarithm", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-4568a482ee", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 63", "location": "Solution of Oblique-Angled Triangles", "latex": "\\sin (c - \\theta) = \\cot A \\tan B \\sin \\theta", "name": null, "statement": "Equation determining c minus theta from its sine; may be ambiguous.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "theta", "meaning": "auxiliary angle with cot theta = cot a / cos B" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" } ], "sympy": "Eq(sin(c - theta), cot(A)*tan(B)*sin(theta))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/cotangent", "concept/side", "concept/sine", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-d04c5b16d7", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 63", "location": "Solution of Oblique-Angled Triangles", "latex": "\\cot \\tfrac{1}{2}C = \\tan A \\cos a", "name": null, "statement": "In the special case a = b (so A = B), half of cot C is tan A times cos a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "C", "meaning": "angle at vertex C" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A, equal to B in this case" }, { "unit": null, "symbol": "a", "meaning": "side opposite A, equal to b" } ], "sympy": "Eq(cot(C/2), tan(A)*cos(a))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/cotangent", "concept/isosceles-triangle", "concept/same-affection", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-c44f373df0", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 63", "location": "Solution of Oblique-Angled Triangles", "latex": "\\tan \\tfrac{1}{2}c = \\tan a \\cos A", "name": null, "statement": "In the special case a = b (so A = B), half of tan c is tan a times cos A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c", "meaning": "side opposite C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A, equal to b" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" } ], "sympy": "Eq(tan(c/2), tan(a)*cos(A))", "physics": false, "states": [], "concepts": [ "concept/isosceles-triangle", "concept/same-affection", "concept/side", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-7a48453b8b", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 64", "location": "Solution of Oblique-Angled Triangles", "latex": "\\sin B = \\dfrac{\\sin b \\sin A}{\\sin a}", "name": null, "statement": "Sine rule giving the two values of B, beta and beta', when two sides and the angle opposite one are given.", "kind": "law", "symbols": [ { "unit": null, "symbol": "B", "meaning": "angle at vertex B" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" } ], "sympy": "Eq(sin(B), sin(b)*sin(A)/sin(a))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/sine", "concept/solution", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-1f53d9bd64", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 64", "location": "Solution of Oblique-Angled Triangles", "latex": "\\beta' = \\pi - \\beta", "name": null, "statement": "The two values of B from the sine equation are supplementary: beta' is pi minus beta.", "kind": "result", "symbols": [ { "unit": "radian", "symbol": "beta", "meaning": "the smaller of the two values of B" }, { "unit": "radian", "symbol": "beta'", "meaning": "the other value of B" } ], "sympy": "Eq(betap, pi - beta)", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/supplementary-angles", "quantity/angle", "quantity/pi" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-8d3e533101", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 69", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan r = \\tan\\dfrac{A}{2} \\sin (s-a)", "name": null, "statement": "The tangent of the angular radius of the small circle inscribed in a spherical triangle equals tan(A/2) times sin(s−a).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r", "meaning": "angular radius of the small circle inscribed in the triangle ABC" }, { "unit": null, "symbol": "A", "meaning": "angle of the triangle at A" }, { "unit": null, "symbol": "a", "meaning": "side BC of the spherical triangle" }, { "unit": null, "symbol": "s", "meaning": "half the sum of the sides, ½(a+b+c)" } ], "sympy": "Eq(tan(r), tan(A/2)*sin(s - a))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/method-finding-the-angular-radius-of-the-inscribed-circle-of-a-spherical-triangle", "concept/side", "concept/sine", "concept/small-circle", "concept/spherical-triangle", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-fb84e284f3", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 70", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan \\dfrac{A}{2} = \\Surd {\\frac{\\sin (s - b) \\sin (s - c)}{\\sin s\\, \\sin (s - a)} }", "name": null, "statement": "The tangent of half an angle of a spherical triangle is the square root of sin(s−b)sin(s−c) divided by sin s sin(s−a), quoted from Article 45.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle of the triangle at A" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" }, { "unit": null, "symbol": "s", "meaning": "half the sum of the sides" } ], "sympy": "Eq(tan(A/2), sqrt(sin(s - b)*sin(s - c)/(sin(s)*sin(s - a))))", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "concept/spherical-triangle", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-04fc2dc9b4", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 70", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan r = \\Surd{\\left\\{ \\dfrac{\\sin (s - a) \\sin (s - b) \\sin (s - c)}{\\sin s} \\right\\}} = \\dfrac{n}{\\sin s}", "name": null, "statement": "The tangent of the inscribed circle's angular radius equals the square root of sin(s−a)sin(s−b)sin(s−c)/sin s, which equals n/sin s.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r", "meaning": "angular radius of the inscribed small circle" }, { "unit": null, "symbol": "n", "meaning": "book's abbreviation: the square root of sin s sin(s−a) sin(s−b) sin(s−c)" }, { "unit": null, "symbol": "s", "meaning": "half the sum of the sides" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(tan(r), sqrt(sin(s - a)*sin(s - b)*sin(s - c)/sin(s)))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/side", "concept/sine", "concept/small-circle", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-62601d0b83", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 70", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan r = \\dfrac{\\sin\\tfrac{1}{2}B \\sin \\tfrac{1}{2}C}{\\cos \\tfrac{1}{2}A} \\sin a", "name": null, "statement": "The tangent of the inscribed circle's angular radius equals sin(B/2)sin(C/2)/cos(A/2) times sin a.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r", "meaning": "angular radius of the inscribed small circle" }, { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "B", "meaning": "angle at B" }, { "unit": null, "symbol": "C", "meaning": "angle at C" }, { "unit": null, "symbol": "a", "meaning": "side BC" } ], "sympy": "Eq(tan(r), sin(B/2)*sin(C/2)/cos(A/2)*sin(a))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/sine", "concept/small-circle", "concept/spherical-triangle", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-768fc7d450", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 70", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan r = \\dfrac{\\surd\\{-\\cos S \\cos (S - A) \\cos (S - B) \\cos (S - C)\\}}{2 \\cos \\tfrac{1}{2}A \\cos \\tfrac{1}{2}B \\cos \\tfrac{1}{2}C}", "name": null, "statement": "The tangent of the inscribed circle's angular radius equals the square root of −cos S cos(S−A)cos(S−B)cos(S−C), divided by 2 cos(A/2)cos(B/2)cos(C/2); the book sets this equal to N over the same denominator.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r", "meaning": "angular radius of the inscribed small circle" }, { "unit": null, "symbol": "S", "meaning": "half the sum of the angles, ½(A+B+C)" }, { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "B", "meaning": "angle at B" }, { "unit": null, "symbol": "C", "meaning": "angle at C" } ], "sympy": "Eq(tan(r), sqrt(-cos(S)*cos(S - A)*cos(S - B)*cos(S - C))/(2*cos(A/2)*cos(B/2)*cos(C/2)))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/cosine", "concept/small-circle", "concept/spherical-triangle", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f0f1cf0ad3", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 70", "location": "Circumscribed and Inscribed Circles", "latex": "4 \\cos\\tfrac{1}{2}A \\cos\\tfrac{1}{2}B \\cos\\tfrac{1}{2}C = \\cos S + \\cos (S - A) + \\cos (S - B) + \\cos (S - C)", "name": null, "statement": "Four times the product of the cosines of half the angles equals the sum of cos S and the three cos(S−A), cos(S−B), cos(S−C).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "B", "meaning": "angle at B" }, { "unit": null, "symbol": "C", "meaning": "angle at C" }, { "unit": null, "symbol": "S", "meaning": "half the sum of the angles, ½(A+B+C)" } ], "sympy": "Eq(4*cos(A/2)*cos(B/2)*cos(C/2), cos(S) + cos(S - A) + cos(S - B) + cos(S - C))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-73e871dd4c", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 70", "location": "Circumscribed and Inscribed Circles", "latex": "\\cot r = \\frac{1}{2N} \\bigl\\{\\cos S + \\cos (S - A) + \\cos (S - B) + \\cos (S - C)\\bigr\\}", "name": null, "statement": "The cotangent of the inscribed circle's angular radius equals the sum cos S + cos(S−A) + cos(S−B) + cos(S−C) divided by 2N.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "angular radius of the inscribed small circle" }, { "unit": null, "symbol": "N", "meaning": "book's abbreviation: the square root of −cos S cos(S−A) cos(S−B) cos(S−C)" }, { "unit": null, "symbol": "S", "meaning": "half the sum of the angles" }, { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "B", "meaning": "angle at B" }, { "unit": null, "symbol": "C", "meaning": "angle at C" } ], "sympy": "Eq(cot(r), (cos(S) + cos(S - A) + cos(S - B) + cos(S - C))/(2*N))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/cosine", "concept/cotangent", "concept/small-circle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-569f2ed9e5", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 71", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan r_1 = \\tan\\dfrac{A}{2}\\sin s", "name": null, "statement": "For the small circle touching BC and the other two sides produced, tan r_1 equals tan(A/2) times sin s.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r_1", "meaning": "angular radius of the small circle touching side BC and AB, AC produced" }, { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "s", "meaning": "half the sum of the sides" } ], "sympy": "Eq(tan(r_1), tan(A/2)*sin(s))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/escribed-circle", "concept/sine", "concept/small-circle", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-da7876b4b3", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 71", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan r_1 = \\Surd{\\left\\{\\dfrac{\\sin s \\sin(s-b)\\sin(s-c)} {\\sin(s-a)}\\right\\}} = \\dfrac{n}{\\sin(s-a)}", "name": null, "statement": "For the escribed circle touching BC, tan r_1 equals the square root of sin s sin(s−b)sin(s−c)/sin(s−a), which equals n/sin(s−a).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r_1", "meaning": "angular radius of the escribed small circle touching BC" }, { "unit": null, "symbol": "n", "meaning": "book's abbreviation: the square root of sin s sin(s−a) sin(s−b) sin(s−c)" }, { "unit": null, "symbol": "s", "meaning": "half the sum of the sides" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(tan(r_1), sqrt(sin(s)*sin(s - b)*sin(s - c)/sin(s - a)))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/escribed-circle", "concept/side", "concept/sine", "concept/small-circle", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-916e3ae166", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 71", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan r_1 = \\dfrac{\\cos\\tfrac{1}{2}B \\cos\\tfrac{1}{2}C} {\\cos\\tfrac{1}{2}A} \\sin a", "name": null, "statement": "For the escribed circle touching BC, tan r_1 equals cos(B/2)cos(C/2)/cos(A/2) times sin a.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r_1", "meaning": "angular radius of the escribed small circle touching BC" }, { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "B", "meaning": "angle at B" }, { "unit": null, "symbol": "C", "meaning": "angle at C" }, { "unit": null, "symbol": "a", "meaning": "side BC" } ], "sympy": "Eq(tan(r_1), cos(B/2)*cos(C/2)/cos(A/2)*sin(a))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/cosine", "concept/escribed-circle", "concept/sine", "concept/small-circle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-9153800a52", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 71", "location": "Circumscribed and Inscribed Circles", "latex": "\\cot r_1 = \\dfrac{1}{2N} \\{-c\\cos S - \\cos(S-A) + \\cos(S-B) + \\cos(S-C) \\}", "name": null, "statement": "The cotangent of the escribed circle's angular radius r_1 equals [−c cos S − cos(S−A) + cos(S−B) + cos(S−C)] divided by 2N, as printed; the leading term reads '−c cos S', which looks like a misprint for −cos S and is FLAGGED, not corrected, pending check against the source.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r_1", "meaning": "angular radius of the escribed small circle touching BC" }, { "unit": null, "symbol": "N", "meaning": "book's abbreviation: the square root of −cos S cos(S−A) cos(S−B) cos(S−C)" }, { "unit": null, "symbol": "c", "meaning": "side AB as printed here (the printed leading term '−c cos S' is flagged as a possible misprint; here c is used as printed, not as the side)" }, { "unit": null, "symbol": "S", "meaning": "half the sum of the angles" }, { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "B", "meaning": "angle at B" }, { "unit": null, "symbol": "C", "meaning": "angle at C" } ], "sympy": "Eq(cot(r_1), (-c*cos(S) - cos(S - A) + cos(S - B) + cos(S - C))/(2*N))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/cosine", "concept/cotangent", "concept/escribed-circle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-d91254e1dc", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 73", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan R = \\dfrac{\\tan\\tfrac{1}{2}a}{\\cos(S-A)}", "name": null, "statement": "The tangent of the angular radius R of the small circle described about the triangle equals tan(a/2) divided by cos(S−A).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "R", "meaning": "angular radius of the small circle described about the triangle ABC (PC in the figure)" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "S", "meaning": "half the sum of the angles" } ], "sympy": "Eq(tan(R), tan(a/2)/cos(S - A))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/cosine", "concept/side", "concept/small-circle", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-74a8f7d01c", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 73", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan R = \\Surd{\\left\\{\\dfrac{-\\cos S}{\\cos(S-A)\\cos(S-B)\\cos(S-C)}\\right\\}} = \\dfrac{\\cos S}{N}", "name": null, "statement": "The tangent of the circumscribed small circle's angular radius R equals the square root of −cos S divided by cos(S−A)cos(S−B)cos(S−C), which equals cos S / N.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "R", "meaning": "angular radius of the small circle described about ABC" }, { "unit": null, "symbol": "S", "meaning": "half the sum of the angles" }, { "unit": null, "symbol": "N", "meaning": "book's abbreviation: the square root of −cos S cos(S−A) cos(S−B) cos(S−C)" } ], "sympy": "Eq(tan(R), sqrt(-cos(S)/(cos(S - A)*cos(S - B)*cos(S - C))))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/cosine", "concept/small-circle", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-1db9267540", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 73", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan R = \\dfrac{\\sin\\tfrac{1}{2}a } {\\sin A\\cos\\tfrac{1}{2}b\\cos\\tfrac{1}{2}c }", "name": null, "statement": "The tangent of the circumscribed small circle's angular radius R equals sin(a/2) divided by sin A cos(b/2) cos(c/2).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "R", "meaning": "angular radius of the small circle described about ABC" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" }, { "unit": null, "symbol": "A", "meaning": "angle at A" } ], "sympy": "Eq(tan(R), sin(a/2)/(sin(A)*cos(b/2)*cos(c/2)))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/cosine", "concept/sine", "concept/small-circle", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-032e2247dc", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 73", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan R = \\dfrac{2\\sin\\tfrac{1}{2}a \\sin\\tfrac{1}{2}b \\sin\\tfrac{1}{2}c} {\\surd{\\left\\{\\sin s\\sin(s-a) \\sin(s-b) \\sin(s-c) \\right\\}}}", "name": null, "statement": "The tangent of the circumscribed small circle's angular radius R equals 2 sin(a/2)sin(b/2)sin(c/2) divided by the square root of sin s sin(s−a)sin(s−b)sin(s−c), which is 2 sin(a/2)sin(b/2)sin(c/2)/n.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "R", "meaning": "angular radius of the small circle described about ABC" }, { "unit": null, "symbol": "n", "meaning": "book's abbreviation: the square root of sin s sin(s−a) sin(s−b) sin(s−c)" }, { "unit": null, "symbol": "s", "meaning": "half the sum of the sides" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(tan(R), 2*sin(a/2)*sin(b/2)*sin(c/2)/sqrt(sin(s)*sin(s - a)*sin(s - b)*sin(s - c)))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/side", "concept/sine", "concept/small-circle", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-1575f54e6c", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 74", "location": "Circumscribed and Inscribed Circles", "latex": "4\\sin\\tfrac{1}{2}a\\sin\\tfrac{1}{2}b\\sin\\tfrac{1}{2}c = \\sin(s-a) + \\sin(s-b) + \\sin(s-c)-\\sin s", "name": null, "statement": "Four times the product of the sines of half the sides equals sin(s−a)+sin(s−b)+sin(s−c)−sin s.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" }, { "unit": null, "symbol": "s", "meaning": "half the sum of the sides" } ], "sympy": "Eq(4*sin(a/2)*sin(b/2)*sin(c/2), sin(s - a) + sin(s - b) + sin(s - c) - sin(s))", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-3a8158aafa", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 74", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan R=\\dfrac{1}{2n}\\{ \\sin(s-a)+\\sin(s-b)+\\sin(s-c)-\\sin s \\}", "name": null, "statement": "The tangent of the circumscribed small circle's angular radius R equals [sin(s−a)+sin(s−b)+sin(s−c)−sin s] divided by 2n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R", "meaning": "angular radius of the small circle described about ABC" }, { "unit": null, "symbol": "n", "meaning": "book's abbreviation: the square root of sin s sin(s−a) sin(s−b) sin(s−c)" }, { "unit": null, "symbol": "s", "meaning": "half the sum of the sides" } ], "sympy": "Eq(tan(R), (sin(s - a) + sin(s - b) + sin(s - c) - sin(s))/(2*n))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/sine", "concept/small-circle", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-a3c3848cce", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 74", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan R_1 = \\frac{\\tan\\frac{1}{2}a}{-\\cos S}", "name": null, "statement": "For the circle described about the associated triangle A'BC, tan R_1 equals tan(a/2) divided by −cos S.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "R_1", "meaning": "angular radius of the small circle described about the associated triangle A'BC" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "S", "meaning": "half the sum of the angles of ABC" } ], "sympy": "Eq(tan(R_1), tan(a/2)/(-cos(S)))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/associated-triangles", "concept/cosine", "concept/small-circle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-69a917517f", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 74", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan R_1 = \\Surd{\\left\\{ \\frac{\\cos(S-A)}{-\\cos S\\cos(S-B)\\cos(S-C)}\\right\\}} = \\frac{\\cos(S-A)}{N}", "name": null, "statement": "For the circle described about the associated triangle A'BC, tan R_1 equals the square root of cos(S−A) divided by −cos S cos(S−B)cos(S−C), which equals cos(S−A)/N.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "R_1", "meaning": "angular radius of the small circle described about A'BC" }, { "unit": null, "symbol": "N", "meaning": "book's abbreviation: the square root of −cos S cos(S−A) cos(S−B) cos(S−C)" }, { "unit": null, "symbol": "S", "meaning": "half the sum of the angles of ABC" }, { "unit": null, "symbol": "A", "meaning": "angle at A" }, { "unit": null, "symbol": "B", "meaning": "angle at B" }, { "unit": null, "symbol": "C", "meaning": "angle at C" } ], "sympy": "Eq(tan(R_1), sqrt(cos(S - A)/(-cos(S)*cos(S - B)*cos(S - C))))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/associated-triangles", "concept/cosine", "concept/small-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-d9ce4adeb9", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 74", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan R_1 = \\frac{\\sin\\frac{1}{2}a} {\\sin A\\sin\\frac{1}{2}b\\sin\\frac{1}{2}c}", "name": null, "statement": "For the circle described about the associated triangle A'BC, tan R_1 equals sin(a/2) divided by sin A sin(b/2) sin(c/2).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "R_1", "meaning": "angular radius of the small circle described about A'BC" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" }, { "unit": null, "symbol": "A", "meaning": "angle at A" } ], "sympy": "Eq(tan(R_1), sin(a/2)/(sin(A)*sin(b/2)*sin(c/2)))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/associated-triangles", "concept/sine", "concept/small-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-10f8472e19", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 74", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan R_1 = \\frac{2\\sin\\frac{1}{2}a\\cos\\frac{1}{2}b\\cos\\frac{1}{2}c} {\\surd{\\{\\sin s\\sin(s-a)\\sin(s-b)\\sin(s-c)\\}}}", "name": null, "statement": "For the circle described about the associated triangle A'BC, tan R_1 equals 2 sin(a/2)cos(b/2)cos(c/2) divided by the square root of sin s sin(s−a)sin(s−b)sin(s−c).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "R_1", "meaning": "angular radius of the small circle described about A'BC" }, { "unit": null, "symbol": "s", "meaning": "half the sum of the sides" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(tan(R_1), 2*sin(a/2)*cos(b/2)*cos(c/2)/sqrt(sin(s)*sin(s - a)*sin(s - b)*sin(s - c)))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/associated-triangles", "concept/cosine", "concept/sine", "concept/small-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f893970be1", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 74", "location": "Circumscribed and Inscribed Circles", "latex": "\\tan R_1 =\\frac{1}{2n} \\{\\sin s - \\sin(s-a) + \\sin(s-b) + \\sin(s-c) \\}", "name": null, "statement": "For the circle described about the associated triangle A'BC, tan R_1 equals [sin s − sin(s−a) + sin(s−b) + sin(s−c)] divided by 2n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R_1", "meaning": "angular radius of the small circle described about A'BC" }, { "unit": null, "symbol": "n", "meaning": "book's abbreviation: the square root of sin s sin(s−a) sin(s−b) sin(s−c)" }, { "unit": null, "symbol": "s", "meaning": "half the sum of the sides" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(tan(R_1), (sin(s) - sin(s - a) + sin(s - b) + sin(s - c))/(2*n))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/associated-triangles", "concept/sine", "concept/small-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-1586a4479e", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 75", "location": "Circumscribed and Inscribed Circles", "latex": "(\\cot r + \\tan R)^2=\\dfrac{1}{4n^2}(\\sin a+\\sin b+\\sin c)^2 -1", "name": null, "statement": "The square of cot r plus tan R equals (sin a + sin b + sin c)² divided by 4n², minus 1 (Article 94 result).", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "angular radius of the inscribed small circle" }, { "unit": null, "symbol": "R", "meaning": "angular radius of the circumscribed small circle" }, { "unit": null, "symbol": "n", "meaning": "book's abbreviation: the square root of sin s sin(s−a) sin(s−b) sin(s−c)" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq((cot(r) + tan(R))**2, (sin(a) + sin(b) + sin(c))**2/(4*n**2) - 1)", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/cotangent", "concept/sine", "concept/small-circle", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-8e0b5f1392", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 75", "location": "Circumscribed and Inscribed Circles", "latex": "(\\cot r_1-\\tan R)^2=\\dfrac{1}{4n^2}(\\sin b+\\sin c-\\sin a)^2 -1", "name": null, "statement": "The square of cot r_1 minus tan R equals (sin b + sin c − sin a)² divided by 4n², minus 1 (Article 94, stated as 'similarly').", "kind": "result", "symbols": [ { "unit": null, "symbol": "r_1", "meaning": "angular radius of the escribed small circle touching BC" }, { "unit": null, "symbol": "R", "meaning": "angular radius of the circumscribed small circle" }, { "unit": null, "symbol": "n", "meaning": "book's abbreviation: the square root of sin s sin(s−a) sin(s−b) sin(s−c)" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq((cot(r_1) - tan(R))**2, (sin(b) + sin(c) - sin(a))**2/(4*n**2) - 1)", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/cotangent", "concept/escribed-circle", "concept/sine", "concept/small-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-beff7f0256", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 75", "location": "Circumscribed and Inscribed Circles", "latex": "4n^2=1-\\cos^2 a-\\cos^2 b-\\cos^2 c+2\\cos a\\cos b\\cos c", "name": null, "statement": "Four times n squared equals 1 minus the squares of the cosines of the sides plus twice the product of the cosines of the sides.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "n", "meaning": "book's abbreviation: the square root of sin s sin(s−a) sin(s−b) sin(s−c)" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(4*n**2, 1 - cos(a)**2 - cos(b)**2 - cos(c)**2 + 2*cos(a)*cos(b)*cos(c))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-1b9000ea4f", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 75", "location": "Circumscribed and Inscribed Circles", "latex": "\\cot r + \\tan R = \\dfrac{1}{2n}\\Bigl\\{\\sin s + \\sin(s-a)+\\sin(s-b)+\\sin (s-c)\\Bigr\\}", "name": null, "statement": "The sum of cot r and tan R equals [sin s + sin(s−a) + sin(s−b) + sin(s−c)] divided by 2n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "angular radius of the inscribed small circle" }, { "unit": null, "symbol": "R", "meaning": "angular radius of the circumscribed small circle" }, { "unit": null, "symbol": "n", "meaning": "book's abbreviation: the square root of sin s sin(s−a) sin(s−b) sin(s−c)" }, { "unit": null, "symbol": "s", "meaning": "half the sum of the sides" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(cot(r) + tan(R), (sin(s) + sin(s - a) + sin(s - b) + sin(s - c))/(2*n))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/cotangent", "concept/sine", "concept/small-circle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-7619587408", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 75", "location": "Circumscribed and Inscribed Circles", "latex": "\\sin^2 s + \\sin^2 (s-a) + \\sin^2 (s-b) + \\sin^2 (s-c) = 2-2 \\cos a \\cos b \\cos c", "name": null, "statement": "The sum of the squared sines of s and of s−a, s−b, s−c equals 2 minus 2 cos a cos b cos c.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "s", "meaning": "half the sum of the sides" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(sin(s)**2 + sin(s - a)**2 + sin(s - b)**2 + sin(s - c)**2, 2 - 2*cos(a)*cos(b)*cos(c))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/sine" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-362e343216", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 75", "location": "Circumscribed and Inscribed Circles", "latex": "PA' = PB' = PC' = \\dfrac{\\pi}{2}-r", "name": null, "statement": "The point P is at angular distance π/2 − r from each vertex A', B', C' of the polar triangle, so the circle about the polar triangle has angular radius complementary to r.", "kind": "result", "symbols": [ { "unit": null, "symbol": "P", "meaning": "pole of the small circle inscribed in ABC" }, { "unit": null, "symbol": "A'", "meaning": "pole of BC, vertex of the polar triangle A'B'C'" }, { "unit": null, "symbol": "B'", "meaning": "vertex of the polar triangle, pole of CA" }, { "unit": null, "symbol": "C'", "meaning": "vertex of the polar triangle, pole of AB" }, { "unit": null, "symbol": "r", "meaning": "angular radius of the inscribed small circle" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/complementary-angles", "concept/polar-triangle", "concept/pole-of-a-circle", "concept/quadrant", "concept/small-circle", "quantity/pi" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-eb8673a8f4", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 77", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "\\dfrac{\\text{area of lune}}{\\text{surface of sphere}} = \\dfrac{A}{2\\pi}\\,", "name": null, "statement": "A lune's area is to the whole sphere's surface as its angle (in circular measure) is to 2π, since lunes are proportional to their angles.", "kind": "law", "symbols": [ { "unit": null, "symbol": "A", "meaning": "circular measure of the lune's angle" } ], "sympy": "Eq(L/S, A/(2*pi))", "physics": false, "states": [], "concepts": [ "concept/circular-measure", "concept/lune", "concept/sphere", "quantity/angle", "quantity/area", "quantity/pi" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-304fee8d60", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 77", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "\\text{area of lune } = \\dfrac{A}{2\\pi} 4\\pi r^2 = 2Ar^2.", "name": null, "statement": "The area of a lune of angle A on a sphere of radius r is 2Ar².", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "circular measure of the lune's angle" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "L", "meaning": "area of the lune" } ], "sympy": "Eq(L, 2*A*r**2)", "physics": false, "states": [], "concepts": [ "concept/lune", "concept/radius-of-a-regular-polygon", "concept/sphere", "quantity/area", "theorem/area-of-a-lune" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-28c67e571a", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 78", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "\\text{triangle } ABC=(A+B+C-\\pi)r^2.", "name": null, "statement": "The area of a spherical triangle equals its spherical excess times r².", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "circular measure of the angle at A" }, { "unit": null, "symbol": "B", "meaning": "circular measure of the angle at B" }, { "unit": null, "symbol": "C", "meaning": "circular measure of the angle at C" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "T", "meaning": "area of the spherical triangle ABC" } ], "sympy": "Eq(T, (A+B+C-pi)*r**2)", "physics": false, "states": [], "concepts": [ "concept/radius-of-a-regular-polygon", "concept/spherical-triangle", "quantity/angle", "quantity/area", "quantity/spherical-excess", "theorem/area-of-a-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-906f30a7ce", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 80", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "E=A+B+C-\\pi", "name": "spherical excess", "statement": "The spherical excess E of a triangle is the sum of its angles minus π.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "E", "meaning": "spherical excess of the triangle" }, { "unit": null, "symbol": "A", "meaning": "circular measure of the angle at A" }, { "unit": null, "symbol": "B", "meaning": "circular measure of the angle at B" }, { "unit": null, "symbol": "C", "meaning": "circular measure of the angle at C" } ], "sympy": "Eq(E, A+B+C-pi)", "physics": false, "states": [ "quantity/spherical-excess" ], "concepts": [ "concept/spherical-triangle", "quantity/angle", "quantity/pi" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-e3c9cf0aee", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 79", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "\\text{area of polygon} = \\Bigl\\{\\Sigma - (n-2)\\pi \\Bigr\\} r^2.", "name": null, "statement": "The area of a spherical polygon with n sides and angle sum Σ is (Σ − (n−2)π) r².", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "number of sides of the polygon" }, { "unit": null, "symbol": "Σ", "meaning": "sum of all the angles of the polygon" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "P", "meaning": "area of the spherical polygon" } ], "sympy": "Eq(P, (Sigma-(n-2)*pi)*r**2)", "physics": false, "states": [], "concepts": [ "concept/radius-of-a-regular-polygon", "concept/spherical-polygon", "concept/sum", "quantity/angle", "quantity/spherical-excess", "theorem/area-of-a-spherical-polygon" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-2207369818", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 80", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "\\sin\\tfrac{1}{2}E = \\dfrac{\\surd\\{\\sin s \\sin(s-a) \\sin(s-b) \\sin(s-c) \\} x} {2\\cos\\tfrac{1}{2}a \\cos\\tfrac{1}{2}b \\cos\\tfrac{1}{2}c }", "name": "Cagnoli's theorem", "statement": "The sine of half the spherical excess equals the square root of sin s sin(s−a) sin(s−b) sin(s−c), divided by 2 cos½a cos½b cos½c. The stated form carries a factor x that the proof in Art. 101 does not produce; this is flagged as a discrepancy between the statement and its derivation (possible transcription error or erratum), not silently corrected.", "kind": "law", "symbols": [ { "unit": null, "symbol": "E", "meaning": "spherical excess" }, { "unit": null, "symbol": "s", "meaning": "semi-perimeter of the triangle (half the sum of its sides)" }, { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle opposite A" }, { "unit": null, "symbol": "b", "meaning": "side of the spherical triangle opposite B" }, { "unit": null, "symbol": "c", "meaning": "side of the spherical triangle opposite C" }, { "unit": null, "symbol": "x", "meaning": "undefined in the chapter as printed; appears in the stated result only" } ], "sympy": "Eq(sin(E/2), sqrt(sin(s)*sin(s-a)*sin(s-b)*sin(s-c))*x/(2*cos(a/2)*cos(b/2)*cos(c/2)))", "physics": false, "states": [ "theorem/cagnoli-s-theorem" ], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/spherical-triangle", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ba76121da9", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 80", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "\\tan\\tfrac{1}{4}E = \\surd\\{\\tan\\tfrac{1}{2}s \\tan\\tfrac{1}{2}(s-a) \\tan\\tfrac{1}{2}(s-b) \\tan\\tfrac{1}{2}(s-c) \\}", "name": "Lhuilier's theorem", "statement": "The tangent of a quarter of the spherical excess equals the square root of the product of the tangents of half of s and of half of s−a, s−b, s−c.", "kind": "law", "symbols": [ { "unit": null, "symbol": "E", "meaning": "spherical excess" }, { "unit": null, "symbol": "s", "meaning": "semi-perimeter of the triangle" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": "Eq(tan(E/4), sqrt(tan(s/2)*tan((s-a)/2)*tan((s-b)/2)*tan((s-c)/2)))", "physics": false, "states": [ "theorem/lhuilier-s-theorem" ], "concepts": [ "concept/side", "concept/tangent-function", "quantity/semi-perimeter", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-298ba1333f", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 81", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "\\sin\\tfrac{1}{2}E = \\sin C \\sin\\tfrac{1}{2}a \\sin\\tfrac{1}{2}b \\sec\\tfrac{1}{2}c;", "name": null, "statement": "Half the spherical excess has sine equal to sin C times sin½a sin½b divided by cos½c.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "E", "meaning": "spherical excess" }, { "unit": null, "symbol": "C", "meaning": "circular measure of the angle at C" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": "Eq(sin(E/2), sin(C)*sin(a/2)*sin(b/2)/cos(c/2))", "physics": false, "states": [], "concepts": [ "concept/secant", "concept/side", "concept/sine", "concept/spherical-angle", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-e24709451a", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 81", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "\\tan\\tfrac{1}{2}E = \\frac{\\sin\\tfrac{1}{2}a \\sin\\tfrac{1}{2}b \\sin C } {\\cos\\tfrac{1}{2}a \\cos\\tfrac{1}{2}b + \\sin\\tfrac{1}{2}a \\sin\\tfrac{1}{2}b \\cos C }", "name": null, "statement": "The tangent of half the spherical excess expressed in the two sides a, b and the angle C between them.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "E", "meaning": "spherical excess" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "C", "meaning": "circular measure of the angle at C" } ], "sympy": "Eq(tan(E/2), sin(a/2)*sin(b/2)*sin(C)/(cos(a/2)*cos(b/2)+sin(a/2)*sin(b/2)*cos(C)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/tangent-function", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-4d203e9088", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 81", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "\\frac{\\cos^2\\tfrac{1}{2}a + \\cos^2\\tfrac{1}{2}b + \\cos^2\\tfrac{1}{2}c-1 } {2\\cos\\tfrac{1}{2}a \\cos\\tfrac{1}{2}b \\cos\\tfrac{1}{2}c }", "name": null, "statement": "Right-hand side of the result for cos½E in terms of the half-sides only (equation (3)); the chapter's chain writes cos½E equal to this expression.", "kind": "result", "symbols": [ { "unit": null, "symbol": "E", "meaning": "spherical excess" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": "(cos(a/2)**2 + cos(b/2)**2 + cos(c/2)**2 - 1)/(2*cos(a/2)*cos(b/2)*cos(c/2))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/spherical-triangle", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-826316e317", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 82", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "\\sin^2\\tfrac{1}{4}E = \\frac{\\sin\\frac{1}{2}s \\sin\\frac{1}{2}(s-a) \\sin\\frac{1}{2}(s-b) \\sin\\frac{1}{2}(s-c) } {\\cos\\frac{1}{2}a \\cos\\frac{1}{2}b \\cos\\frac{1}{2}c }", "name": null, "statement": "The square of the sine of a quarter of the spherical excess equals the product of sines of half s and of half (s−a), (s−b), (s−c), over the product of cosines of half the sides.", "kind": "result", "symbols": [ { "unit": null, "symbol": "E", "meaning": "spherical excess" }, { "unit": null, "symbol": "s", "meaning": "semi-perimeter of the triangle" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": "Eq(sin(E/4)**2, sin(s/2)*sin((s-a)/2)*sin((s-b)/2)*sin((s-c)/2)/(cos(a/2)*cos(b/2)*cos(c/2)))", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "quantity/semi-perimeter", "quantity/spherical-excess", "theorem/lhuilier-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-abaf048ff8", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 82", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "\\cos^2\\tfrac{1}{4}E = \\frac{\\cos\\frac{1}{2}s \\cos\\frac{1}{2}(s-a) \\cos\\frac{1}{2}(s-b) \\cos\\frac{1}{2}(s-c) } {\\cos\\frac{1}{2}a \\cos\\frac{1}{2}b \\cos\\frac{1}{2}c }", "name": null, "statement": "The square of the cosine of a quarter of the spherical excess equals the product of cosines of half s and of half (s−a), (s−b), (s−c), over the product of cosines of half the sides.", "kind": "result", "symbols": [ { "unit": null, "symbol": "E", "meaning": "spherical excess" }, { "unit": null, "symbol": "s", "meaning": "semi-perimeter of the triangle" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": "Eq(cos(E/4)**2, cos(s/2)*cos((s-a)/2)*cos((s-b)/2)*cos((s-c)/2)/(cos(a/2)*cos(b/2)*cos(c/2)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "quantity/semi-perimeter", "quantity/spherical-excess", "theorem/lhuilier-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-6da0507ea5", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 82", "location": "Area of a Spherical Triangle. Spherical Excess", "latex": "\\sin(C-\\tfrac12E) = \\frac{\\surd\\{\\sin s \\sin(s-a) \\sin(s-b) \\sin(s-c) \\} } {2\\sin\\frac12a \\sin\\frac12b \\cos\\frac12c }", "name": null, "statement": "The sine of (C − ½E) equals the square root of sin s sin(s−a) sin(s−b) sin(s−c), over 2 sin½a sin½b cos½c.", "kind": "result", "symbols": [ { "unit": null, "symbol": "E", "meaning": "spherical excess" }, { "unit": null, "symbol": "C", "meaning": "circular measure of the angle at C" }, { "unit": null, "symbol": "s", "meaning": "semi-perimeter of the triangle" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": "Eq(sin(C-E/2), sqrt(sin(s)*sin(s-a)*sin(s-b)*sin(s-c))/(2*sin(a/2)*sin(b/2)*cos(c/2)))", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "concept/spherical-angle", "quantity/semi-perimeter", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-7ff2e03c9e", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 85", "location": "On certain approximate Formul\\ae", "latex": "\\cos EF = \\sin \\frac{1}{2} b \\sin \\frac{1}{2} c + \\cos \\frac{1}{2} b \\cos \\frac{1}{2} c \\cos A", "name": null, "statement": "The cosine of the chord-angle side EF equals a combination of the half-sides b/2 and c/2 and the angle A between the two sides of the triangle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "EF", "meaning": "side of the spherical triangle DEF, the arc joining E and F on the sphere described about A" }, { "unit": null, "symbol": "b", "meaning": "side AC of the spherical triangle ABC" }, { "unit": null, "symbol": "c", "meaning": "side AB of the spherical triangle ABC" }, { "unit": null, "symbol": "A", "meaning": "angle of the spherical triangle at A, equal to the angle EDF" } ], "sympy": "Eq(cos(EF), sin(b/2)*sin(c/2) + cos(b/2)*cos(c/2)*cos(A))", "physics": false, "states": [], "concepts": [ "concept/chord", "concept/chordal-triangle", "concept/cosine", "concept/side", "concept/spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-3061b14a0f", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 85", "location": "On certain approximate Formul\\ae", "latex": "\\cos EF=\\cos DE \\cos DF + \\sin DE \\sin DF \\cos A", "name": null, "statement": "The spherical cosine rule applied to the spherical triangle DEF, with angle A at D.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "EF", "meaning": "side of the triangle DEF opposite the angle at D" }, { "unit": null, "symbol": "DE", "meaning": "arc from D to E on the auxiliary sphere" }, { "unit": null, "symbol": "DF", "meaning": "arc from D to F on the auxiliary sphere" }, { "unit": null, "symbol": "A", "meaning": "angle EDF, equal to the inclination of the planes OAB and OAC" } ], "sympy": "Eq(cos(EF), cos(DE)*cos(DF) + sin(DE)*sin(DF)*cos(A))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/inclination-of-two-lines", "concept/side", "concept/spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-6330ffbec6", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 86", "location": "On certain approximate Formul\\ae", "latex": "\\theta = \\tan\\tfrac12A \\sin^2\\tfrac14(b + c) - \\cot\\tfrac12A \\sin^2\\tfrac14(b - c)", "name": null, "statement": "The small angle theta by which EF differs from A is approximately tan(A/2) sin^2((b+c)/4) minus cot(A/2) sin^2((b-c)/4); this gives its circular measure.", "kind": "approximation", "symbols": [ { "unit": "circular measure", "symbol": "theta", "meaning": "small difference EF = A - theta; its circular measure is given by the formula" }, { "unit": null, "symbol": "A", "meaning": "angle of the spherical triangle at A" }, { "unit": null, "symbol": "b", "meaning": "side AC" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(theta, tan(A/2)*sin((b+c)/4)**2 - cot(A/2)*sin((b-c)/4)**2)", "physics": false, "states": [], "concepts": [ "concept/angle-between-the-chords", "concept/approximation", "concept/chordal-triangle", "concept/circular-measure", "concept/cotangent", "concept/spherical-triangle", "concept/tangent" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-2fe41806f3", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 86", "location": "On certain approximate Formul\\ae", "latex": "2r\\sin\\dfrac{\\alpha}{2r}", "name": null, "statement": "The length of a side of the chordal triangle, from the arc alpha of the sphere of radius r.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "alpha", "meaning": "length of the arc corresponding to side a" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/chord", "concept/chordal-triangle", "concept/radius-of-a-regular-polygon", "concept/sine", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-aae4e1aee6", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 87", "location": "On certain approximate Formul\\ae", "latex": "\\cos A = \\frac{\\cos a - \\cos b\\cos c}{\\sin b \\sin c}", "name": null, "statement": "The spherical cosine rule giving the cosine of angle A from the three sides a, b, c.", "kind": "law", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle of the spherical triangle at A" }, { "unit": null, "symbol": "a", "meaning": "side BC of the spherical triangle" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(cos(A), (cos(a) - cos(b)*cos(c))/(sin(b)*sin(c)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-cfb1c6ec64", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 87", "location": "On certain approximate Formul\\ae", "latex": "1 - \\frac{\\alpha^2}{2r^2} + \\frac{\\alpha^4}{24r^4} - \\ldots", "name": null, "statement": "Series expansion of cos a in powers of alpha/r, where alpha is the arc corresponding to side a.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle (angular length)" }, { "unit": null, "symbol": "alpha", "meaning": "length of the arc corresponding to side a" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/cosine", "concept/radius-of-the-sphere", "concept/series-expansion", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-d4ab6f42aa", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 87", "location": "On certain approximate Formul\\ae", "latex": "\\frac{\\alpha}{r} - \\frac{\\alpha^3}{6r^3} +\\ldots", "name": null, "statement": "Series expansion of sin a in powers of alpha/r, where alpha is the arc corresponding to side a.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle (angular length)" }, { "unit": null, "symbol": "alpha", "meaning": "length of the arc corresponding to side a" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/radius-of-the-sphere", "concept/series-expansion", "concept/sine", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-9f08f7c802", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 87", "location": "On certain approximate Formul\\ae", "latex": "\\theta = \\frac{\\beta \\gamma \\sin A'}{6r^2}=\\frac{S}{3r^2}", "name": null, "statement": "Legendre's correction: the excess of the spherical angle A over the plane angle A' equals S/(3r^2), where S is the plane triangle's area.", "kind": "result", "symbols": [ { "unit": "circular measure", "symbol": "theta", "meaning": "A - A', the excess of the spherical angle over the plane angle" }, { "unit": null, "symbol": "beta", "meaning": "length of the arc corresponding to side b" }, { "unit": null, "symbol": "gamma", "meaning": "length of the arc corresponding to side c" }, { "unit": null, "symbol": "A'", "meaning": "angle of the plane triangle with sides alpha, beta, gamma, opposite alpha" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "S", "meaning": "area of the plane triangle whose sides are alpha, beta, gamma" } ], "sympy": "Eq(theta, beta*gamma*sin(A_p)/(6*r**2))", "physics": false, "states": [], "concepts": [ "concept/plane-triangle", "concept/radius-of-a-regular-polygon", "concept/spherical-triangle", "quantity/angle", "quantity/area", "theorem/legendre-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-8fae053762", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 88", "location": "On certain approximate Formul\\ae", "latex": "B = B' + \\frac{S}{3r^2}", "name": null, "statement": "Each spherical angle exceeds the corresponding plane angle by S/(3r^2); the same holds for B.", "kind": "result", "symbols": [ { "unit": null, "symbol": "B", "meaning": "angle of the spherical triangle at B" }, { "unit": null, "symbol": "B'", "meaning": "angle of the plane triangle at B'" }, { "unit": null, "symbol": "S", "meaning": "area of the plane triangle" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": "Eq(B, B_p + S/(3*r**2))", "physics": false, "states": [], "concepts": [ "concept/plane-triangle", "concept/radius-of-a-regular-polygon", "quantity/angle", "quantity/area", "theorem/legendre-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-25e34dbffd", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 88", "location": "On certain approximate Formul\\ae", "latex": "A+B+C = A'+B'+C'+\\frac{S}{r^2} = \\pi + \\frac{S}{r^2}", "name": null, "statement": "The spherical excess (sum of the spherical angles minus pi) is approximately S/r^2, the plane area divided by the square of the radius.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A, B, C", "meaning": "angles of the spherical triangle" }, { "unit": null, "symbol": "A', B', C'", "meaning": "angles of the plane triangle" }, { "unit": null, "symbol": "S", "meaning": "area of the plane triangle" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": "Eq(A + B + C, A_p + B_p + C_p + S/r**2)", "physics": false, "states": [], "concepts": [ "concept/plane-triangle", "concept/radius-of-a-regular-polygon", "quantity/angle", "quantity/area", "quantity/pi", "quantity/spherical-excess", "theorem/legendre-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f8dde2dd0a", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 88", "location": "On certain approximate Formul\\ae", "latex": "S = \\tfrac{1}{2}\\beta\\gamma\\sin A' = \\tfrac{1}{2}\\beta\\gamma\\sin A", "name": null, "statement": "The area of the plane triangle with two sides and the included angle, equal to the spherical one approximately.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "S", "meaning": "area of the plane triangle" }, { "unit": null, "symbol": "beta", "meaning": "length of the arc corresponding to side b" }, { "unit": null, "symbol": "gamma", "meaning": "length of the arc corresponding to side c" }, { "unit": null, "symbol": "A'", "meaning": "plane-triangle angle" }, { "unit": null, "symbol": "A", "meaning": "spherical angle" } ], "sympy": "Eq(S, sin(A)*beta*gamma/2)", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/plane-triangle", "concept/sine", "quantity/angle", "quantity/area" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-bbe33dc6e2", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 89", "location": "On certain approximate Formul\\ae", "latex": "\\sin B' = \\frac{\\beta}{\\alpha}\\sin A' = \\frac{\\beta}{\\alpha}\\sin A", "name": null, "statement": "The sine rule for the plane triangle, used to find B' from A, alpha and beta.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "B'", "meaning": "angle of the plane triangle at B'" }, { "unit": null, "symbol": "A'", "meaning": "angle of the plane triangle at A'" }, { "unit": null, "symbol": "alpha", "meaning": "length of the arc corresponding to side a" }, { "unit": null, "symbol": "beta", "meaning": "length of the arc corresponding to side b" } ], "sympy": "Eq(sin(B_p), beta/alpha*sin(A_p))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/plane-triangle", "concept/sine", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-611426c391", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 89", "location": "On certain approximate Formul\\ae", "latex": "S=\\frac{\\gamma^2 \\sin A' \\sin B'}{2\\sin(A'+B')}", "name": null, "statement": "The area of the plane triangle expressed from two angles and the included side gamma.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "S", "meaning": "area of the plane triangle" }, { "unit": null, "symbol": "gamma", "meaning": "length of the arc corresponding to side c, the included side" }, { "unit": null, "symbol": "A'", "meaning": "plane-triangle angle at A'" }, { "unit": null, "symbol": "B'", "meaning": "plane-triangle angle at B'" } ], "sympy": "Eq(S, gamma**2*sin(A_p)*sin(B_p)/(2*sin(A_p + B_p)))", "physics": false, "states": [], "concepts": [ "concept/plane-triangle", "concept/side", "concept/sine", "quantity/angle", "quantity/area" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-047d92b0ae", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 89", "location": "On certain approximate Formul\\ae", "latex": "S=\\frac{\\alpha^2\\sin B' \\sin C'}{2\\sin(B'+C')}", "name": null, "statement": "The area of the plane triangle expressed from two angles and the side alpha opposite one of them.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "S", "meaning": "area of the plane triangle" }, { "unit": null, "symbol": "alpha", "meaning": "length of the arc corresponding to side a" }, { "unit": null, "symbol": "B'", "meaning": "plane-triangle angle at B'" }, { "unit": null, "symbol": "C'", "meaning": "plane-triangle angle at C'" } ], "sympy": "Eq(S, alpha**2*sin(B_p)*sin(C_p)/(2*sin(B_p + C_p)))", "physics": false, "states": [], "concepts": [ "concept/plane-triangle", "concept/side", "concept/sine", "quantity/angle", "quantity/area" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ff1920fea4", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 88", "location": "On certain approximate Formul\\ae", "latex": "A = A' + \\frac{S}{3r^2}", "name": null, "statement": "The spherical angle A equals the plane-triangle angle A' plus S/(3r^2).", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle of the spherical triangle at A" }, { "unit": null, "symbol": "A'", "meaning": "angle of the plane triangle at A'" }, { "unit": null, "symbol": "S", "meaning": "area of the plane triangle" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": "Eq(A, A_p + S/(3*r**2))", "physics": false, "states": [], "concepts": [ "concept/plane-triangle", "concept/radius-of-a-regular-polygon", "quantity/angle", "quantity/area", "theorem/legendre-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-7b05326c6c", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 90", "location": "On certain approximate Formul\\ae", "latex": "\\sin\\frac{1}{2}E = \\frac{\\sin\\tfrac{1}{2}a \\sin\\tfrac{1}{2}b \\sin C} {\\cos\\tfrac{1}{2}c}", "name": null, "statement": "The sine of half the spherical excess E in terms of two sides and the included angle C.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "E", "meaning": "spherical excess of the triangle" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" }, { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle at C" } ], "sympy": "Eq(sin(E/2), sin(a/2)*sin(b/2)*sin(C)/cos(c/2))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/spherical-triangle", "quantity/angle", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-8c133698d2", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 90", "location": "On certain approximate Formul\\ae", "latex": "\\sin C' \\frac{\\alpha\\beta}{2r^2} \\left( 1 + \\frac{\\alpha^2+\\beta^2+\\gamma^2}{24r^2} \\right)", "name": null, "statement": "A closer approximation to the spherical excess E, the plane-triangle sine term with a correction of order 1/r^2.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "E", "meaning": "spherical excess of the triangle" }, { "unit": null, "symbol": "C'", "meaning": "angle of the plane triangle at C'" }, { "unit": null, "symbol": "alpha", "meaning": "length of the arc corresponding to side a" }, { "unit": null, "symbol": "beta", "meaning": "length of the arc corresponding to side b" }, { "unit": null, "symbol": "gamma", "meaning": "length of the arc corresponding to side c" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/plane-triangle", "concept/radius-of-a-regular-polygon", "concept/sine", "quantity/area", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-b07d086024", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 90", "location": "On certain approximate Formul\\ae", "latex": "\\frac{\\operatorname{Sin} A}{\\operatorname{Sin} B} = \\frac{\\sin a}{\\sin b}", "name": null, "statement": "The sine rule for a spherical triangle: the ratio of the sines of two angles equals the ratio of the sines of the opposite sides.", "kind": "law", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle of the spherical triangle at A" }, { "unit": null, "symbol": "B", "meaning": "angle of the spherical triangle at B" }, { "unit": null, "symbol": "a", "meaning": "side BC opposite A" }, { "unit": null, "symbol": "b", "meaning": "side CA opposite B" } ], "sympy": "Eq(sin(A)/sin(B), sin(a)/sin(b))", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "concept/spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-c9d4bebbcf", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 91", "location": "On certain approximate Formul\\ae", "latex": "\\frac{\\alpha}{\\beta} \\left\\{1 + \\frac{\\beta^2 - \\alpha^2}{6r^2} \\left(1 + \\frac{7\\beta^2-3\\alpha^2}{60r^2}\\right)\\right\\}", "name": null, "statement": "Approximate value of sin A / sin B in terms of the arcs alpha and beta and the radius r.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle of the spherical triangle at A" }, { "unit": null, "symbol": "B", "meaning": "angle of the spherical triangle at B" }, { "unit": null, "symbol": "alpha", "meaning": "length of the arc corresponding to side a" }, { "unit": null, "symbol": "beta", "meaning": "length of the arc corresponding to side b" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": "Eq(sin(A)/sin(B), alpha/beta*(1 + (beta**2 - alpha**2)/(6*r**2)*(1 + (7*beta**2 - 3*alpha**2)/(60*r**2))))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/radius-of-a-regular-polygon", "concept/sine", "quantity/angle", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ce2a1367b1", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 91", "location": "On certain approximate Formul\\ae", "latex": "= \\frac{\\alpha^2-\\beta^2}{\\alpha\\gamma\\sin B} \\left( 1 - \\frac{\\beta^2+\\gamma^2-\\alpha^2}{12r^2} \\right)", "name": null, "statement": "Approximate value of cot B minus cot A in terms of the sides and the radius.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle of the spherical triangle at A" }, { "unit": null, "symbol": "B", "meaning": "angle of the spherical triangle at B" }, { "unit": null, "symbol": "alpha", "meaning": "length of the arc corresponding to side a" }, { "unit": null, "symbol": "beta", "meaning": "length of the arc corresponding to side b" }, { "unit": null, "symbol": "gamma", "meaning": "length of the arc corresponding to side c" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/cotangent", "concept/radius-of-a-regular-polygon", "concept/sine", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-70dc21c690", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 93", "location": "On certain approximate Formul\\ae", "latex": "x = \\alpha - \\dfrac{\\beta\\sin A}{\\sin B} - \\dfrac{\\mu (\\alpha^2-\\beta^2) }{\\gamma\\sin B}", "name": null, "statement": "The error x in the calculated side alpha, when the side is found by the approximate formula with the spherical excess 3 mu.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "error in the length of side alpha computed by Legendre's theorem" }, { "unit": null, "symbol": "alpha", "meaning": "true length of the side required" }, { "unit": null, "symbol": "beta", "meaning": "known side" }, { "unit": null, "symbol": "A", "meaning": "angle of the spherical triangle at A" }, { "unit": null, "symbol": "B", "meaning": "angle of the spherical triangle at B" }, { "unit": null, "symbol": "gamma", "meaning": "length of the arc corresponding to side c" }, { "unit": null, "symbol": "mu", "meaning": "one third of the spherical excess adopted in the approximation" } ], "sympy": "Eq(x, alpha - beta*sin(A)/sin(B) - mu*(alpha**2 - beta**2)/(gamma*sin(B)))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/error-of-measurement", "concept/side", "concept/sine", "quantity/spherical-excess", "theorem/legendre-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f26e7db759", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 93", "location": "On certain approximate Formul\\ae", "latex": "\\mu=\\dfrac{\\alpha\\gamma\\sin B}{6r^2}", "name": null, "statement": "Choice of the adopted spherical excess 3 mu that makes the error in the side vanish to the order r^-2, written from the Legendre formula.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "mu", "meaning": "one third of the spherical excess adopted" }, { "unit": null, "symbol": "alpha", "meaning": "length of the arc corresponding to side a" }, { "unit": null, "symbol": "gamma", "meaning": "length of the arc corresponding to side c" }, { "unit": null, "symbol": "B", "meaning": "angle of the spherical triangle at B" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": "Eq(mu, alpha*gamma*sin(B)/(6*r**2))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/error-of-measurement", "concept/radius-of-a-regular-polygon", "concept/sine", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-fcf71dbcfb", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 93", "location": "On certain approximate Formul\\ae", "latex": "x=\\frac{\\alpha(\\beta^2-\\alpha^2)(3\\alpha^2-7\\beta^2)}{360r^4}", "name": null, "statement": "The error in the calculated side when mu is taken from the Legendre formula, to order r^-4.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "error in the length of side alpha" }, { "unit": null, "symbol": "alpha", "meaning": "true length of the side required" }, { "unit": null, "symbol": "beta", "meaning": "known side" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": "Eq(x, alpha*(beta**2 - alpha**2)*(3*alpha**2 - 7*beta**2)/(360*r**4))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/error-of-measurement", "concept/radius-of-a-regular-polygon", "concept/side" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-a9b000eac4", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 93", "location": "On certain approximate Formul\\ae", "latex": "= \\frac{\\alpha (\\beta^2-\\alpha^2) (\\alpha^2 + \\beta^2 - 5\\gamma^2)} {720r^4}", "name": null, "statement": "The error in the calculated side when mu is taken from an equation corresponding to (1) of Art. 109, to order r^-4.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "error in the length of side alpha" }, { "unit": null, "symbol": "alpha", "meaning": "true length of the side required" }, { "unit": null, "symbol": "beta", "meaning": "known side" }, { "unit": null, "symbol": "gamma", "meaning": "length of the arc corresponding to side c" }, { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": "Eq(x, alpha*(beta**2 - alpha**2)*(alpha**2 + beta**2 - 5*gamma**2)/(720*r**4))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/error-of-measurement", "concept/radius-of-a-regular-polygon", "concept/side" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-a74d6af74a", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 98", "location": "Geodetical Operations", "latex": "s=Er^2", "name": null, "statement": "The area s of a spherical triangle on the Earth's surface equals the circular measure E of its spherical excess times the square of the Earth's radius r.", "kind": "formula", "symbols": [ { "unit": "square feet", "symbol": "s", "meaning": "number of square feet in the area of the triangle" }, { "unit": null, "symbol": "E", "meaning": "circular measure of the spherical excess" }, { "unit": "feet", "symbol": "r", "meaning": "number of feet in the radius of the Earth" } ], "sympy": "Eq(s, E*r**2)", "physics": true, "states": [], "concepts": [ "concept/area", "concept/circular-measure", "concept/radius", "concept/spherical-triangle", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-9463a7c87c", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 98", "location": "Geodetical Operations", "latex": "E=\\frac{n\\pi}{180\\centerdot 60\\centerdot 60}", "name": null, "statement": "The circular measure E of an angle equals n seconds converted through 180 degrees of π radians and 60 minutes of 60 seconds each.", "kind": "definition", "symbols": [ { "unit": "radian", "symbol": "E", "meaning": "circular measure of the spherical excess" }, { "unit": "second of arc", "symbol": "n", "meaning": "number of seconds in the spherical excess" } ], "sympy": "Eq(E, n*pi/(180*60*60))", "physics": false, "states": [], "concepts": [ "concept/circular-measure", "quantity/pi", "quantity/spherical-excess", "unit/degree-of-angle", "unit/second-of-arc" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-67dfd61a4c", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 98", "location": "Geodetical Operations", "latex": "= \\frac{n}{206265}\\ \\text{ approximately;}", "name": null, "statement": "Approximately, the circular measure E equals n divided by 206265, the number of seconds in one radian.", "kind": "approximation", "symbols": [ { "unit": "radian", "symbol": "E", "meaning": "circular measure of the spherical excess" }, { "unit": "second of arc", "symbol": "n", "meaning": "number of seconds in the spherical excess" } ], "sympy": "Eq(E, n/206265)", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/circular-measure", "quantity/spherical-excess", "unit/second-of-arc" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-fa88a49409", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 98", "location": "Geodetical Operations", "latex": "s=\\frac{nr^2}{206265}\\,", "name": "General Roy's rule (intermediate form)", "statement": "The area s of the spherical triangle, in square feet, equals n r squared divided by 206265, where n is the spherical excess in seconds and r the Earth's radius in feet.", "kind": "formula", "symbols": [ { "unit": "square feet", "symbol": "s", "meaning": "number of square feet in the area of the triangle" }, { "unit": "second of arc", "symbol": "n", "meaning": "number of seconds in the spherical excess" }, { "unit": "feet", "symbol": "r", "meaning": "number of feet in the radius of the Earth" } ], "sympy": "Eq(s, n*r**2/206265)", "physics": true, "states": [ "theorem/general-roy-s-rule-intermediate-form" ], "concepts": [ "concept/approximation", "concept/area", "concept/radius", "concept/spherical-triangle", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-26ec02c88b", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 98", "location": "Geodetical Operations", "latex": "\\frac{\\pi r}{180}=365155", "name": null, "statement": "The length of one degree of arc on the Earth's surface, πr/180, equals the measured value 365155 feet, which fixes the Earth's radius r.", "kind": "result", "symbols": [ { "unit": "feet", "symbol": "r", "meaning": "number of feet in the radius of the Earth" } ], "sympy": "Eq(pi*r/180, 365155)", "physics": true, "states": [], "concepts": [ "concept/figure-of-the-earth", "concept/radius", "quantity/arc-of-a-circle", "unit/degree-of-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-3cb795b6e7", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 98", "location": "Geodetical Operations", "latex": "\\log n = \\log s - 9.326774", "name": "General Roy's rule", "statement": "The logarithm of the spherical excess in seconds equals the logarithm of the triangle's area in square feet minus 9.326774, so n is found once s is known.", "kind": "rule", "symbols": [ { "unit": "second of arc", "symbol": "n", "meaning": "number of seconds in the spherical excess" }, { "unit": "square feet", "symbol": "s", "meaning": "number of square feet in the area of the triangle" } ], "sympy": "Eq(log(n), log(s) - 9.326774)", "physics": true, "states": [ "method/general-roy-s-rule" ], "concepts": [ "concept/area", "concept/logarithm", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-5bd06f71ac", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 100", "location": "Geodetical Operations", "latex": "\\delta A + \\delta B + \\delta C = 0", "name": null, "statement": "The errors in the three observed angles A, B, C sum to zero, because the altered angles are supposed to sum correctly.", "kind": "result", "symbols": [ { "unit": "radian", "symbol": "δA", "meaning": "error of the angle A" }, { "unit": "radian", "symbol": "δB", "meaning": "error of the angle B" }, { "unit": "radian", "symbol": "δC", "meaning": "error of the angle C" } ], "sympy": "Eq(dA + dB + dC, 0)", "physics": false, "states": [], "concepts": [ "concept/error-of-measurement", "concept/observational-error", "concept/spherical-triangle", "concept/sum" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-c49c01a15d", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 100", "location": "Geodetical Operations", "latex": "\\dfrac{a \\sin (C + \\delta C)}{\\sin (A + \\delta A)}", "name": null, "statement": "Treating the triangle as approximately plane, the true side c is a sin(C + δC) divided by sin(A + δA), where a is the known side opposite A.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "c", "meaning": "side of the triangle opposite the angle C" }, { "unit": null, "symbol": "a", "meaning": "known side opposite the angle A" }, { "unit": "degree of angle", "symbol": "A", "meaning": "observed angle A, altered if necessary" }, { "unit": "degree of angle", "symbol": "C", "meaning": "observed angle C, altered if necessary" }, { "unit": "radian", "symbol": "δA", "meaning": "error of the angle A" }, { "unit": "radian", "symbol": "δC", "meaning": "error of the angle C" } ], "sympy": "Eq(c, a*sin(C + dC)/sin(A + dA))", "physics": false, "states": [], "concepts": [ "concept/error-of-measurement", "concept/plane-triangle", "concept/side", "concept/sine" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-72fb5d1823", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 100", "location": "Geodetical Operations", "latex": "\\sin (C + \\delta C) = \\sin C + \\delta C \\cos C", "name": null, "statement": "For a small error δC, sin(C + δC) is approximately sin C plus δC times cos C.", "kind": "approximation", "symbols": [ { "unit": "radian", "symbol": "δC", "meaning": "error of the angle C" }, { "unit": "degree of angle", "symbol": "C", "meaning": "observed angle C" } ], "sympy": "Eq(sin(C + dC), sin(C) + dC*cos(C))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/cosine", "concept/error-of-measurement", "concept/infinitesimal", "concept/sine" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-3fe6422f17", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 100", "location": "Geodetical Operations", "latex": "\\sin (A - \\delta B - \\delta C) = \\sin A - (\\delta B + \\delta C) \\cos A", "name": null, "statement": "For small errors, sin(A − δB − δC) is approximately sin A minus (δB + δC) times cos A.", "kind": "approximation", "symbols": [ { "unit": "radian", "symbol": "δB", "meaning": "error of the angle B" }, { "unit": "radian", "symbol": "δC", "meaning": "error of the angle C" }, { "unit": "degree of angle", "symbol": "A", "meaning": "observed angle A" } ], "sympy": "Eq(sin(A - dB - dC), sin(A) - (dB + dC)*cos(A))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/cosine", "concept/error-of-measurement", "concept/sine" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-57583349f0", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 100", "location": "Geodetical Operations", "latex": "\\cot C + \\cot A = \\dfrac{\\sin (A + C)}{\\sin A \\sin C} = \\dfrac{\\sin B}{\\sin A \\sin C}", "name": null, "statement": "The sum of the cotangents of A and C equals sin B divided by sin A sin C, using A + C = 180° − B; the book states this holds approximately for the nearly plane triangle.", "kind": "identity", "symbols": [ { "unit": "degree of angle", "symbol": "A", "meaning": "angle A of the triangle" }, { "unit": "degree of angle", "symbol": "B", "meaning": "angle B of the triangle" }, { "unit": "degree of angle", "symbol": "C", "meaning": "angle C of the triangle" } ], "sympy": "Eq(cot(C) + cot(A), sin(A + C)/(sin(A)*sin(C)))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/cotangent", "concept/plane-triangle", "concept/sine", "concept/triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-0a019d6e23", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 100", "location": "Geodetical Operations", "latex": "\\frac{a \\sin B}{\\sin^2 A} \\delta C + \\frac{a \\sin C \\cos A}{\\sin^2 A)} \\delta B", "name": null, "statement": "The approximate error in the side c due to the angle errors δB and δC is a sin B δC over sin²A plus a sin C cos A δB over sin²A; the source prints a stray closing parenthesis in the second denominator, which is reproduced here as printed.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "c", "meaning": "side opposite the angle C" }, { "unit": null, "symbol": "a", "meaning": "known side opposite the angle A" }, { "unit": "radian", "symbol": "δB", "meaning": "error of the angle B" }, { "unit": "radian", "symbol": "δC", "meaning": "error of the angle C" } ], "sympy": "Eq(delta_c, a*sin(B)/sin(A)**2*dC + a*sin(C)*cos(A)/sin(A)**2*dB)", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/cosine", "concept/error-of-measurement", "concept/side", "concept/sine" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-bba8f1585a", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 100", "location": "Geodetical Operations", "latex": "\\frac{a \\sin C}{\\sin^2 A} \\delta B + \\frac{a \\sin B \\cos A}{\\sin^2 A} \\delta C", "name": null, "statement": "The approximate error in the side b due to the angle errors δB and δC is a sin C δB over sin²A plus a sin B cos A δC over sin²A.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "b", "meaning": "side opposite the angle B" }, { "unit": null, "symbol": "a", "meaning": "known side opposite the angle A" }, { "unit": "radian", "symbol": "δB", "meaning": "error of the angle B" }, { "unit": "radian", "symbol": "δC", "meaning": "error of the angle C" } ], "sympy": "Eq(delta_b, a*sin(C)/sin(A)**2*dB + a*sin(B)*cos(A)/sin(A)**2*dC)", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/cosine", "concept/error-of-measurement", "concept/side", "concept/sine" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-0b4b0082ab", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 102", "location": "Geodetical Operations", "latex": "\\cos (\\theta + x)=\\frac{\\cos \\theta-\\sin h \\,\\sin k}{\\cos h \\,\\cos k}", "name": null, "statement": "The cosine of the reduced horizontal angle θ + x equals (cos θ − sin h sin k) divided by cos h cos k; this formula is exact.", "kind": "formula", "symbols": [ { "unit": "degree of angle", "symbol": "θ", "meaning": "observed angle AOB between the two points" }, { "unit": "radian", "symbol": "x", "meaning": "correction from the angle at the observer to the horizon angle" }, { "unit": "radian", "symbol": "h", "meaning": "angle AOC, the elevation or depression of the first point" }, { "unit": "radian", "symbol": "k", "meaning": "angle BOD, the elevation or depression of the second point" } ], "sympy": "Eq(cos(theta + x), (cos(theta) - sin(h)*sin(k))/(cos(h)*cos(k)))", "physics": true, "states": [], "concepts": [ "concept/cosine", "concept/horizontal-angle", "concept/spherical-triangle", "method/reduction-to-the-horizon", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-cd94f9a08e", "chapter": "todhunter-spherical-trigonometry-1886/ch-geodetical-operations", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 102", "location": "Geodetical Operations", "latex": "\\cos \\theta-x \\sin \\theta=\\frac{\\cos \\theta-hk} {1-\\frac{1}{2}(h^2+k^2)}", "name": null, "statement": "To first order in the small quantities x, h and k, cos θ − x sin θ equals (cos θ − hk) divided by 1 − ½(h² + k²).", "kind": "approximation", "symbols": [ { "unit": "degree of angle", "symbol": "θ", "meaning": "observed angle AOB between the two points" }, { "unit": "radian", "symbol": "x", "meaning": "correction from the angle at the observer to the horizon angle" }, { "unit": "radian", "symbol": "h", "meaning": "angle AOC, the elevation or depression of the first point" }, { "unit": "radian", "symbol": "k", "meaning": "angle BOD, the elevation or depression of the second point" } ], "sympy": "Eq(cos(theta) - x*sin(theta), (cos(theta) - h*k)/(1 - (h**2 + k**2)/2))", "physics": true, "states": [], "concepts": [ "concept/approximation", "concept/cosine", "concept/horizontal-angle", "concept/infinitesimal", "concept/sine", "method/reduction-to-the-horizon" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-6a7b9d112f", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 103", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\cos c = \\cos a \\,\\cos b + \\sin a \\,\\sin b \\,\\cos C", "name": "spherical law of cosines for sides", "statement": "The cosine of side c equals the cosine of a times the cosine of b plus the sine of a times the sine of b times the cosine of the angle C, for a spherical triangle.", "kind": "law", "symbols": [ { "unit": null, "symbol": "c", "meaning": "side of the spherical triangle opposite the angle C" }, { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle opposite the angle A" }, { "unit": null, "symbol": "b", "meaning": "side of the spherical triangle opposite the angle B" }, { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle at the vertex between sides a and b" } ], "sympy": "Eq(cos(c), cos(a)*cos(b) + sin(a)*sin(b)*cos(C))", "physics": false, "states": [ "law/spherical-law-of-cosines-for-sides" ], "concepts": [ "concept/cosine", "concept/side", "concept/spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-31ee1ad3c2", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 104", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta a \\,\\cos B + \\delta b \\,\\cos A = 0", "name": null, "statement": "The small variations of the sides a and b, when C and c are constant, satisfy this relation with the cosines of the opposite angles B and A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta a", "meaning": "small variation of the side a" }, { "unit": null, "symbol": "\\delta b", "meaning": "small variation of the side b" }, { "unit": null, "symbol": "A", "meaning": "angle opposite the side a" }, { "unit": null, "symbol": "B", "meaning": "angle opposite the side b" } ], "sympy": "Eq(da*cos(B) + db*cos(A), 0)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/small-variation", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-768a831f1c", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 104", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta A \\,\\cos b + \\delta B \\,\\cos a = 0", "name": null, "statement": "By the polar triangle, the small variations of the angles A and B are connected by the cosines of the sides b and a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta A", "meaning": "small variation of the angle A" }, { "unit": null, "symbol": "\\delta B", "meaning": "small variation of the angle B" }, { "unit": null, "symbol": "a", "meaning": "side opposite the angle A" }, { "unit": null, "symbol": "b", "meaning": "side opposite the angle B" } ], "sympy": "Eq(dA*cos(b) + dB*cos(a), 0)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/polar-triangle", "concept/side", "concept/small-variation", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-16b4d8c2b3", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 103", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\cos (a + \\delta a) = \\cos a - \\sin a \\,\\delta a", "name": null, "statement": "To first order in the small increment, the cosine of a side increased by delta a is the cosine minus sine times delta a.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "\\delta a", "meaning": "small increment of the side a" }, { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle" } ], "sympy": "Eq(cos(a + da), cos(a) - sin(a)*da)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/small-variation" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-61e2903576", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 103", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\sin (a + \\delta a) = \\sin a + \\cos a \\,\\delta a", "name": null, "statement": "To first order in the small increment, the sine of a side increased by delta a is the sine plus cosine times delta a.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "\\delta a", "meaning": "small increment of the side a" }, { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle" } ], "sympy": "Eq(sin(a + da), sin(a) + cos(a)*da)", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "concept/small-variation" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-0b08793202", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 104", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\sin A \\sin c = \\sin C \\,\\sin a", "name": "sine rule for spherical triangles", "statement": "The sines of the angles of a spherical triangle are proportional to the sines of the opposite sides.", "kind": "law", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle opposite the side a" }, { "unit": null, "symbol": "c", "meaning": "side opposite the angle C" }, { "unit": null, "symbol": "C", "meaning": "angle opposite the side c" }, { "unit": null, "symbol": "a", "meaning": "side opposite the angle A" } ], "sympy": "Eq(sin(A)*sin(c), sin(C)*sin(a))", "physics": false, "states": [ "law/sine-rule-for-spherical-triangles" ], "concepts": [ "concept/side", "concept/sine", "concept/spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-bc495b38a7", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 104", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta A \\cot A = \\delta a \\cot a", "name": null, "statement": "For constant side c and opposite angle C, the small variations of the side a and its opposite angle A are related through their cotangents.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta A", "meaning": "small variation of the angle A" }, { "unit": null, "symbol": "\\delta a", "meaning": "small variation of the side a" }, { "unit": null, "symbol": "A", "meaning": "angle opposite the side a" }, { "unit": null, "symbol": "a", "meaning": "side opposite the angle A" } ], "sympy": "Eq(dA*cot(A), da*cot(a))", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/side", "concept/small-variation", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ee63d785e7", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\cot C \\sin B = \\cot c \\sin a - \\cos B \\cos a", "name": null, "statement": "A four-part relation among the sides c, a and the angles C, B of a spherical triangle, involving cotangents, sines and cosines.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle opposite the side c" }, { "unit": null, "symbol": "B", "meaning": "angle opposite the side b" }, { "unit": null, "symbol": "c", "meaning": "side opposite the angle C" }, { "unit": null, "symbol": "a", "meaning": "side opposite the angle A" } ], "sympy": "Eq(cot(C)*sin(B), cot(c)*sin(a) - cos(B)*cos(a))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cotangent", "concept/side", "concept/sine", "concept/spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-260df06816", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta B \\cos A = - \\delta a \\cot b \\sin B", "name": null, "statement": "With C and c constant, the small variation of the angle B is tied to that of the side a by cosine of A and cotangent of b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta B", "meaning": "small variation of the angle B" }, { "unit": null, "symbol": "\\delta a", "meaning": "small variation of the side a" }, { "unit": null, "symbol": "A", "meaning": "angle opposite the side a" }, { "unit": null, "symbol": "b", "meaning": "side opposite the angle B" }, { "unit": null, "symbol": "B", "meaning": "angle opposite the side b" } ], "sympy": "Eq(dB*cos(A), -da*cot(b)*sin(B))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cotangent", "concept/side", "concept/sine", "concept/small-variation", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-df708001e4", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "-\\frac{\\cos A}{\\sin C} \\delta B = \\frac{\\cos b}{\\sin c} \\delta a", "name": null, "statement": "The intermediate relation between the small variations of the angle B and the side a, with the sines and cosines of the triangle's parts, from which the result is obtained.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta B", "meaning": "small variation of the angle B" }, { "unit": null, "symbol": "\\delta a", "meaning": "small variation of the side a" }, { "unit": null, "symbol": "A", "meaning": "angle opposite the side a" }, { "unit": null, "symbol": "C", "meaning": "angle opposite the side c" }, { "unit": null, "symbol": "b", "meaning": "side opposite the angle B" }, { "unit": null, "symbol": "c", "meaning": "side opposite the angle C" } ], "sympy": "Eq(-cos(A)/sin(C)*dB, cos(b)/sin(c)*da)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/small-variation", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-33c75a520f", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\frac{\\delta a}{\\surd{(1 - n^2 \\sin^2 a)}} + \\frac{\\delta b}{\\surd{(1 - n^2 \\sin^2 b)}} = 0", "name": null, "statement": "Example 1: if C and c are constant, the small increments of a and b, scaled by the square roots involving n, sum to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta a", "meaning": "small increment of the side a" }, { "unit": null, "symbol": "\\delta b", "meaning": "small increment of the side b" }, { "unit": null, "symbol": "n", "meaning": "ratio sin C / sin c" }, { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle" }, { "unit": null, "symbol": "b", "meaning": "side of the spherical triangle" } ], "sympy": "Eq(da/sqrt(1 - n**2*sin(a)**2) + db/sqrt(1 - n**2*sin(b)**2), 0)", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "concept/small-variation", "concept/surd" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-22071d4562", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "n = \\frac{\\sin C}{\\sin c}\\,", "name": null, "statement": "Definition of n as the ratio of sin C to sin c, used in Example 1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "n", "meaning": "ratio sin C / sin c" }, { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle" }, { "unit": null, "symbol": "c", "meaning": "side of the spherical triangle" } ], "sympy": "Eq(n, sin(C)/sin(c))", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/sine" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-66bd77ebfc", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\sin C \\delta b = \\sin a \\delta B", "name": null, "statement": "Example 3: with A and c constant, the small variations of b and B are related through sin C and sin a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta b", "meaning": "small variation of the side b" }, { "unit": null, "symbol": "\\delta B", "meaning": "small variation of the angle B" }, { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle" }, { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle" } ], "sympy": "Eq(sin(C)*db, sin(a)*dB)", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "concept/small-variation", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-2e09b66a59", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta b \\sin C = -\\delta C \\tan a", "name": null, "statement": "Example 3: with A and c constant, the small variations of b and C are related through sin C and tan a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta b", "meaning": "small variation of the side b" }, { "unit": null, "symbol": "\\delta C", "meaning": "small variation of the angle C" }, { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle" }, { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle" } ], "sympy": "Eq(db*sin(C), -dC*tan(a))", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "concept/small-variation", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-c60b71bab7", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta a \\tan C = \\delta B \\sin a", "name": null, "statement": "Example 3: with A and c constant, the small variations of a and B are related through tan C and sin a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta a", "meaning": "small variation of the side a" }, { "unit": null, "symbol": "\\delta B", "meaning": "small variation of the angle B" }, { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle" }, { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle" } ], "sympy": "Eq(da*tan(C), dB*sin(a))", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "concept/small-variation", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f54a77482f", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta a \\tan C = -\\delta C \\tan a", "name": null, "statement": "Example 3: with A and c constant, the small variations of a and C are related through tan C and tan a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta a", "meaning": "small variation of the side a" }, { "unit": null, "symbol": "\\delta C", "meaning": "small variation of the angle C" }, { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle" }, { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle" } ], "sympy": "Eq(da*tan(C), -dC*tan(a))", "physics": false, "states": [], "concepts": [ "concept/side", "concept/small-variation", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-6e45612880", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta b \\cos C = \\delta a", "name": null, "statement": "Example 3: with A and c constant, the small variations of b and a are related through cos C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta b", "meaning": "small variation of the side b" }, { "unit": null, "symbol": "\\delta a", "meaning": "small variation of the side a" }, { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(db*cos(C), da)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/small-variation", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-2697c6b93d", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta B \\cos a = -\\delta C", "name": null, "statement": "Example 3: with A and c constant, the small variations of B and C are related through cos a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta B", "meaning": "small variation of the angle B" }, { "unit": null, "symbol": "\\delta C", "meaning": "small variation of the angle C" }, { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle" } ], "sympy": "Eq(dB*cos(a), -dC)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/small-variation", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-c8ef95b0b4", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta B \\tan C = \\delta C \\tan B", "name": null, "statement": "Example 4: with b and c constant, the small variations of B and C are related through their tangents.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta B", "meaning": "small variation of the angle B" }, { "unit": null, "symbol": "\\delta C", "meaning": "small variation of the angle C" }, { "unit": null, "symbol": "B", "meaning": "angle of the spherical triangle" }, { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(dB*tan(C), dC*tan(B))", "physics": false, "states": [], "concepts": [ "concept/small-variation", "concept/tangent-function", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-6e3a6a4ea2", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta a \\cot C = -\\delta B \\sin a", "name": null, "statement": "Example 4: with b and c constant, the small variations of a and B are related through cot C and sin a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta a", "meaning": "small variation of the side a" }, { "unit": null, "symbol": "\\delta B", "meaning": "small variation of the angle B" }, { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle" }, { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle" } ], "sympy": "Eq(da*cot(C), -dB*sin(a))", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/side", "concept/sine", "concept/small-variation", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-5c59a9ea1e", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta a = \\delta A \\sin c \\sin B", "name": null, "statement": "Example 4: with b and c constant, the small variation of a equals that of A times sin c sin B.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta a", "meaning": "small variation of the side a" }, { "unit": null, "symbol": "\\delta A", "meaning": "small variation of the angle A" }, { "unit": null, "symbol": "c", "meaning": "side of the spherical triangle" }, { "unit": null, "symbol": "B", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(da, dA*sin(c)*sin(B))", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "concept/small-variation", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-b620e7d2fe", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 105", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta A \\sin B \\cos C = -\\delta B \\sin A", "name": null, "statement": "Example 4: with b and c constant, the small variations of A and B are related through sines and the cosine of C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta A", "meaning": "small variation of the angle A" }, { "unit": null, "symbol": "\\delta B", "meaning": "small variation of the angle B" }, { "unit": null, "symbol": "A", "meaning": "angle of the spherical triangle" }, { "unit": null, "symbol": "B", "meaning": "angle of the spherical triangle" }, { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(dA*sin(B)*cos(C), -dB*sin(A))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/sine", "concept/small-variation", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-dd3b1e5e9a", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 106", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta b \\tan c = \\delta c \\tan b", "name": null, "statement": "Example 5: with B and C constant, the small variations of b and c are related through their tangents.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta b", "meaning": "small variation of the side b" }, { "unit": null, "symbol": "\\delta c", "meaning": "small variation of the side c" }, { "unit": null, "symbol": "b", "meaning": "side of the spherical triangle" }, { "unit": null, "symbol": "c", "meaning": "side of the spherical triangle" } ], "sympy": "Eq(db*tan(c), dc*tan(b))", "physics": false, "states": [], "concepts": [ "concept/side", "concept/small-variation", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-3bab42ac05", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 106", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta A \\cot c = \\delta b \\sin A", "name": null, "statement": "Example 5: with B and C constant, the small variations of A and b are related through cot c and sin A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta A", "meaning": "small variation of the angle A" }, { "unit": null, "symbol": "\\delta b", "meaning": "small variation of the side b" }, { "unit": null, "symbol": "c", "meaning": "side of the spherical triangle" }, { "unit": null, "symbol": "A", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(dA*cot(c), db*sin(A))", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/side", "concept/sine", "concept/small-variation", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-8772bfd74d", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 106", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta A = \\delta a \\sin b \\sin C", "name": null, "statement": "Example 5: with B and C constant, the small variation of A equals that of a times sin b sin C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta A", "meaning": "small variation of the angle A" }, { "unit": null, "symbol": "\\delta a", "meaning": "small variation of the side a" }, { "unit": null, "symbol": "b", "meaning": "side of the spherical triangle" }, { "unit": null, "symbol": "C", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(dA, da*sin(b)*sin(C))", "physics": false, "states": [], "concepts": [ "concept/side", "concept/sine", "concept/small-variation", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-97c62aeed0", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 106", "location": "On small variations in the parts of a Spherical Triangle", "latex": "\\delta a \\sin B \\cos c = \\delta b \\sin A", "name": null, "statement": "Example 5: with B and C constant, the small variations of a and b are related through sines and the cosine of c.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\delta a", "meaning": "small variation of the side a" }, { "unit": null, "symbol": "\\delta b", "meaning": "small variation of the side b" }, { "unit": null, "symbol": "B", "meaning": "angle of the spherical triangle" }, { "unit": null, "symbol": "A", "meaning": "angle of the spherical triangle" }, { "unit": null, "symbol": "c", "meaning": "side of the spherical triangle" } ], "sympy": "Eq(da*sin(B)*cos(c), db*sin(A))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/small-variation", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-8f42d22be2", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 112", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\cos(\\rho+r_1) = \\cos\\alpha_1\\cos\\beta + \\sin\\alpha_1\\sin\\beta\\cos\\gamma", "name": null, "statement": "Condition that the touching circle touches the escribed circle of angular radius r1 externally.", "kind": "law", "symbols": [ { "unit": null, "symbol": "rho", "meaning": "angular radius of the touching circle" }, { "unit": null, "symbol": "r_1", "meaning": "angular radius of the escribed circle" }, { "unit": null, "symbol": "alpha_1", "meaning": "distance from A of the pole of that escribed circle" }, { "unit": null, "symbol": "beta", "meaning": "distance from A of the pole of the touching circle" }, { "unit": null, "symbol": "gamma", "meaning": "angle at A between the arcs to the poles" } ], "sympy": "Eq(cos(rho + r_1), cos(alpha_1)*cos(beta) + sin(alpha_1)*sin(beta)*cos(gamma))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/cosine", "concept/escribed-circle", "concept/tangent" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-90526d7351", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 107", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\cos A = \\frac{\\cos a - \\cos b \\cos c }{\\sin b \\sin c}", "name": null, "statement": "Spherical cosine rule for angle A, quoted as the starting formula from which the plane result is deduced by letting the sphere's radius grow without limit.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "A", "meaning": "spherical angle of the triangle at A" }, { "unit": null, "symbol": "a", "meaning": "side of the spherical triangle opposite A (an arc)" }, { "unit": null, "symbol": "b", "meaning": "side of the spherical triangle opposite B (an arc)" }, { "unit": null, "symbol": "c", "meaning": "side of the spherical triangle opposite C (an arc)" } ], "sympy": "Eq(cos(A), (cos(a) - cos(b)*cos(c))/(sin(b)*sin(c)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/formula", "concept/side", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-85a23ecd78", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 107", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\cos A = \\frac{\\beta^2 + \\gamma^2 - \\alpha^2}{2 \\beta \\gamma}\\,", "name": null, "statement": "Cosine of an angle of a plane triangle in terms of its three sides, obtained as the limit of the spherical formula when the radius becomes infinite.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle of the plane triangle" }, { "unit": null, "symbol": "alpha", "meaning": "length of the side opposite A" }, { "unit": null, "symbol": "beta", "meaning": "length of the side opposite B" }, { "unit": null, "symbol": "gamma", "meaning": "length of the side opposite C" } ], "sympy": "Eq(cos(A), (beta**2 + gamma**2 - alpha**2)/(2*beta*gamma))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/limiting-case", "concept/plane-triangle", "concept/side", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-69c081ae40", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 107", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\frac{\\sin A}{\\sin B} = \\frac{\\sin a}{\\sin b}", "name": "law of sines (spherical)", "statement": "On a sphere the sines of the sides of a spherical triangle are proportional to the sines of the opposite angles; quoted as the starting formula.", "kind": "law", "symbols": [ { "unit": null, "symbol": "A", "meaning": "spherical angle at A" }, { "unit": null, "symbol": "B", "meaning": "spherical angle at B" }, { "unit": null, "symbol": "a", "meaning": "side opposite A (an arc)" }, { "unit": null, "symbol": "b", "meaning": "side opposite B (an arc)" } ], "sympy": "Eq(sin(A)/sin(B), sin(a)/sin(b))", "physics": false, "states": [ "law/law-of-sines-spherical" ], "concepts": [ "concept/formula", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-76fb85a5cc", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 107", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\frac{\\sin A}{\\sin B} = \\frac{\\alpha}{\\beta}", "name": "law of sines (plane)", "statement": "In a plane triangle the sides are as the sines of the opposite angles, the limiting form of the spherical law as the radius becomes infinite.", "kind": "law", "symbols": [ { "unit": null, "symbol": "A", "meaning": "angle of the plane triangle at A" }, { "unit": null, "symbol": "B", "meaning": "angle of the plane triangle at B" }, { "unit": null, "symbol": "alpha", "meaning": "length of the side opposite A" }, { "unit": null, "symbol": "beta", "meaning": "length of the side opposite B" } ], "sympy": "Eq(sin(A)/sin(B), alpha/beta)", "physics": false, "states": [ "law/law-of-sines-plane" ], "concepts": [ "concept/limiting-case", "concept/plane-triangle", "concept/side", "concept/sine", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-be7f1476db", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 108", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\cos r = \\cos \\alpha \\cos \\theta + \\sin \\alpha \\sin \\theta \\cos (\\phi - \\beta)", "name": null, "statement": "Equation of a small circle of angular radius r on the sphere, in angular co-ordinates of its pole and of a general point of the circle.", "kind": "law", "symbols": [ { "unit": null, "symbol": "r", "meaning": "angular radius of the small circle (OP)" }, { "unit": null, "symbol": "alpha", "meaning": "distance OS from the fixed point S to the pole O" }, { "unit": null, "symbol": "beta", "meaning": "angle OSX fixing the pole's longitude" }, { "unit": null, "symbol": "theta", "meaning": "distance PS of a point P on the circle from S" }, { "unit": null, "symbol": "phi", "meaning": "angle PSX of point P from the fixed great circle" } ], "sympy": "Eq(cos(r), cos(alpha)*cos(theta) + sin(alpha)*sin(theta)*cos(phi - beta))", "physics": false, "states": [], "concepts": [ "concept/angular-co-ordinates-on-a-sphere", "concept/angular-radius", "concept/cosine", "concept/small-circle", "concept/sphere" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-8d706c5694", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 108", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "0 = \\cos \\alpha \\cos \\theta + \\sin \\alpha \\sin \\theta \\cos (\\phi - \\beta)", "name": null, "statement": "Equation of a great circle on the sphere, the special case r = pi/2 of the small circle equation.", "kind": "law", "symbols": [ { "unit": null, "symbol": "alpha", "meaning": "distance OS from S to the pole O" }, { "unit": null, "symbol": "beta", "meaning": "angle OSX" }, { "unit": null, "symbol": "theta", "meaning": "distance PS of a point P from S" }, { "unit": null, "symbol": "phi", "meaning": "angle PSX of point P" } ], "sympy": "Eq(0, cos(alpha)*cos(theta) + sin(alpha)*sin(theta)*cos(phi - beta))", "physics": false, "states": [], "concepts": [ "concept/angular-co-ordinates-on-a-sphere", "concept/circle", "concept/cosine", "concept/great-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-e593165e50", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 108", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\tan \\frac{\\theta_1}{2} \\tan \\frac{\\theta_2}{2}= \\frac{\\cos r - \\cos \\alpha}{\\cos r + \\cos \\alpha}", "name": null, "statement": "The product of the two roots of the quadratic in tan(theta/2) is independent of phi, the spherical analogue of an Euclid III.36 property.", "kind": "result", "symbols": [ { "unit": null, "symbol": "theta_1", "meaning": "first root angle of the quadratic (distance from S)" }, { "unit": null, "symbol": "theta_2", "meaning": "second root angle of the quadratic" }, { "unit": null, "symbol": "r", "meaning": "angular radius of the small circle" }, { "unit": null, "symbol": "alpha", "meaning": "distance of the pole from S" } ], "sympy": "Eq(tan(theta_1/2)*tan(theta_2/2), (cos(r) - cos(alpha))/(cos(r) + cos(alpha)))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/circle", "concept/cosine", "concept/quadratic-equation", "concept/small-circle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ae226de395", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 108", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\frac{\\cos r - \\cos \\alpha}{\\cos r + \\cos \\alpha} = \\tan \\frac{\\alpha + r}{2} \\tan \\frac{\\alpha - r}{2}", "name": null, "statement": "The cosine ratio equals the product of tangents of half-sums and half-differences of alpha and r.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "r", "meaning": "angular radius" }, { "unit": null, "symbol": "alpha", "meaning": "distance of the pole from S" } ], "sympy": "Eq((cos(r) - cos(alpha))/(cos(r) + cos(alpha)), tan((alpha + r)/2)*tan((alpha - r)/2))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-cfe0db7585", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 109", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\sin PM = \\sin OP \\sin AOB", "name": null, "statement": "Perpendicular from P to the arc OA is found from the sine of OP times the sine of the angle AOB (right-angled triangle relation, Art. 65).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "PM", "meaning": "arc perpendicular from P to OA" }, { "unit": null, "symbol": "OP", "meaning": "arc from O to P" }, { "unit": null, "symbol": "AOB", "meaning": "angle between arcs OA and OB" } ], "sympy": "Eq(sin(PM), sin(OP)*sin(AOB))", "physics": false, "states": [], "concepts": [ "concept/perpendicular", "concept/sine", "concept/spherical-angle", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-349dd6820f", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 109", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\frac{\\sin PM}{\\sin PN} = \\frac{\\sin AOB}{\\sin COB}", "name": null, "statement": "The ratio of the sines of the perpendiculars from any point P on OB is independent of P's position.", "kind": "result", "symbols": [ { "unit": null, "symbol": "PM", "meaning": "arc perpendicular from P to OA" }, { "unit": null, "symbol": "PN", "meaning": "arc perpendicular from P to OC" }, { "unit": null, "symbol": "AOB", "meaning": "angle between OA and OB" }, { "unit": null, "symbol": "COB", "meaning": "angle between OC and OB" } ], "sympy": "Eq(sin(PM)/sin(PN), sin(AOB)/sin(COB))", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/perpendicular", "concept/sine", "concept/spherical-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-4b0def777d", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 109", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\sin x = \\frac{\\sin \\theta_2}{\\sin(\\theta_1 + \\theta_2)} \\sin x_1 + \\frac{\\sin \\theta_1}{\\sin(\\theta_1 + \\theta_2)} \\sin x_2", "name": null, "statement": "Three-point relation: the sine of the perpendicular from P is a weighted sum of the sines of the perpendiculars from P1 and P2 to the fixed arc.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "perpendicular from P to the fixed arc" }, { "unit": null, "symbol": "x_1", "meaning": "perpendicular from P1 to the fixed arc" }, { "unit": null, "symbol": "x_2", "meaning": "perpendicular from P2 to the fixed arc" }, { "unit": null, "symbol": "theta_1", "meaning": "arc PP1" }, { "unit": null, "symbol": "theta_2", "meaning": "arc PP2" } ], "sympy": "Eq(sin(x), sin(theta_2)/sin(theta_1 + theta_2)*sin(x_1) + sin(theta_1)/sin(theta_1 + theta_2)*sin(x_2))", "physics": false, "states": [], "concepts": [ "concept/great-circle", "concept/perpendicular", "concept/sine", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-fb1c5f1a1d", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 110", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\cos B = \\cos CF \\sin FCB", "name": null, "statement": "Cosine of an angle of the triangle equals the cosine of the perpendicular CF times the sine of the angle FCB (Art. 65).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "B", "meaning": "spherical angle at B" }, { "unit": null, "symbol": "CF", "meaning": "arc of the perpendicular from C to AB" }, { "unit": null, "symbol": "FCB", "meaning": "angle at C between CF and CB" } ], "sympy": "Eq(cos(B), cos(CF)*sin(FCB))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/perpendicular", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-661c96c233", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 111", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\frac{\\sin x}{\\cos B \\cos C} = \\frac{\\sin y}{\\cos C \\cos A} = \\frac{\\sin z}{\\cos A \\cos B}", "name": null, "statement": "The three perpendiculars from the point of concurrence of the altitudes, divided by the products of cosines of the angles, are equal; this fixes the point where the perpendiculars meet.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "perpendicular from the concurrence point to side a" }, { "unit": null, "symbol": "y", "meaning": "perpendicular from the concurrence point to side b" }, { "unit": null, "symbol": "z", "meaning": "perpendicular from the concurrence point to side c" }, { "unit": null, "symbol": "A", "meaning": "spherical angle at A" }, { "unit": null, "symbol": "B", "meaning": "spherical angle at B" }, { "unit": null, "symbol": "C", "meaning": "spherical angle at C" } ], "sympy": "Eq(sin(x)/(cos(B)*cos(C)), sin(y)/(cos(C)*cos(A)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/perpendicular", "concept/spherical-angle", "concept/spherical-triangle", "theorem/concurrence-of-altitudes" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ae89c15e6c", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 111", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\frac{\\sin x}{\\sin B \\sin C} = \\frac{\\sin y}{\\sin C \\sin A} = \\frac{\\sin z}{\\sin A \\sin B}", "name": null, "statement": "For the point where the arcs to the midpoints of the opposite sides meet, the perpendiculars divided by sine-products of the angles are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "perpendicular from the point to side a" }, { "unit": null, "symbol": "y", "meaning": "perpendicular from the point to side b" }, { "unit": null, "symbol": "z", "meaning": "perpendicular from the point to side c" }, { "unit": null, "symbol": "A", "meaning": "spherical angle at A" }, { "unit": null, "symbol": "B", "meaning": "spherical angle at B" }, { "unit": null, "symbol": "C", "meaning": "spherical angle at C" } ], "sympy": "Eq(sin(x)/(sin(B)*sin(C)), sin(y)/(sin(C)*sin(A)))", "physics": false, "states": [], "concepts": [ "concept/median", "concept/perpendicular", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-6da907218a", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 112", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\cos(\\rho-r) = \\cos\\alpha\\cos\\beta + \\sin\\alpha\\sin\\beta\\cos\\gamma", "name": null, "statement": "Condition that the touching circle of angular radius rho touches the inscribed circle internally.", "kind": "law", "symbols": [ { "unit": null, "symbol": "rho", "meaning": "angular radius of the touching circle" }, { "unit": null, "symbol": "r", "meaning": "angular radius of the inscribed circle" }, { "unit": null, "symbol": "alpha", "meaning": "distance from A of the pole of the inscribed circle" }, { "unit": null, "symbol": "beta", "meaning": "distance from A of the pole of the touching circle" }, { "unit": null, "symbol": "gamma", "meaning": "angle at A between the arcs to the two poles" } ], "sympy": "Eq(cos(rho - r), cos(alpha)*cos(beta) + sin(alpha)*sin(beta)*cos(gamma))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/cosine", "concept/inscribed-circle", "concept/small-circle", "concept/tangent" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-5216e40cfd", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 112", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\cos(\\rho+r_2) = \\cos\\alpha_2\\cos\\beta + \\sin\\alpha_2\\sin\\beta \\cos\\left(\\frac{\\pi}{2}-\\gamma\\right)", "name": null, "statement": "Condition that the touching circle touches the second escribed circle externally, with the angle replaced by pi/2 minus gamma.", "kind": "law", "symbols": [ { "unit": null, "symbol": "rho", "meaning": "angular radius of the touching circle" }, { "unit": null, "symbol": "r_2", "meaning": "angular radius of the second escribed circle" }, { "unit": null, "symbol": "alpha_2", "meaning": "distance from A of its pole" }, { "unit": null, "symbol": "beta", "meaning": "distance from A of the pole of the touching circle" }, { "unit": null, "symbol": "gamma", "meaning": "angle between the arcs to the poles" } ], "sympy": "Eq(cos(rho + r_2), cos(alpha_2)*cos(beta) + sin(alpha_2)*sin(beta)*cos(pi/2 - gamma))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/cosine", "concept/escribed-circle", "quantity/pi" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ca9cd2c245", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 112", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\cos(\\rho+r_3) = \\cos\\alpha_3\\cos\\beta + \\sin\\alpha_3\\sin\\beta \\cos\\left(\\frac{\\pi}{2}+\\gamma\\right)", "name": null, "statement": "Condition that the touching circle touches the third escribed circle externally, with the angle replaced by pi/2 plus gamma.", "kind": "law", "symbols": [ { "unit": null, "symbol": "rho", "meaning": "angular radius of the touching circle" }, { "unit": null, "symbol": "r_3", "meaning": "angular radius of the third escribed circle" }, { "unit": null, "symbol": "alpha_3", "meaning": "distance from A of its pole" }, { "unit": null, "symbol": "beta", "meaning": "distance from A of the pole of the touching circle" }, { "unit": null, "symbol": "gamma", "meaning": "angle between the arcs to the poles" } ], "sympy": "Eq(cos(rho + r_3), cos(alpha_3)*cos(beta) + sin(alpha_3)*sin(beta)*cos(pi/2 + gamma))", "physics": false, "states": [], "concepts": [ "concept/angular-radius", "concept/cosine", "concept/escribed-circle", "quantity/pi" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-4a37097765", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 113", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "2\\cos\\rho \\sin\\frac{a}{2}\\cos\\frac{b+c}{2} + 2n\\sin\\rho=\\cos\\beta\\sin a", "name": null, "statement": "Elimination of cos gamma between the first two touching conditions gives this relation between rho, beta and the sides.", "kind": "result", "symbols": [ { "unit": null, "symbol": "rho", "meaning": "angular radius of the touching circle" }, { "unit": null, "symbol": "beta", "meaning": "distance from A of the pole of the touching circle" }, { "unit": null, "symbol": "a", "meaning": "side BC of the triangle" }, { "unit": null, "symbol": "b", "meaning": "side CA of the triangle" }, { "unit": null, "symbol": "c", "meaning": "side AB of the triangle" }, { "unit": null, "symbol": "n", "meaning": "quantity defined in Art. 46" } ], "sympy": "Eq(2*cos(rho)*sin(a/2)*cos((b + c)/2) + 2*n*sin(rho), cos(beta)*sin(a))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/small-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-39ddfa0aba", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 113", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "2\\cos\\rho\\sin\\frac{a}{2}\\cos\\frac{b-c}{2}-2n\\sin\\rho=\\cos\\beta\\sin a", "name": null, "statement": "Companion relation from the elimination of sin gamma, with b - c in place of b + c and the sign of the n term reversed.", "kind": "result", "symbols": [ { "unit": null, "symbol": "rho", "meaning": "angular radius of the touching circle" }, { "unit": null, "symbol": "beta", "meaning": "distance from A of the pole of the touching circle" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" }, { "unit": null, "symbol": "n", "meaning": "quantity defined in Art. 46" } ], "sympy": "Eq(2*cos(rho)*sin(a/2)*cos((b - c)/2) - 2*n*sin(rho), cos(beta)*sin(a))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/sine" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ec40b21ace", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 113", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\tan\\rho= \\frac{\\sin\\dfrac{a}{2}\\sin\\dfrac{b}{2}\\sin\\dfrac{c}{2}}{n} = \\frac{1}{2}\\tan R", "name": null, "statement": "The angular radius of the touching circle is determined by tan rho, which is half tan R (the circumradius relation of Art. 92).", "kind": "result", "symbols": [ { "unit": null, "symbol": "rho", "meaning": "angular radius of the touching circle" }, { "unit": null, "symbol": "R", "meaning": "radius of the circumscribing circle of the triangle" }, { "unit": null, "symbol": "n", "meaning": "quantity defined in Art. 46" } ], "sympy": "Eq(tan(rho), sin(a/2)*sin(b/2)*sin(c/2)/n)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/radius", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f514b47875", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 113", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\cos\\beta= \\dfrac{\\cos\\dfrac{b}{2}\\cos\\dfrac{c}{2}\\cos\\rho}{\\cos\\dfrac{a}{2}}", "name": null, "statement": "The distance beta of the touching circle's pole from A is fixed by cos beta.", "kind": "result", "symbols": [ { "unit": null, "symbol": "beta", "meaning": "distance from A of the pole of the touching circle" }, { "unit": null, "symbol": "rho", "meaning": "angular radius of the touching circle" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(cos(beta), cos(b/2)*cos(c/2)*cos(rho)/cos(a/2))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/pole-of-a-circle", "concept/small-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-c78ecbf863", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 117", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\tan\\frac{\\lambda}{2} \\tan\\frac{\\mu}{2} = \\frac{\\cos\\rho - \\cos\\beta}{\\cos\\rho + \\cos\\beta}", "name": null, "statement": "Product of the tangents of half the distances from A to the two points where the touching circle meets side AB (Art. 134 applied).", "kind": "result", "symbols": [ { "unit": null, "symbol": "lam", "meaning": "distance from A of one intersection point on AB" }, { "unit": null, "symbol": "mu", "meaning": "distance from A of the other intersection point on AB" }, { "unit": null, "symbol": "rho", "meaning": "angular radius of the touching circle" }, { "unit": null, "symbol": "beta", "meaning": "distance from A of the pole of the touching circle" } ], "sympy": "Eq(tan(lam/2)*tan(mu/2), (cos(rho) - cos(beta))/(cos(rho) + cos(beta)))", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/cosine", "concept/secant", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-6bf6267c57", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 118", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\tan\\frac{\\lambda}{2} = \\frac{\\cos \\dfrac{a}{2} - \\cos \\dfrac{b}{2} \\cos \\dfrac{c}{2} } {\\cos \\dfrac{b}{2} \\sin \\dfrac{c}{2} }", "name": null, "statement": "Tangent of half the distance lambda from A to the first intersection point of the touching circle with AB.", "kind": "result", "symbols": [ { "unit": null, "symbol": "lam", "meaning": "distance from A of the first intersection point on AB" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(tan(lam/2), (cos(a/2) - cos(b/2)*cos(c/2))/(cos(b/2)*sin(c/2)))", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/side", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-963e982eec", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 118", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\tan\\frac{\\mu}{2} = \\frac{\\cos \\dfrac{b}{2} \\sin \\dfrac{c}{2} } {\\cos \\dfrac{a}{2} + \\cos \\dfrac{b}{2} \\cos \\dfrac{c}{2} }", "name": null, "statement": "Tangent of half the distance mu from A to the second intersection point of the touching circle with AB.", "kind": "result", "symbols": [ { "unit": null, "symbol": "mu", "meaning": "distance from A of the second intersection point on AB" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(tan(mu/2), cos(b/2)*sin(c/2)/(cos(a/2) + cos(b/2)*cos(c/2)))", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/side", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-1476ffaf7f", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 120", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\sin z = \\frac{\\cos \\rho}{n} \\sin\\frac{a}{2} \\sin\\frac{b}{2} \\sin\\frac{c}{2} \\cos(A - B)", "name": null, "statement": "The perpendicular z from the pole of the touching circle to side AB equals sin rho times cos(A - B), written in terms of the sides.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "perpendicular from the pole of the touching circle on AB" }, { "unit": null, "symbol": "rho", "meaning": "angular radius of the touching circle" }, { "unit": null, "symbol": "n", "meaning": "quantity defined in Art. 46" }, { "unit": null, "symbol": "A", "meaning": "spherical angle at A" }, { "unit": null, "symbol": "B", "meaning": "spherical angle at B" } ], "sympy": "Eq(sin(z), cos(rho)/n*sin(a/2)*sin(b/2)*sin(c/2)*cos(A - B))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/perpendicular", "concept/sine", "concept/spherical-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-2df09c5e30", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 120", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\frac{\\sin x}{\\cos(B - C)} = \\frac{\\sin y}{\\cos(C - A)} = \\frac{\\sin z}{\\cos(A - B)}", "name": null, "statement": "Perpendiculars from the pole of the touching circle, divided by cosines of angle differences, are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "perpendicular from the pole N to side a" }, { "unit": null, "symbol": "y", "meaning": "perpendicular from N to side b" }, { "unit": null, "symbol": "z", "meaning": "perpendicular from N to side c" }, { "unit": null, "symbol": "A", "meaning": "spherical angle at A" }, { "unit": null, "symbol": "B", "meaning": "spherical angle at B" }, { "unit": null, "symbol": "C", "meaning": "spherical angle at C" } ], "sympy": "Eq(sin(x)/cos(B - C), sin(y)/cos(C - A))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/perpendicular", "concept/spherical-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-e95ecb044c", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 121", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "z = \\dfrac12 R \\cos(A - B)", "name": null, "statement": "Plane-geometry limit of the perpendicular to the sides: the Nine points circle property, obtained when the sphere's radius becomes infinite.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "perpendicular from the nine-points-circle centre to side c" }, { "unit": null, "symbol": "R", "meaning": "radius of the circumscribing circle" }, { "unit": null, "symbol": "A", "meaning": "angle at A of the plane triangle" }, { "unit": null, "symbol": "B", "meaning": "angle at B of the plane triangle" } ], "sympy": "Eq(z, R*cos(A - B)/2)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/nine-points-circle", "concept/plane-triangle", "concept/radius" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-770ab45613", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 121", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\lambda = \\dfrac{b^2 + c^2 - a^2}{2c}", "name": null, "statement": "Plane limit of equation (4): the nine-points circle passes through the feet of the altitudes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "lam", "meaning": "distance from A of the intersection point on AB (plane limit)" }, { "unit": null, "symbol": "a", "meaning": "side BC" }, { "unit": null, "symbol": "b", "meaning": "side CA" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(lam, (b**2 + c**2 - a**2)/(2*c))", "physics": false, "states": [], "concepts": [ "concept/nine-points-circle", "concept/plane-triangle", "concept/side" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-09d63ce8ae", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 121", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\mu = \\dfrac{c}{2}", "name": null, "statement": "Plane limit of equation (5): the nine-points circle passes through the midpoints of the sides.", "kind": "result", "symbols": [ { "unit": null, "symbol": "mu", "meaning": "distance from A of the second intersection point on AB (plane limit)" }, { "unit": null, "symbol": "c", "meaning": "side AB" } ], "sympy": "Eq(mu, c/2)", "physics": false, "states": [], "concepts": [ "concept/limiting-case", "concept/nine-points-circle", "concept/plane-triangle", "concept/side" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-e72d80c6db", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 118", "location": "On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry", "latex": "\\sin z = \\sin \\beta \\sin \\left(\\frac{A}{2} + \\gamma\\right)", "name": null, "statement": "The perpendicular z from the pole of the touching circle to AB, written through the angles beta and gamma (Art. 145).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "z", "meaning": "perpendicular from the pole of the touching circle on AB" }, { "unit": null, "symbol": "beta", "meaning": "distance from A of the pole of the touching circle" }, { "unit": null, "symbol": "gamma", "meaning": "angle at A between the arcs to the poles" }, { "unit": null, "symbol": "A", "meaning": "spherical angle at A" } ], "sympy": "Eq(sin(z), sin(beta)*sin(A/2 + gamma))", "physics": false, "states": [], "concepts": [ "concept/perpendicular", "concept/sine", "concept/small-circle", "concept/spherical-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-79e7ded3c6", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 124", "location": "Polyhedrons", "latex": "\\mathrm{S+F=E+2}", "name": "Euler's polyhedral formula", "statement": "In any polyhedron, the number of solid angles plus the number of faces equals the number of edges plus two.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "number of solid angles of the polyhedron" }, { "unit": null, "symbol": "F", "meaning": "number of faces of the polyhedron" }, { "unit": null, "symbol": "E", "meaning": "number of edges of the polyhedron" } ], "sympy": "Eq(S+F, E+2)", "physics": false, "states": [ "theorem/euler-s-polyhedral-formula" ], "concepts": [ "concept/edge", "concept/face", "concept/polyhedral-angle", "concept/polyhedron", "theorem/polyhedral-formula" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-0fd343d679", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 124", "location": "Polyhedrons", "latex": "r^2\\{s-(m-2)\\pi\\}", "name": null, "statement": "The area of a spherical polygon with m sides and angle sum s, on a sphere of radius r, is r squared times s minus (m-2) pi.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r", "meaning": "radius of the sphere" }, { "unit": null, "symbol": "s", "meaning": "sum of the angles of the polygon" }, { "unit": null, "symbol": "m", "meaning": "number of sides of the polygon" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/area", "concept/polygon", "concept/sphere", "quantity/pi" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-52f362833c", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 124", "location": "Polyhedrons", "latex": "4\\pi r^2", "name": null, "statement": "The surface area of a sphere of radius r is 4 pi r squared.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r", "meaning": "radius of the sphere" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/sphere", "quantity/pi", "theorem/area-of-a-sphere" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-5eac6c4aae", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 126", "location": "Polyhedrons", "latex": "2(S-2)\\pi", "name": null, "statement": "The sum of all the plane angles forming the solid angles of any polyhedron is 2(S-2) pi.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "number of solid angles of the polyhedron" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/plane-angle", "concept/polyhedral-angle", "concept/polyhedron", "quantity/pi" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-189d427919", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 125", "location": "Polyhedrons", "latex": "mF=nS=2E", "name": null, "statement": "For a regular polyhedron, the total number of plane angles counted by faces (m F), by solid angles (n S), or by edges (2E) is the same.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "number of sides in each face" }, { "unit": null, "symbol": "F", "meaning": "number of faces" }, { "unit": null, "symbol": "n", "meaning": "number of plane angles forming each solid angle" }, { "unit": null, "symbol": "S", "meaning": "number of solid angles" }, { "unit": null, "symbol": "E", "meaning": "number of edges" } ], "sympy": "Eq(m*F, n*S)", "physics": false, "states": [], "concepts": [ "concept/edge", "concept/face", "concept/plane-angle", "concept/polyhedral-angle", "concept/regular-polyhedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-20359ce215", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 125", "location": "Polyhedrons", "latex": "S = \\frac{4 m}{2(m+n)-mn}", "name": null, "statement": "The number of solid angles of a regular polyhedron is 4m divided by 2(m+n) minus mn.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "number of solid angles" }, { "unit": null, "symbol": "m", "meaning": "number of sides in each face" }, { "unit": null, "symbol": "n", "meaning": "number of plane angles forming each solid angle" } ], "sympy": "Eq(S, 4*m/(2*(m+n)-m*n))", "physics": false, "states": [], "concepts": [ "concept/polyhedral-angle", "concept/regular-polyhedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-b9a3b34cea", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 125", "location": "Polyhedrons", "latex": "E = \\frac{2mn}{2(m+n)-mn}", "name": null, "statement": "The number of edges of a regular polyhedron is 2mn divided by 2(m+n) minus mn.", "kind": "result", "symbols": [ { "unit": null, "symbol": "E", "meaning": "number of edges" }, { "unit": null, "symbol": "m", "meaning": "number of sides in each face" }, { "unit": null, "symbol": "n", "meaning": "number of plane angles forming each solid angle" } ], "sympy": "Eq(E, 2*m*n/(2*(m+n)-m*n))", "physics": false, "states": [], "concepts": [ "concept/edge", "concept/regular-polyhedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-fabebb2c1f", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 125", "location": "Polyhedrons", "latex": "F = \\frac{4 n}{2(m+n)-mn}", "name": null, "statement": "The number of faces of a regular polyhedron is 4n divided by 2(m+n) minus mn.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F", "meaning": "number of faces" }, { "unit": null, "symbol": "m", "meaning": "number of sides in each face" }, { "unit": null, "symbol": "n", "meaning": "number of plane angles forming each solid angle" } ], "sympy": "Eq(F, 4*n/(2*(m+n)-m*n))", "physics": false, "states": [], "concepts": [ "concept/face", "concept/regular-polyhedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-125ecfb5bb", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 125", "location": "Polyhedrons", "latex": "\\frac{1}{m}+\\frac{1}{n} \\text{ must be greater than } \\frac{1}{2}", "name": null, "statement": "For a regular polyhedron to exist, 1/m + 1/n must be greater than 1/2.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "m", "meaning": "number of sides in each face" }, { "unit": null, "symbol": "n", "meaning": "number of plane angles forming each solid angle" } ], "sympy": "Gt(1/m + 1/n, Rational(1,2))", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/integer", "concept/regular-polyhedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-048dde3aef", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 127", "location": "Polyhedrons", "latex": "\\sin{\\frac{I}{2}}=\\frac{\\cos \\dfrac{\\pi}{n}} {\\sin{\\dfrac{\\pi}{m}}}", "name": null, "statement": "The sine of half the inclination of two adjacent faces of a regular polyhedron equals cos(pi/n) divided by sin(pi/m).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "I", "meaning": "inclination of two adjacent faces (the angle CED)" }, { "unit": null, "symbol": "m", "meaning": "number of sides in each face" }, { "unit": null, "symbol": "n", "meaning": "number of plane angles forming each solid angle" } ], "sympy": "Eq(sin(I/2), cos(pi/n)/sin(pi/m))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/face", "concept/inclination-of-two-lines", "concept/regular-polyhedron", "concept/sine", "quantity/pi" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-1b8bbfe0ac", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 127", "location": "Polyhedrons", "latex": "CE = AE\\cot{ACE} = \\frac{a}{2}\\cot{\\frac{\\pi}{m}}", "name": null, "statement": "The perpendicular CE from the centre of a face to the midpoint E of an edge equals half the edge times the cotangent of pi/m.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "edge of the polyhedron (AB)" }, { "unit": null, "symbol": "m", "meaning": "number of sides in each face" } ], "sympy": "Eq(CE, a/2*cot(pi/m))", "physics": false, "states": [], "concepts": [ "concept/centre-of-a-regular-polygon", "concept/cotangent", "concept/edge", "quantity/pi" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ac9b3748cf", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 127", "location": "Polyhedrons", "latex": "r = CE\\tan{CEO} = CE\\tan{\\frac{I}{2}} = \\frac{a}{2}\\cot{\\frac{\\pi}{m}}\\tan{\\frac{I}{2}}", "name": null, "statement": "The radius r of the sphere inscribed in a regular polyhedron equals (a/2) cot(pi/m) tan(I/2).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r", "meaning": "radius of the inscribed sphere (OC)" }, { "unit": null, "symbol": "a", "meaning": "edge of the polyhedron (AB)" }, { "unit": null, "symbol": "m", "meaning": "number of sides in each face" }, { "unit": null, "symbol": "I", "meaning": "inclination of two adjacent faces" } ], "sympy": "Eq(r, a/2*cot(pi/m)*tan(I/2))", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/inclination-of-two-lines", "concept/radius-of-the-sphere", "concept/sphere-inscribed-in-a-polyhedron", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-fd85cd3a6f", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 127", "location": "Polyhedrons", "latex": "r = R\\cos{aOc} = R\\cot{eca}\\cot{eac} = R\\cot{\\frac{\\pi}{m}} \\cot{\\frac{\\pi}{n}}", "name": null, "statement": "The inscribed-sphere radius r equals the circumscribed-sphere radius R times cot(pi/m) times cot(pi/n).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r", "meaning": "radius of the inscribed sphere" }, { "unit": null, "symbol": "R", "meaning": "radius of the circumscribed sphere (OA)" }, { "unit": null, "symbol": "m", "meaning": "number of sides in each face" }, { "unit": null, "symbol": "n", "meaning": "number of plane angles forming each solid angle" } ], "sympy": "Eq(r, R*cot(pi/m)*cot(pi/n))", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/polyhedron-inscribed-in-a-sphere", "concept/radius-of-the-sphere", "concept/sphere-inscribed-in-a-polyhedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-121ebd5f42", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 127", "location": "Polyhedrons", "latex": "R = r\\tan{\\frac{\\pi}{m}} \\tan{\\frac{\\pi}{n}} = \\frac{a}{2} \\tan{\\frac{I}{2}} \\tan{\\frac{\\pi}{n}}", "name": null, "statement": "The circumscribed-sphere radius R of a regular polyhedron equals r tan(pi/m) tan(pi/n), or (a/2) tan(I/2) tan(pi/n).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "R", "meaning": "radius of the circumscribed sphere" }, { "unit": null, "symbol": "r", "meaning": "radius of the inscribed sphere" }, { "unit": null, "symbol": "a", "meaning": "edge of the polyhedron" }, { "unit": null, "symbol": "m", "meaning": "number of sides in each face" }, { "unit": null, "symbol": "n", "meaning": "number of plane angles forming each solid angle" }, { "unit": null, "symbol": "I", "meaning": "inclination of two adjacent faces" } ], "sympy": "Eq(R, r*tan(pi/m)*tan(pi/n))", "physics": false, "states": [], "concepts": [ "concept/polyhedron-inscribed-in-a-sphere", "concept/radius-of-the-sphere", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-732d278f02", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 127", "location": "Polyhedrons", "latex": "\\dfrac{ma^2}{4}\\cot{\\dfrac{\\pi}{m}}", "name": null, "statement": "The area of one face of a regular polyhedron with m sides and edge a is (m a^2 /4) cot(pi/m).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "m", "meaning": "number of sides in each face" }, { "unit": null, "symbol": "a", "meaning": "edge of the polyhedron" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/area", "concept/cotangent", "concept/face", "concept/regular-polygon" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-4f8a7f8e26", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 127", "location": "Polyhedrons", "latex": "\\dfrac{mFa^2}{4}\\cot{\\dfrac{\\pi}{m}}", "name": null, "statement": "The surface of a regular polyhedron equals (m F a^2/4) cot(pi/m).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "m", "meaning": "number of sides in each face" }, { "unit": null, "symbol": "F", "meaning": "number of faces" }, { "unit": null, "symbol": "a", "meaning": "edge of the polyhedron" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/area", "concept/face", "concept/regular-polyhedron", "concept/surface-of-a-polyhedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-e9ce095f9f", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 127", "location": "Polyhedrons", "latex": "\\dfrac{mFra^2}{12}\\cot{\\dfrac{\\pi}{m}}", "name": null, "statement": "The volume of a regular polyhedron equals (m F r a^2 /12) cot(pi/m), with r the radius of the inscribed sphere.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the polyhedron" }, { "unit": null, "symbol": "m", "meaning": "number of sides in each face" }, { "unit": null, "symbol": "F", "meaning": "number of faces" }, { "unit": null, "symbol": "r", "meaning": "radius of the inscribed sphere" }, { "unit": null, "symbol": "a", "meaning": "edge of the polyhedron" } ], "sympy": "Eq(V, m*F*r*a**2/12*cot(pi/m))", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/regular-polyhedron", "concept/sphere-inscribed-in-a-polyhedron", "quantity/volume" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-68cce3b6e6", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 128", "location": "Polyhedrons", "latex": "= abc\\surd{( 1 - \\cos^2{\\alpha} - \\cos^2{\\beta} - \\cos^2{\\gamma} + 2\\cos{\\alpha} \\cos{\\beta} \\cos{\\gamma} )}", "name": null, "statement": "The volume of a parallelepiped with edges a, b, c from one vertex and inclinations alpha, beta, gamma is abc times the square root of 1 minus the squared cosines plus twice the product of the cosines.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the parallelepiped" }, { "unit": null, "symbol": "a", "meaning": "edge OA" }, { "unit": null, "symbol": "b", "meaning": "edge OB" }, { "unit": null, "symbol": "c", "meaning": "edge OC" }, { "unit": null, "symbol": "alpha", "meaning": "inclination of the edges OB and OC (angle BOC)" }, { "unit": null, "symbol": "beta", "meaning": "inclination of the edges OC and OA (angle COA)" }, { "unit": null, "symbol": "gamma", "meaning": "inclination of the edges OA and OB (angle AOB)" } ], "sympy": "Eq(V, a*b*c*sqrt(1 - cos(alpha)**2 - cos(beta)**2 - cos(gamma)**2 + 2*cos(alpha)*cos(beta)*cos(gamma)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/edge", "concept/inclination-of-two-lines", "concept/parallelepiped", "theorem/volume-of-a-parallelepiped" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-64f572916f", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 129", "location": "Polyhedrons", "latex": "OD^2 = a^2+b^2+c^2 + 2ab\\cos{\\gamma} + 2bc\\cos{\\alpha} + 2ca\\cos{\\beta}", "name": null, "statement": "The squared length of the diagonal OD of a parallelepiped equals the sum of the squares of the three edges plus twice each product of two edges times the cosine of their inclination.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "OD", "meaning": "diagonal of the parallelepiped from vertex O" }, { "unit": null, "symbol": "a", "meaning": "edge OA" }, { "unit": null, "symbol": "b", "meaning": "edge OB" }, { "unit": null, "symbol": "c", "meaning": "edge OC" }, { "unit": null, "symbol": "alpha", "meaning": "inclination BOC" }, { "unit": null, "symbol": "beta", "meaning": "inclination COA" }, { "unit": null, "symbol": "gamma", "meaning": "inclination AOB" } ], "sympy": "Eq(OD**2, a**2+b**2+c**2+2*a*b*cos(gamma)+2*b*c*cos(alpha)+2*c*a*cos(beta))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/diagonal", "concept/edge", "concept/inclination-of-two-lines", "concept/parallelepiped" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-a8bb94e492", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 130", "location": "Polyhedrons", "latex": "144\\hspace{3pt}V^2=2a^6", "name": null, "statement": "For a regular tetrahedron of edge a, 144 V squared equals 2 a to the sixth power.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the tetrahedron" }, { "unit": null, "symbol": "a", "meaning": "edge of the regular tetrahedron" } ], "sympy": "Eq(144*V**2, 2*a**6)", "physics": false, "states": [], "concepts": [ "concept/regular-polyhedron", "concept/tetrahedron", "theorem/volume-of-a-tetrahedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-b05425a14d", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 130", "location": "Polyhedrons", "latex": "144\\hspace{3pt}V^2 = -a'^2 b'^2 c'^2 + a^2a'^2 (b'^2+c'^2-a'^2) + b^2b'^2 (c'^2+a'^2-b'^2) + c^2c'^2 (a'^2+b'^2-c'^2) - a'^2 (a^2-b^2) (a^2-c^2) - b'^2 (b^2-c^2) (b^2-a^2) - c'^2 (c^2-a^2) (c^2-b^2)", "name": null, "statement": "The volume of a tetrahedron is expressed in terms of its six edges by this relation, with a, b, c the three edges at one vertex and a', b', c' the opposite edges.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the tetrahedron" }, { "unit": null, "symbol": "a", "meaning": "edge OA of the tetrahedron" }, { "unit": null, "symbol": "b", "meaning": "edge OB" }, { "unit": null, "symbol": "c", "meaning": "edge OC" }, { "unit": null, "symbol": "a'", "meaning": "edge BC, opposite to a" }, { "unit": null, "symbol": "b'", "meaning": "edge CA, opposite to b" }, { "unit": null, "symbol": "c'", "meaning": "edge AB, opposite to c" } ], "sympy": "Eq(144*V**2, -ap**2*bp**2*cp**2 + a**2*ap**2*(bp**2+cp**2-ap**2) + b**2*bp**2*(cp**2+ap**2-bp**2) + c**2*cp**2*(ap**2+bp**2-cp**2) - ap**2*(a**2-b**2)*(a**2-c**2) - bp**2*(b**2-c**2)*(b**2-a**2) - cp**2*(c**2-a**2)*(c**2-b**2))", "physics": false, "states": [], "concepts": [ "concept/edge", "concept/tetrahedron", "theorem/volume-of-a-tetrahedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-89ffab1455", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 130", "location": "Polyhedrons", "latex": "\\cos{ADB}=\\dfrac{a'^2+b'^2-c^2}{2a'b'}", "name": "law of cosines", "statement": "In triangle ADB, the cosine of the angle at D equals (a'^2 + b'^2 - c^2) divided by 2a'b', where c is the side AB.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side BC of triangle ABC" }, { "unit": null, "symbol": "b", "meaning": "side CA of triangle ABC" }, { "unit": null, "symbol": "c", "meaning": "side AB of triangle ABC" }, { "unit": null, "symbol": "a'", "meaning": "straight line DA" }, { "unit": null, "symbol": "b'", "meaning": "straight line DB" } ], "sympy": "Eq(cos(ADB), (ap**2+bp**2-c**2)/(2*ap*bp))", "physics": false, "states": [ "theorem/law-of-cosines" ], "concepts": [ "concept/cosine", "concept/plane-angle", "concept/triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-df1d298afc", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 130", "location": "Polyhedrons", "latex": "1=\\cos^2{ADB}+\\cos^2{BDC}+\\cos^2{CDA}-2\\cos{ADB}\\cos{BDC}\\cos{CDA}", "name": null, "statement": "For four points in a plane with D joined to A, B, C, the squared cosines of the three angles at D, minus twice their product, equal 1.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "ADB", "meaning": "angle at D between DA and DB" }, { "unit": null, "symbol": "BDC", "meaning": "angle at D between DB and DC" }, { "unit": null, "symbol": "CDA", "meaning": "angle at D between DC and DA" } ], "sympy": "Eq(1, cos(ADB)**2+cos(BDC)**2+cos(CDA)**2-2*cos(ADB)*cos(BDC)*cos(CDA))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/plane-angle", "concept/point" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-778dfb0254", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 131", "location": "Polyhedrons", "latex": "0=-a^2b^2c^2 + a'^2a^2(b^2+c^2-a^2) + b'^2b^2(c^2+a^2-b^2) + c'^2c^2(a^2+b^2-c^2) - a^2(a'^2-b'^2)(a'^2-c'^2) - b^2(b'^2-c'^2)(b'^2-a'^2) - c^2(c'^2-a'^2)(c'^2-b'^2)", "name": null, "statement": "The six straight lines joining four points taken arbitrarily in a plane satisfy this relation.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "length BC" }, { "unit": null, "symbol": "b", "meaning": "length CA" }, { "unit": null, "symbol": "c", "meaning": "length AB" }, { "unit": null, "symbol": "a'", "meaning": "length DA" }, { "unit": null, "symbol": "b'", "meaning": "length DB" }, { "unit": null, "symbol": "c'", "meaning": "length DC" } ], "sympy": "Eq(0, -a**2*b**2*c**2 + ap**2*a**2*(b**2+c**2-a**2) + bp**2*b**2*(c**2+a**2-b**2) + cp**2*c**2*(a**2+b**2-c**2) - a**2*(ap**2-bp**2)*(ap**2-cp**2) - b**2*(bp**2-cp**2)*(bp**2-ap**2) - c**2*(cp**2-ap**2)*(cp**2-bp**2))", "physics": false, "states": [], "concepts": [ "concept/point", "concept/relation-between-variables", "concept/triangle", "theorem/volume-of-a-tetrahedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-22c8b9899a", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 131", "location": "Polyhedrons", "latex": "p^2(2a^2b^2+2b^2c^2+2c^2a^2 -a^4-b^4-c^4) = -a^2b^2c^2 + a'^2a^2(b^2+c^2-a^2) + b'^2b^2(c^2+a^2-b^2) + c'^2c^2(a^2+b^2-c^2) - a^2(a'^2-b'^2)(a'^2-c'^2) - b^2(b'^2-c'^2)(b'^2-a'^2) - c^2(c'^2-a'^2)(c'^2-b'^2)", "name": null, "statement": "With p the altitude of a tetrahedron on a base triangle of sides a, b, c and lateral edges a', b', c', the left side equals 144 V squared, giving the volume in terms of the six edges.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "perpendicular from the vertex of the tetrahedron to its base" }, { "unit": null, "symbol": "a", "meaning": "side BC of the base triangle ABC" }, { "unit": null, "symbol": "b", "meaning": "side CA of the base triangle" }, { "unit": null, "symbol": "c", "meaning": "side AB of the base triangle" }, { "unit": null, "symbol": "a'", "meaning": "lateral edge from the vertex to A" }, { "unit": null, "symbol": "b'", "meaning": "lateral edge from the vertex to B" }, { "unit": null, "symbol": "c'", "meaning": "lateral edge from the vertex to C" } ], "sympy": "Eq(p**2*(2*a**2*b**2+2*b**2*c**2+2*c**2*a**2-a**4-b**4-c**4), -a**2*b**2*c**2 + ap**2*a**2*(b**2+c**2-a**2) + bp**2*b**2*(c**2+a**2-b**2) + cp**2*c**2*(a**2+b**2-c**2) - a**2*(ap**2-bp**2)*(ap**2-cp**2) - b**2*(bp**2-cp**2)*(bp**2-ap**2) - c**2*(cp**2-ap**2)*(cp**2-bp**2))", "physics": false, "states": [], "concepts": [ "concept/altitude-of-a-pyramid", "concept/tetrahedron", "theorem/area-of-a-triangle", "theorem/volume-of-a-tetrahedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-bccf076a5d", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 132", "location": "Polyhedrons", "latex": "\\cos{ADB}=\\dfrac{\\cos{\\gamma}-\\cos{\\alpha'}\\cos{\\beta'}} {\\sin{\\alpha'}\\sin{\\beta}'}", "name": "spherical law of cosines (sides)", "statement": "On the sphere, the cosine of the angle ADB equals (cos gamma minus cos alpha' cos beta') divided by sin alpha' sin beta'.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "alpha", "meaning": "arc BC of a great circle" }, { "unit": null, "symbol": "beta", "meaning": "arc CA" }, { "unit": null, "symbol": "gamma", "meaning": "arc AB" }, { "unit": null, "symbol": "alpha'", "meaning": "arc DA" }, { "unit": null, "symbol": "beta'", "meaning": "arc DB" } ], "sympy": "Eq(cos(ADB), (cos(gamma)-cos(ap)*cos(bp))/(sin(ap)*sin(bp)))", "physics": false, "states": [ "theorem/spherical-law-of-cosines-sides" ], "concepts": [ "concept/cosine", "concept/great-circle", "concept/sine", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-bf65169ac5", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 132", "location": "Polyhedrons", "latex": "\\cos{\\alpha}=1-2\\sin^2{\\frac{\\alpha}{2}}", "name": null, "statement": "The cosine of an angle equals one minus twice the squared sine of half that angle.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "alpha", "meaning": "an arc (angle subtended at the centre of the sphere)" } ], "sympy": "Eq(cos(alpha), 1 - 2*sin(alpha/2)**2)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/sine", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-be81949a9f", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 132", "location": "Polyhedrons", "latex": "aa'+bb'+cc'=2\\sigma", "name": null, "statement": "The quantity sigma is defined as half the sum of the products aa', bb', cc'.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "sigma", "meaning": "half the sum of aa', bb', cc'" }, { "unit": null, "symbol": "a", "meaning": "edge OA" }, { "unit": null, "symbol": "a'", "meaning": "edge opposite a" } ], "sympy": "Eq(a*ap+b*bp+c*cp, 2*sigma)", "physics": false, "states": [], "concepts": [ "concept/edge", "concept/tetrahedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-1e1de4d61b", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 132", "location": "Polyhedrons", "latex": "36\\hspace{3pt}V^2r^2=\\sigma(\\sigma-aa')(\\sigma-bb')(\\sigma-cc')", "name": null, "statement": "The product 36 V squared r squared equals sigma times the three factors sigma minus aa', bb', cc', by analogy with Heron's formula.", "kind": "result", "symbols": [ { "unit": null, "symbol": "V", "meaning": "volume of the tetrahedron" }, { "unit": null, "symbol": "r", "meaning": "radius of the circumscribing sphere" }, { "unit": null, "symbol": "sigma", "meaning": "half the sum aa'+bb'+cc'" } ], "sympy": "Eq(36*V**2*r**2, sigma*(sigma-a*ap)*(sigma-b*bp)*(sigma-c*cp))", "physics": false, "states": [], "concepts": [ "concept/radius-of-the-sphere", "concept/tetrahedron", "theorem/area-of-a-triangle", "theorem/volume-of-a-tetrahedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-1fc7892b52", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 135", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "\\cos^2{TA}+\\cos^2{TB}+\\cos^2{TC}=1", "name": null, "statement": "For any point T on the sphere, the squares of the cosines of the arcs from T to the three vertices of the quadrantal triangle ABC sum to 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "T", "meaning": "any point on the surface of the sphere" }, { "unit": null, "symbol": "A", "meaning": "vertex of the spherical triangle ABC having all its sides quadrants" }, { "unit": null, "symbol": "B", "meaning": "vertex of the spherical triangle ABC having all its sides quadrants" }, { "unit": null, "symbol": "C", "meaning": "vertex of the spherical triangle ABC having all its sides quadrants" }, { "unit": null, "symbol": "TA", "meaning": "arc joining T to A" } ], "sympy": "Eq(cos(TA)**2 + cos(TB)**2 + cos(TC)**2, 1)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/quadrant", "concept/sphere", "concept/spherical-triangle", "concept/square", "concept/sum", "quantity/right-angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-2366db96d7", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 136", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "\\cos TU = \\cos TA \\cos UA + \\cos TB \\cos UB + \\cos TC \\cos UC", "name": null, "statement": "The cosine of the arc TU equals the sum of the products of the cosines of the arcs from T and from U to each vertex of the quadrantal triangle ABC.", "kind": "result", "symbols": [ { "unit": null, "symbol": "T", "meaning": "any point on the surface of the sphere" }, { "unit": null, "symbol": "U", "meaning": "any point on the surface of the sphere" }, { "unit": null, "symbol": "A", "meaning": "vertex of the quadrantal triangle ABC" }, { "unit": null, "symbol": "B", "meaning": "vertex of the quadrantal triangle ABC" }, { "unit": null, "symbol": "C", "meaning": "vertex of the quadrantal triangle ABC" } ], "sympy": "Eq(cos(TU), cos(TA)*cos(UA) + cos(TB)*cos(UB) + cos(TC)*cos(UC))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/product", "concept/quadrant", "concept/sphere", "concept/spherical-triangle", "concept/sum", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-20392bb642", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 137", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "\\Sigma = \\cos TH_1 + \\cos TH_2 + \\cos TH_3 + \\ldots", "name": null, "statement": "Sigma is defined as the sum of the cosines of the arcs joining T to the fixed points H1, H2, H3, and so on.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Sigma", "meaning": "sum of the cosines of the arcs joining T with the fixed points" }, { "unit": null, "symbol": "T", "meaning": "any point on the surface of the sphere" }, { "unit": null, "symbol": "H_1", "meaning": "first fixed point on the surface of the sphere" }, { "unit": null, "symbol": "H_2", "meaning": "second fixed point on the surface of the sphere" }, { "unit": null, "symbol": "H_3", "meaning": "third fixed point on the surface of the sphere" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/point", "concept/sphere", "concept/sum" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-6e52058936", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 138", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "G=\\surd{(P^2+Q^2+R^2)}", "name": null, "statement": "G is defined as the square root of the sum of the squares of P, Q and R.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "G", "meaning": "square root of P^2 + Q^2 + R^2" }, { "unit": null, "symbol": "P", "meaning": "sum of squares-of-cosine coefficients for A" }, { "unit": null, "symbol": "Q", "meaning": "coefficient for B" }, { "unit": null, "symbol": "R", "meaning": "coefficient for C" } ], "sympy": "Eq(G, sqrt(P**2 + Q**2 + R**2))", "physics": false, "states": [], "concepts": [ "concept/square", "concept/sum", "concept/surd" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-d26aa9aeb5", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 138", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "\\cos \\alpha = \\frac{P}{G}", "name": null, "statement": "The arc alpha is defined by the cosine equal to P divided by G (the source line, as transcribed, omits the equals sign before Q/G and R/G in the same display).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "alpha", "meaning": "arc determined by cos alpha = P/G" }, { "unit": null, "symbol": "P", "meaning": "coefficient defined in Art. 168" }, { "unit": null, "symbol": "G", "meaning": "square root of P^2 + Q^2 + R^2" } ], "sympy": "Eq(cos(alpha), P/G)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/quotient", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-4cc7277b9c", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 138", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "\\cos^2\\alpha +\\cos^2\\beta+\\cos^2\\gamma = 1", "name": null, "statement": "The squares of the cosines of the three arcs alpha, beta, gamma sum to 1.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "alpha", "meaning": "arc with cos alpha = P/G" }, { "unit": null, "symbol": "beta", "meaning": "arc with cos beta = Q/G" }, { "unit": null, "symbol": "gamma", "meaning": "arc with cos gamma = R/G" } ], "sympy": "Eq(cos(alpha)**2 + cos(beta)**2 + cos(gamma)**2, 1)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/square", "concept/sum", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-cf7fe690be", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 138", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "\\Sigma=G \\cos TU", "name": null, "statement": "Sigma, the sum of the cosines of arcs from T to the fixed points, varies as the cosine of the arc from T to a fixed point U, with factor G.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Sigma", "meaning": "sum of the cosines of arcs from T to the fixed points" }, { "unit": null, "symbol": "G", "meaning": "square root of P^2 + Q^2 + R^2" }, { "unit": null, "symbol": "TU", "meaning": "arc joining T to the fixed point U" } ], "sympy": "Eq(Sigma, G*cos(TU))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/point", "concept/sum" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-049a870f1f", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 138", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "\\cos TU= \\lambda \\cos \\alpha + \\mu \\cos \\beta + \\nu \\cos \\gamma", "name": null, "statement": "The cosine of the arc TU equals the sum of the products of the cosines of T's arcs to A, B, C with the cosines of alpha, beta, gamma.", "kind": "result", "symbols": [ { "unit": null, "symbol": "TU", "meaning": "arc joining T to the point U with arcs alpha, beta, gamma to A, B, C" }, { "unit": null, "symbol": "lambda", "meaning": "cosine of the arc joining T with A" }, { "unit": null, "symbol": "mu", "meaning": "cosine of the arc joining T with B" }, { "unit": null, "symbol": "nu", "meaning": "cosine of the arc joining T with C" }, { "unit": null, "symbol": "alpha", "meaning": "arc joining U to A" }, { "unit": null, "symbol": "beta", "meaning": "arc joining U to B" }, { "unit": null, "symbol": "gamma", "meaning": "arc joining U to C" } ], "sympy": "Eq(cos(TU), lambda_*cos(alpha) + mu*cos(beta) + nu*cos(gamma))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/product", "concept/sum", "quantity/arc-of-a-circle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-c2dc6f3362", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 139", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "G = 0", "name": null, "statement": "For a regular polyhedron inscribed in a sphere, or a rectangular parallelepiped inscribed in a sphere, G must be zero, since symmetry gives more than one position of T with the greatest value of Sigma.", "kind": "result", "symbols": [ { "unit": null, "symbol": "G", "meaning": "square root of P^2 + Q^2 + R^2" } ], "sympy": "Eq(G, 0)", "physics": false, "states": [], "concepts": [ "concept/polyhedron", "concept/regular-polyhedron", "concept/sum", "concept/symmetry", "concept/zero-displacement" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-7596a277f8", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 139", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "\\Sigma = \\cos^2 TH_1 + \\cos^2 TH_2 + \\cos^2 TH_3 + \\dots", "name": null, "statement": "Sigma is defined as the sum of the squares of the cosines of the arcs joining T with the fixed points.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Sigma", "meaning": "sum of the squares of the cosines of the arcs joining T with the fixed points" }, { "unit": null, "symbol": "T", "meaning": "any point on the surface of the sphere" }, { "unit": null, "symbol": "H_1", "meaning": "first fixed point on the surface of the sphere" }, { "unit": null, "symbol": "H_2", "meaning": "second fixed point on the surface of the sphere" }, { "unit": null, "symbol": "H_3", "meaning": "third fixed point on the surface of the sphere" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/point", "concept/square", "concept/sum" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-0feaf39c63", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 139", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "\\Sigma = P\\lambda^2+Q\\mu^2+R\\nu^2+2p\\mu\\nu+2q\\nu\\lambda+2r\\lambda \\mu", "name": null, "statement": "The sum of the squares of the cosines to the fixed points is a quadratic form in lambda, mu, nu, with cross-coefficients p, q, r.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "Sigma", "meaning": "sum of the squares of the cosines of the arcs joining T with the fixed points" }, { "unit": null, "symbol": "P", "meaning": "sum of squares of the l's: l_1^2 + l_2^2 + ..." }, { "unit": null, "symbol": "Q", "meaning": "sum of squares of the m's" }, { "unit": null, "symbol": "R", "meaning": "sum of squares of the n's" }, { "unit": null, "symbol": "p", "meaning": "sum of m_i n_i over the fixed points" }, { "unit": null, "symbol": "q", "meaning": "sum of n_i l_i over the fixed points" }, { "unit": null, "symbol": "r", "meaning": "sum of l_i m_i over the fixed points" }, { "unit": null, "symbol": "lambda", "meaning": "cosine of the arc joining T with A" }, { "unit": null, "symbol": "mu", "meaning": "cosine of the arc joining T with B" }, { "unit": null, "symbol": "nu", "meaning": "cosine of the arc joining T with C" } ], "sympy": "Eq(Sigma, P*lambda**2 + Q*mu**2 + R*nu**2 + 2*p*mu*nu + 2*q*nu*lambda + 2*r*lambda*mu)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/product", "concept/square", "concept/sum", "concept/variable" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-30146f55e2", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 139", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "\\Sigma=P\\lambda^2+Q\\mu^2+R\\nu^2", "name": null, "statement": "With the triangle ABC placed so that p, q, r vanish, the sum of the squares of the cosines equals P lambda^2 + Q mu^2 + R nu^2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Sigma", "meaning": "sum of the squares of the cosines of the arcs joining T with the fixed points" }, { "unit": null, "symbol": "P", "meaning": "sum of squares of the l's" }, { "unit": null, "symbol": "Q", "meaning": "sum of squares of the m's" }, { "unit": null, "symbol": "R", "meaning": "sum of squares of the n's" }, { "unit": null, "symbol": "lambda", "meaning": "cosine of the arc joining T with A" }, { "unit": null, "symbol": "mu", "meaning": "cosine of the arc joining T with B" }, { "unit": null, "symbol": "nu", "meaning": "cosine of the arc joining T with C" } ], "sympy": "Eq(Sigma, P*lambda**2 + Q*mu**2 + R*nu**2)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/spherical-triangle", "concept/square", "concept/sum" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-5f37530b5a", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 143", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "0 = 2p\\mu\\nu + 2q\\nu\\lambda + 2r\\lambda\\mu", "name": null, "statement": "When P=Q=R, the cross-term relation forces the cross-coefficients p, q, r each to vanish for every position of T.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "sum of m_i n_i over the fixed points" }, { "unit": null, "symbol": "q", "meaning": "sum of n_i l_i over the fixed points" }, { "unit": null, "symbol": "r", "meaning": "sum of l_i m_i over the fixed points" }, { "unit": null, "symbol": "lambda", "meaning": "cosine of the arc joining T with A" }, { "unit": null, "symbol": "mu", "meaning": "cosine of the arc joining T with B" }, { "unit": null, "symbol": "nu", "meaning": "cosine of the arc joining T with C" } ], "sympy": "Eq(0, 2*p*mu*nu + 2*q*nu*lambda + 2*r*lambda*mu)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/product", "concept/sum", "concept/zero-displacement" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ecec46ec0a", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 144", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "\\lambda\\lambda'P + \\mu\\mu'Q + \\nu\\nu'R", "name": null, "statement": "The sum of the products of the corresponding cosines to two points T and U equals lambda lambda' P + mu mu' Q + nu nu' R when p, q, r vanish.", "kind": "result", "symbols": [ { "unit": null, "symbol": "lambda", "meaning": "cosine of the arc joining T with A" }, { "unit": null, "symbol": "lambda'", "meaning": "cosine of the arc joining U with A" }, { "unit": null, "symbol": "mu", "meaning": "cosine of the arc joining T with B" }, { "unit": null, "symbol": "mu'", "meaning": "cosine of the arc joining U with B" }, { "unit": null, "symbol": "nu", "meaning": "cosine of the arc joining T with C" }, { "unit": null, "symbol": "nu'", "meaning": "cosine of the arc joining U with C" }, { "unit": null, "symbol": "P", "meaning": "sum of squares of the l's" }, { "unit": null, "symbol": "Q", "meaning": "sum of squares of the m's" }, { "unit": null, "symbol": "R", "meaning": "sum of squares of the n's" } ], "sympy": "Eq(Sigma_TU, lambda*lambda_p*P + mu*mu_p*Q + nu*nu_p*R)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/product", "concept/sum" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-308ea681d3", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 144", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "P = Q = R = \\dfrac{S}{3}", "name": null, "statement": "For a regular polyhedron inscribed in a sphere, P, Q and R are each one third of S, the number of solid angles of the polyhedron.", "kind": "result", "symbols": [ { "unit": null, "symbol": "P", "meaning": "sum of squares of the l's" }, { "unit": null, "symbol": "Q", "meaning": "sum of squares of the m's" }, { "unit": null, "symbol": "R", "meaning": "sum of squares of the n's" }, { "unit": null, "symbol": "S", "meaning": "number of solid angles of the regular polyhedron" } ], "sympy": "Eq(P, Q, R, S/3)", "physics": false, "states": [], "concepts": [ "concept/equality", "concept/polyhedral-angle", "concept/regular-polyhedron", "concept/sum", "concept/symmetry" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-1d85f9938a", "chapter": "todhunter-spherical-trigonometry-1886/ch-arcs-drawn-to-fixed-points-on-the-surface-of-a-sphere", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 144", "location": "Arcs drawn to fixed points on the Surface of a Sphere", "latex": "\\dfrac{S}{3} (\\lambda\\lambda' + \\mu\\mu' + \\nu\\nu') = \\dfrac{S}{3} \\cos TU", "name": null, "statement": "For a regular polyhedron inscribed in a sphere, the sum of products of corresponding cosines equals one third of the number of solid angles times cos TU.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "number of solid angles of the regular polyhedron" }, { "unit": null, "symbol": "lambda", "meaning": "cosine of the arc joining T with A" }, { "unit": null, "symbol": "lambda'", "meaning": "cosine of the arc joining U with A" }, { "unit": null, "symbol": "mu", "meaning": "cosine of the arc joining T with B" }, { "unit": null, "symbol": "mu'", "meaning": "cosine of the arc joining U with B" }, { "unit": null, "symbol": "nu", "meaning": "cosine of the arc joining T with C" }, { "unit": null, "symbol": "nu'", "meaning": "cosine of the arc joining U with C" }, { "unit": null, "symbol": "TU", "meaning": "arc joining T and U" } ], "sympy": "Eq(S/3*(lambda*lambda_p + mu*mu_p + nu*nu_p), S/3*cos(TU))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/polyhedral-angle", "concept/product", "concept/regular-polyhedron", "concept/sum" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-52db6629ab", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 157", "location": "Miscellaneous Propositions", "latex": "\\cos A' = \\sin (S - A) \\cos \\frac{a}{2}", "name": null, "statement": "The cosine of an angle of the chordal triangle equals the sine of S minus the corresponding angle times the cosine of half the side (example 13).", "kind": "result", "symbols": [ { "unit": null, "symbol": "A'", "meaning": "angle of the chordal triangle corresponding to angle A" }, { "unit": null, "symbol": "S", "meaning": "semi-sum of the angles of the spherical triangle" }, { "unit": null, "symbol": "a", "meaning": "side opposite A" } ], "sympy": "Eq(cos(A_p), sin(S - A)*cos(a/2))", "physics": false, "states": [], "concepts": [ "concept/chordal-triangle", "concept/cosine", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-5c0a940a72", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 158", "location": "Miscellaneous Propositions", "latex": "\\dfrac{\\sin Pa \\cos PA}{\\sin Aa} + \\dfrac{\\sin Pb \\cos PB}{\\sin Bb} + \\dfrac{\\sin Pc \\cos PC}{\\sin Cc} = 1", "name": null, "statement": "For a point P inside a spherical triangle with cevian great circles through P, the three weighted sine-cosine ratios sum to one (example 16).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "P", "meaning": "interior point of the spherical triangle ABC" }, { "unit": null, "symbol": "a, b, c", "meaning": "points where the great circles through P meet the opposite sides" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/great-circle", "concept/side", "concept/sine", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-7b4d659cac", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 145", "location": "Miscellaneous Propositions", "latex": "\\cot \\tfrac{1}{2}E = \\cot \\tfrac{1}{2}\\theta \\cot \\tfrac{1}{2}c \\operatorname{cosec} \\phi + \\cot \\phi", "name": null, "statement": "The cotangent of half the spherical excess equals a combination of the cotangents of half the sides and the angle at A, for a triangle with given base and area.", "kind": "result", "symbols": [ { "unit": null, "symbol": "E", "meaning": "spherical excess of the triangle" }, { "unit": null, "symbol": "\\theta", "meaning": "arc AC, the side from the vertex A to the variable vertex C" }, { "unit": null, "symbol": "c", "meaning": "the given base AB" }, { "unit": null, "symbol": "\\phi", "meaning": "angle BAC at the vertex A" } ], "sympy": "Eq(cot(E/2), cot(theta/2)*cot(c/2)*csc(phi) + cot(phi))", "physics": false, "states": [], "concepts": [ "concept/area", "concept/cosecant", "concept/cotangent", "concept/locus", "concept/spherical-triangle", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-d4633f403d", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 145", "location": "Miscellaneous Propositions", "latex": "\\cos \\theta \\cot \\tfrac{1}{2}c \\sin \\tfrac{1}{2} E + \\sin \\theta \\cos \\left(\\phi - \\tfrac{1}{2}E + \\dfrac{\\pi}{2}\\right) = -\\cot \\tfrac{1}{2} c \\sin \\tfrac{1}{2} E", "name": null, "statement": "The vertex of a spherical triangle of given base and area satisfies a linear trigonometric equation in theta and phi, which shows its locus is a circle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\theta", "meaning": "arc AC from the fixed vertex A to the variable vertex C" }, { "unit": null, "symbol": "\\phi", "meaning": "angle BAC" }, { "unit": null, "symbol": "c", "meaning": "the given base AB" }, { "unit": null, "symbol": "E", "meaning": "spherical excess (fixed by the given area)" } ], "sympy": "Eq(cos(theta)*cot(c/2)*sin(E/2) + sin(theta)*cos(phi - E/2 + pi/2), -cot(c/2)*sin(E/2))", "physics": false, "states": [], "concepts": [ "concept/area", "concept/circle", "concept/cosecant", "concept/cosine", "concept/cotangent", "concept/locus", "concept/sine", "concept/spherical-triangle", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-c2afe48dd1", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 145", "location": "Miscellaneous Propositions", "latex": "\\beta = \\tfrac{1}{2} E - \\frac{\\pi}{2}", "name": null, "statement": "The angular co-ordinate beta of the pole of the locus circle equals half the spherical excess minus a right angle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\beta", "meaning": "angular co-ordinate of the pole of the locus circle" }, { "unit": null, "symbol": "E", "meaning": "spherical excess" } ], "sympy": "Eq(beta, E/2 - pi/2)", "physics": false, "states": [], "concepts": [ "concept/angular-coordinates-on-a-sphere", "concept/pole-of-a-circle", "quantity/right-angle", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-cb9be38ba7", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 145", "location": "Miscellaneous Propositions", "latex": "0 = \\cos \\theta \\cos \\left(\\frac{\\pi}{2} - \\frac{c}{2}\\right) + \\sin \\theta \\sin \\left(\\frac{\\pi}{2} - \\frac{c}{2}\\right) \\cos (\\phi - \\pi)", "name": null, "statement": "The equation of the great circle that bisects the base AB at right angles, in the polar co-ordinates theta and phi.", "kind": "law", "symbols": [ { "unit": null, "symbol": "\\theta", "meaning": "arc of the great circle drawn from the fixed vertex" }, { "unit": null, "symbol": "\\phi", "meaning": "angle at the fixed vertex" }, { "unit": null, "symbol": "c", "meaning": "the given base AB" } ], "sympy": "Eq(0, cos(theta)*cos(pi/2 - c/2) + sin(theta)*sin(pi/2 - c/2)*cos(phi - pi))", "physics": false, "states": [], "concepts": [ "concept/angular-coordinates-on-a-sphere", "concept/cosine", "concept/great-circle", "concept/perpendicular", "concept/sine", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-df24f3de26", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 146", "location": "Miscellaneous Propositions", "latex": "\\cos PAQ = cos\\tfrac{1}{2}(B-C)", "name": null, "statement": "The angle between the arcs from A to the poles P and Q of the inscribed and circumscribed circles equals the cosine of half the difference of the angles B and C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "P", "meaning": "pole of the inscribed circle of triangle ABC" }, { "unit": null, "symbol": "Q", "meaning": "pole of the circumscribed circle of triangle ABC" }, { "unit": null, "symbol": "A, B, C", "meaning": "angles of the spherical triangle ABC" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/circumscribed-circle", "concept/cosine", "concept/inscribed-circle", "concept/pole-of-a-circle", "concept/spherical-angle", "concept/spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-fff9b4163b", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 146", "location": "Miscellaneous Propositions", "latex": "\\cos PQ = \\cos PA \\cos QA + \\sin PA \\sin QA \\cos \\tfrac{1}{2}(B-C)", "name": null, "statement": "The spherical cosine rule applied to the triangle with vertices P, A, Q gives the cosine of the distance PQ between the two poles.", "kind": "law", "symbols": [ { "unit": null, "symbol": "PQ", "meaning": "angular distance between the pole of the inscribed circle and the pole of the circumscribed circle" }, { "unit": null, "symbol": "PA", "meaning": "arc from P to vertex A" }, { "unit": null, "symbol": "QA", "meaning": "arc from Q to vertex A" } ], "sympy": "Eq(cos(PQ), cos(PA)*cos(QA) + sin(PA)*sin(QA)*cos((B - C)/2))", "physics": false, "states": [], "concepts": [ "concept/circumscribed-circle", "concept/cosine", "concept/inscribed-circle", "concept/pole-of-a-circle", "concept/sine", "concept/spherical-distance", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-fcbe7e2519", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 146", "location": "Miscellaneous Propositions", "latex": "\\sin PA = \\frac{\\sin PE}{\\sin PAE} = \\frac{\\sin r}{\\sin\\tfrac{1}{2}A}", "name": null, "statement": "The sine of the distance PA from the vertex to the pole of the inscribed circle equals the sine of the inradius divided by the sine of half the angle A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "PA", "meaning": "arc from the inscribed-circle pole P to vertex A" }, { "unit": null, "symbol": "r", "meaning": "radius of the inscribed circle (as an arc)" }, { "unit": null, "symbol": "A", "meaning": "angle of the triangle at vertex A" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/inscribed-circle", "concept/pole-of-a-circle", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-57fd2af42d", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 146", "location": "Miscellaneous Propositions", "latex": "\\cos PQ = \\cos R \\cos r \\cos(s-a) + \\sin R \\sin r \\sin \\tfrac{1}{2}(b+c) \\operatorname{cosec} \\tfrac{1}{2}a", "name": null, "statement": "The cosine of the distance between the two poles is expressed in terms of the inradius r, circumradius R, and the sides of the triangle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "PQ", "meaning": "angular distance between the poles of the inscribed and circumscribed circles" }, { "unit": null, "symbol": "R", "meaning": "radius of the circumscribed circle (as an arc)" }, { "unit": null, "symbol": "r", "meaning": "radius of the inscribed circle (as an arc)" }, { "unit": null, "symbol": "s", "meaning": "semi-perimeter of the triangle" }, { "unit": null, "symbol": "a, b, c", "meaning": "sides of the triangle" } ], "sympy": "Eq(cos(PQ), cos(R)*cos(r)*cos(s - a) + sin(R)*sin(r)*sin((b + c)/2)*csc(a/2))", "physics": false, "states": [], "concepts": [ "concept/circumscribed-circle", "concept/cosecant", "concept/cosine", "concept/inscribed-circle", "concept/pole-of-a-circle", "concept/sine", "concept/spherical-distance", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f4a8b7299f", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 146", "location": "Miscellaneous Propositions", "latex": "\\cot r = \\frac{\\sin s}{n}", "name": null, "statement": "The cotangent of the inradius equals the sine of the semi-perimeter divided by n, where n is a quantity defined from the sides.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "r", "meaning": "radius of the inscribed circle (as an arc)" }, { "unit": null, "symbol": "s", "meaning": "semi-perimeter" }, { "unit": null, "symbol": "n", "meaning": "auxiliary quantity of the triangle, n = sqrt(sin s sin(s-a) sin(s-b) sin(s-c))" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/inscribed-circle", "concept/sine", "concept/spherical-triangle", "quantity/spherical-excess" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-cd0a423266", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 146", "location": "Miscellaneous Propositions", "latex": "\\tan R = \\frac{2 \\sin \\frac{1}{2} a \\sin \\tfrac{1}{2} b \\sin \\tfrac{1}{2} c}{n}", "name": null, "statement": "The tangent of the circumradius R equals twice the product of the sines of half the sides, divided by n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "R", "meaning": "radius of the circumscribed circle (as an arc)" }, { "unit": null, "symbol": "a, b, c", "meaning": "sides of the triangle" }, { "unit": null, "symbol": "n", "meaning": "auxiliary quantity of the triangle" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/circumscribed-circle", "concept/sine", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-3ff02703cc", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 146", "location": "Miscellaneous Propositions", "latex": "\\frac{\\cos PQ}{\\cos R \\sin r} = \\cot r \\cos(s-a) + \\tan R \\sin \\frac{1}{2}(b+c) \\operatorname{cosec} \\frac{1}{2}a", "name": null, "statement": "Dividing the cosine formula for PQ by cos R sin r gives a form in cotangents and tangents of the radii.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "PQ", "meaning": "angular distance between the two poles" }, { "unit": null, "symbol": "R", "meaning": "circumradius (as an arc)" }, { "unit": null, "symbol": "r", "meaning": "inradius (as an arc)" }, { "unit": null, "symbol": "s", "meaning": "semi-perimeter" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosecant", "concept/cosine", "concept/cotangent", "concept/sine", "concept/spherical-distance", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-88a959a7cc", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 146", "location": "Miscellaneous Propositions", "latex": "\\cos^2 PQ = \\cos^2 R \\sin^2 r + \\cos^2 (R-r)", "name": null, "statement": "The squared cosine of the distance between the inscribed-circle and circumscribed-circle poles is given in terms of the circumradius R and inradius r.", "kind": "result", "symbols": [ { "unit": null, "symbol": "PQ", "meaning": "angular distance between the pole of the inscribed circle and the pole of the circumscribed circle" }, { "unit": null, "symbol": "R", "meaning": "circumradius (as an arc)" }, { "unit": null, "symbol": "r", "meaning": "inradius (as an arc)" } ], "sympy": "Eq(cos(PQ)**2, cos(R)**2*sin(r)**2 + cos(R - r)**2)", "physics": false, "states": [], "concepts": [ "concept/circumscribed-circle", "concept/cosine", "concept/inscribed-circle", "concept/pole-of-a-circle", "concept/sine", "concept/spherical-distance" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-8d9436e32a", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 146", "location": "Miscellaneous Propositions", "latex": "\\sin^2 PQ = \\sin^2 (R-r) - \\cos^2 R \\sin^2 r", "name": null, "statement": "The squared sine of the distance between the poles of the inscribed and circumscribed circles is given in terms of R and r.", "kind": "result", "symbols": [ { "unit": null, "symbol": "PQ", "meaning": "angular distance between the two poles" }, { "unit": null, "symbol": "R", "meaning": "circumradius (as an arc)" }, { "unit": null, "symbol": "r", "meaning": "inradius (as an arc)" } ], "sympy": "Eq(sin(PQ)**2, sin(R - r)**2 - cos(R)**2*sin(r)**2)", "physics": false, "states": [], "concepts": [ "concept/circumscribed-circle", "concept/cosine", "concept/inscribed-circle", "concept/pole-of-a-circle", "concept/sine", "concept/spherical-distance" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-d6651eee52", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 147", "location": "Miscellaneous Propositions", "latex": "\\cos QQ_1 = \\cos R \\cos r_1 \\cos (s-c) - \\sin R \\sin r_1 \\sin \\tfrac{1}{2}(C-A) \\sec \\tfrac{1}{2}B", "name": null, "statement": "The cosine of the distance between the circumscribed-circle pole Q and the escribed-circle pole Q1 is given by a spherical cosine relation.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "QQ_1", "meaning": "angular distance between the pole of the circumscribed circle and the pole of the escribed circle opposite A" }, { "unit": null, "symbol": "R", "meaning": "circumradius (as an arc)" }, { "unit": null, "symbol": "r_1", "meaning": "radius of the escribed circle opposite A (as an arc)" }, { "unit": null, "symbol": "s", "meaning": "semi-perimeter" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/circumscribed-circle", "concept/cosine", "concept/escribed-circle", "concept/pole-of-a-circle", "concept/secant", "concept/sine", "concept/spherical-distance" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-dfc4828fcb", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 147", "location": "Miscellaneous Propositions", "latex": "\\cos^2 QQ_1 = \\cos^2 R \\sin^2 r_1 + \\cos^2 (R + r_1)", "name": null, "statement": "The squared cosine of the distance QQ1 between the circumscribed and escribed poles in terms of R and r1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "QQ_1", "meaning": "angular distance between the two poles" }, { "unit": null, "symbol": "R", "meaning": "circumradius (as an arc)" }, { "unit": null, "symbol": "r_1", "meaning": "radius of the escribed circle (as an arc)" } ], "sympy": "Eq(cos(QQ1)**2, cos(R)**2*sin(r1)**2 + cos(R + r1)**2)", "physics": false, "states": [], "concepts": [ "concept/circumscribed-circle", "concept/cosine", "concept/escribed-circle", "concept/pole-of-a-circle", "concept/sine", "concept/spherical-distance" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-bab61a2e0c", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 147", "location": "Miscellaneous Propositions", "latex": "\\sin^2 QQ_1 = \\sin^2 (R + r_1) - \\cos^2 R \\sin^2 r_1", "name": null, "statement": "The squared sine of the distance QQ1 between the circumscribed and escribed poles in terms of R and r1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "QQ_1", "meaning": "angular distance between the two poles" }, { "unit": null, "symbol": "R", "meaning": "circumradius (as an arc)" }, { "unit": null, "symbol": "r_1", "meaning": "radius of the escribed circle (as an arc)" } ], "sympy": "Eq(sin(QQ1)**2, sin(R + r1)**2 - cos(R)**2*sin(r1)**2)", "physics": false, "states": [], "concepts": [ "concept/circumscribed-circle", "concept/cosine", "concept/escribed-circle", "concept/sine", "concept/spherical-distance" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-37de993ee3", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 148", "location": "Miscellaneous Propositions", "latex": "\\sin BQ = \\sin CQ", "name": null, "statement": "The sines of the arcs from B and from C to the point Q, where the arc through the midpoints meets BC produced, are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "BQ", "meaning": "arc from B to the fixed point Q" }, { "unit": null, "symbol": "CQ", "meaning": "arc from C to Q" } ], "sympy": "Eq(sin(BQ), sin(CQ))", "physics": false, "states": [], "concepts": [ "concept/great-circle", "concept/midpoint", "concept/sine", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-6086171e27", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 148", "location": "Miscellaneous Propositions", "latex": "BQ + CQ = \\pi", "name": null, "statement": "The arcs BQ and CQ together make a half great circle, so the fixed point Q lies a quadrant beyond the midpoint of the base.", "kind": "result", "symbols": [ { "unit": null, "symbol": "BQ", "meaning": "arc from B to Q" }, { "unit": null, "symbol": "CQ", "meaning": "arc from C to Q" } ], "sympy": "Eq(BQ + CQ, pi)", "physics": false, "states": [], "concepts": [ "concept/great-circle", "concept/quadrant", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-cd8bdfda20", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 148", "location": "Miscellaneous Propositions", "latex": "DQ = \\tfrac{1}{2} (BQ + CQ) = \\tfrac{1}{2} \\pi", "name": null, "statement": "The distance from the midpoint D of BC to the fixed point Q is a quadrant.", "kind": "result", "symbols": [ { "unit": null, "symbol": "DQ", "meaning": "arc from the midpoint D of BC to Q" }, { "unit": null, "symbol": "BQ, CQ", "meaning": "arcs from B and C to Q" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/midpoint", "concept/quadrant", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-a791c31890", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 148", "location": "Miscellaneous Propositions", "latex": "\\sin BD \\sin CE \\sin AF= \\sin CD \\sin AE \\sin BF", "name": null, "statement": "If three arcs from the vertices of a spherical triangle pass through a common point, the products of the sines of alternate segments of the sides are equal.", "kind": "law", "symbols": [ { "unit": null, "symbol": "BD, CD", "meaning": "arcs of side BC divided by the cevian from A at D" }, { "unit": null, "symbol": "CE, AE", "meaning": "arcs of side CA divided at E" }, { "unit": null, "symbol": "AF, BF", "meaning": "arcs of side AB divided at F" } ], "sympy": "Eq(sin(BD)*sin(CE)*sin(AF), sin(CD)*sin(AE)*sin(BF))", "physics": false, "states": [], "concepts": [ "concept/concurrence-of-arcs", "concept/side", "concept/sine", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-83d4ebb4d8", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 148", "location": "Miscellaneous Propositions", "latex": "\\dfrac{\\sin BD}{\\sin CD}\\, \\dfrac{\\sin CE}{\\sin AE}\\, \\dfrac{\\sin AF}{\\sin BF} = 1", "name": null, "statement": "The product of the three sine ratios of the divided sides equals one (Ceva-type condition for concurrent arcs on a sphere).", "kind": "law", "symbols": [ { "unit": null, "symbol": "BD, CD, CE, AE, AF, BF", "meaning": "arcs of the sides of the spherical triangle divided by the three arcs through P" } ], "sympy": "Eq(sin(BD)/sin(CD)*sin(CE)/sin(AE)*sin(AF)/sin(BF), 1)", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/concurrence-of-arcs", "concept/side", "concept/sine", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-160d7d9c36", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 150", "location": "Miscellaneous Propositions", "latex": "\\surd(1-\\cos^2 \\alpha-\\cos^2 \\beta-\\cos^2 \\gamma+2\\cos \\alpha \\cos \\beta \\cos \\gamma)", "name": "sine of a solid angle", "statement": "The sine of a solid angle formed by three plane angles alpha, beta, gamma is the square root of the stated expression, which lies between zero and one.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\alpha, \\beta, \\gamma", "meaning": "angles contained by the three straight lines forming the solid angle, taken in pairs" } ], "sympy": null, "physics": false, "states": [ "concept/sine-of-a-solid-angle" ], "concepts": [ "concept/cosine", "concept/polyhedral-angle", "concept/sine" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-26519385f6", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 152", "location": "Miscellaneous Propositions", "latex": "2E=3a+4b+5c+6d+ \\ldots\\ldots", "name": null, "statement": "Each edge belongs to two faces, so twice the number of edges equals the sum of sides over all faces.", "kind": "result", "symbols": [ { "unit": null, "symbol": "E", "meaning": "number of edges of the polyhedron" }, { "unit": null, "symbol": "a, b, c, d", "meaning": "numbers of faces that are triangles, quadrilaterals, pentagons, hexagons" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/edge", "concept/face", "concept/polyhedron", "concept/real-number" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-982541c041", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 152", "location": "Miscellaneous Propositions", "latex": "2E= 3 \\alpha + 4 \\beta + 5 \\gamma + 6 \\delta + \\ldots\\ldots", "name": null, "statement": "Each edge terminates at two solid angles, so twice the number of edges equals the sum of plane-angle counts over all solid angles.", "kind": "result", "symbols": [ { "unit": null, "symbol": "E", "meaning": "number of edges" }, { "unit": null, "symbol": "\\alpha, \\beta, \\gamma, \\delta", "meaning": "numbers of solid angles formed with three, four, five, six plane angles" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/edge", "concept/polyhedral-angle", "concept/real-number", "concept/vertex" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ab005abd91", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 152", "location": "Miscellaneous Propositions", "latex": "F=a+b+c+d+ \\ldots\\ldots", "name": null, "statement": "The total number of faces is the sum of the numbers of faces with each number of sides.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "F", "meaning": "number of faces of the polyhedron" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/face", "concept/polyhedron", "concept/real-number" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-615696e1eb", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 152", "location": "Miscellaneous Propositions", "latex": "S= \\alpha + \\beta + \\gamma + \\delta + \\ldots\\ldots", "name": null, "statement": "The total number of solid angles is the sum of the numbers of solid angles with each number of plane angles.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "S", "meaning": "number of solid angles (corner points)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/polyhedral-angle", "concept/real-number", "concept/vertex" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f3337c6b61", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 152", "location": "Miscellaneous Propositions", "latex": "2E-3F = b + 2c + 3d + \\ldots\\ldots", "name": null, "statement": "Twice the number of edges minus three times the number of faces equals a sum with nonnegative coefficients, so 2E cannot be less than 3F.", "kind": "result", "symbols": [ { "unit": null, "symbol": "E", "meaning": "number of edges" }, { "unit": null, "symbol": "F", "meaning": "number of faces" }, { "unit": null, "symbol": "b, c, d", "meaning": "numbers of faces that are quadrilaterals, pentagons, hexagons" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/edge", "concept/face", "concept/inequality", "concept/polyhedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-42242cebb6", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 152", "location": "Miscellaneous Propositions", "latex": "2E-3S = \\beta + 2 \\gamma + 3 \\delta + \\ldots\\ldots", "name": null, "statement": "Twice the number of edges minus three times the number of solid angles equals a sum with nonnegative coefficients, so 2E cannot be less than 3S.", "kind": "result", "symbols": [ { "unit": null, "symbol": "E", "meaning": "number of edges" }, { "unit": null, "symbol": "S", "meaning": "number of solid angles" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/edge", "concept/inequality", "concept/polyhedral-angle", "concept/polyhedron" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-9753ddcc87", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 153", "location": "Miscellaneous Propositions", "latex": "2F + 2S=4 + 2E", "name": null, "statement": "Twice the sum of faces and solid angles equals four plus twice the number of edges, which is Euler's relation rewritten.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F", "meaning": "number of faces" }, { "unit": null, "symbol": "S", "meaning": "number of solid angles" }, { "unit": null, "symbol": "E", "meaning": "number of edges" } ], "sympy": "Eq(2*F + 2*S, 4 + 2*E)", "physics": false, "states": [], "concepts": [ "concept/edge", "concept/face", "concept/polyhedral-angle", "concept/polyhedron", "theorem/euler-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-74caa3a88b", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 153", "location": "Miscellaneous Propositions", "latex": "2 (\\alpha + \\beta + \\gamma + \\delta + \\ldots) - (a + 2b + 3c + 4d + \\ldots) = 4", "name": null, "statement": "Combining the face and solid-angle counts gives an identity relating solid-angle counts to face-side counts.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a, b, c, d", "meaning": "numbers of triangular, quadrilateral, pentagonal, hexagonal faces" }, { "unit": null, "symbol": "\\alpha, \\beta, \\gamma, \\delta", "meaning": "numbers of solid angles with three, four, five, six plane angles" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/face", "concept/polyhedral-angle", "concept/polyhedron", "concept/real-number" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-c0969412fd", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 153", "location": "Miscellaneous Propositions", "latex": "2 (a + b + c + d + \\ldots) - (\\alpha + 2 \\beta + 3 \\gamma + 4 \\delta + \\ldots) = 4", "name": null, "statement": "The dual identity relating face counts to solid-angle counts holds for any polyhedron.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a, b, c, d", "meaning": "numbers of faces with three, four, five, six sides" }, { "unit": null, "symbol": "\\alpha, \\beta, \\gamma, \\delta", "meaning": "numbers of solid angles with three, four, five, six plane angles" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/face", "concept/polyhedral-angle", "concept/polyhedron", "concept/real-number" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-bfde24d6fd", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 153", "location": "Miscellaneous Propositions", "latex": "a + \\alpha - (c + \\gamma) - 2 (d + \\delta) - 3 (e + \\epsilon) - \\ldots\\ldots = 8", "name": null, "statement": "Adding the two identities shows the number of triangular faces plus triangular solid angles is at least eight.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "number of triangular faces" }, { "unit": null, "symbol": "\\alpha", "meaning": "number of solid angles formed with three plane angles" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/face", "concept/inequality", "concept/polyhedral-angle", "concept/polyhedron", "concept/real-number" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-697a3bf215", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 153", "location": "Miscellaneous Propositions", "latex": "3a + 2b + c - e - 2f - \\ldots\\ldots -2\\beta - 4\\gamma - \\ldots\\ldots = 12", "name": null, "statement": "Eliminating alpha gives a relation showing 3a + 2b + c cannot be less than 12.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, b, c", "meaning": "numbers of triangular, quadrilateral, pentagonal faces" }, { "unit": null, "symbol": "\\beta, \\gamma", "meaning": "numbers of solid angles with four and five plane angles" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/face", "concept/inequality", "concept/polyhedral-angle", "concept/polyhedron", "concept/real-number" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-8b99ce8883", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 151", "location": "Miscellaneous Propositions", "latex": "\\mathrm{F + S = E + 1 }", "name": "Cauchy's network theorem", "statement": "For a network of rectilineal figures not forming a closed surface, faces plus corner points equals edges plus one.", "kind": "result", "symbols": [ { "unit": null, "symbol": "E", "meaning": "number of edges of the network" }, { "unit": null, "symbol": "F", "meaning": "number of figures in the network" }, { "unit": null, "symbol": "S", "meaning": "number of corner points" } ], "sympy": "Eq(F + S, E + 1)", "physics": false, "states": [ "theorem/cauchy-s-network-theorem" ], "concepts": [ "concept/edge", "concept/polygon", "concept/vertex", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-38d65d06bc", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 152", "location": "Miscellaneous Propositions", "latex": "F-1+S = E + 1", "name": null, "statement": "Removing one face of a polyhedron yields a network to which Cauchy's theorem applies.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F", "meaning": "number of faces of the polyhedron" }, { "unit": null, "symbol": "S", "meaning": "number of solid angles" }, { "unit": null, "symbol": "E", "meaning": "number of edges" } ], "sympy": "Eq(F - 1 + S, E + 1)", "physics": false, "states": [], "concepts": [ "concept/face", "concept/polyhedron", "theorem/euler-s-theorem", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-652302df59", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 152", "location": "Miscellaneous Propositions", "latex": "F + S = E + 2", "name": "Euler's theorem", "statement": "For any polyhedron, faces plus solid angles equals edges plus two.", "kind": "law", "symbols": [ { "unit": null, "symbol": "F", "meaning": "number of faces" }, { "unit": null, "symbol": "S", "meaning": "number of solid angles (corner points)" }, { "unit": null, "symbol": "E", "meaning": "number of edges" } ], "sympy": "Eq(F + S, E + 2)", "physics": false, "states": [ "theorem/euler-s-theorem" ], "concepts": [ "concept/edge", "concept/face", "concept/polyhedron", "concept/vertex", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-6dc53ac76f", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 154", "location": "Miscellaneous Propositions", "latex": "1 + e + e' = s + s' + F", "name": null, "statement": "Splitting edges, corners and faces into those on the bounding contour and those within it gives a relation for the network.", "kind": "result", "symbols": [ { "unit": null, "symbol": "e, e'", "meaning": "numbers of edges in and within the bounding contour" }, { "unit": null, "symbol": "s, s'", "meaning": "numbers of corners in and within the bounding contour" }, { "unit": null, "symbol": "F", "meaning": "number of figures" } ], "sympy": "Eq(1 + e + e_p, s + s_p + F)", "physics": false, "states": [], "concepts": [ "concept/closed-contour", "concept/edge", "concept/vertex", "theorem/cauchy-s-network-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-5bde5974a3", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 154", "location": "Miscellaneous Propositions", "latex": "1 + e' = s' + F", "name": null, "statement": "Since the contour edges and corners are equal in number, the interior edges and corners satisfy 1 + e' = s' + F.", "kind": "result", "symbols": [ { "unit": null, "symbol": "e'", "meaning": "number of edges within the bounding contour" }, { "unit": null, "symbol": "s'", "meaning": "number of corners within the bounding contour" }, { "unit": null, "symbol": "F", "meaning": "number of figures" } ], "sympy": "Eq(1 + e_p, s_p + F)", "physics": false, "states": [], "concepts": [ "concept/closed-contour", "concept/edge", "concept/vertex", "theorem/cauchy-s-network-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-7db4fb0f5a", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 154", "location": "Miscellaneous Propositions", "latex": "\\mathrm{S + F = E + P + 1 }", "name": "Cauchy's extension of Euler's theorem", "statement": "If a polyhedron is decomposed into P polyhedrons, solid angles plus faces equals edges plus P plus one.", "kind": "result", "symbols": [ { "unit": null, "symbol": "P", "meaning": "number of polyhedrons into which the polyhedron is decomposed" }, { "unit": null, "symbol": "S", "meaning": "number of solid angles" }, { "unit": null, "symbol": "F", "meaning": "number of faces" }, { "unit": null, "symbol": "E", "meaning": "number of edges" } ], "sympy": "Eq(S + F, E + P + 1)", "physics": false, "states": [ "theorem/cauchy-s-extension-of-euler-s-theorem" ], "concepts": [ "concept/edge", "concept/face", "concept/polyhedral-angle", "concept/polyhedron", "theorem/euler-s-theorem", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-01380bf85c", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 156", "location": "Miscellaneous Propositions", "latex": "\\tan c = \\frac{\\cot A \\cot a + \\cot B \\cot b} {\\cot a \\cot b - \\cos A \\cos B}", "name": null, "statement": "The tangent of side c of a spherical triangle is given in terms of two angles and their opposite sides (example 4).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a, b, c", "meaning": "sides of the spherical triangle" }, { "unit": null, "symbol": "A, B", "meaning": "angles opposite sides a and b" } ], "sympy": "Eq(tan(c), (cot(A)*cot(a) + cot(B)*cot(b))/(cot(a)*cot(b) - cos(A)*cos(B)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cotangent", "concept/side", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-e0d5206458", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 156", "location": "Miscellaneous Propositions", "latex": "\\cos \\theta \\sin(b - c) + \\cos \\phi \\sin(c - a) + \\cos \\psi \\sin(a - b) = 0", "name": null, "statement": "For the point where the angle bisectors of a spherical triangle meet, the stated weighted cosine sum vanishes (example 5).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "\\theta, \\phi, \\psi", "meaning": "distances from the vertices A, B, C to the point where the bisecting arcs meet" }, { "unit": null, "symbol": "a, b, c", "meaning": "sides of the triangle" } ], "sympy": "Eq(cos(theta)*sin(b - c) + cos(phi)*sin(c - a) + cos(psi)*sin(a - b), 0)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-e0f5c63ad2", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 156", "location": "Miscellaneous Propositions", "latex": "\\cos PA \\cos BC = \\cos PB \\cos CA = \\cos PC \\cos AB", "name": null, "statement": "For the point P where the great circles through the vertices and the poles of the opposite sides meet, the products of cosines are equal (example 6).", "kind": "result", "symbols": [ { "unit": null, "symbol": "P", "meaning": "the common point of the great circles AA', BB', CC'" }, { "unit": null, "symbol": "PA, PB, PC", "meaning": "arcs from P to the vertices" }, { "unit": null, "symbol": "BC, CA, AB", "meaning": "sides of the triangle" } ], "sympy": "Eq(cos(PA)*cos(BC), cos(PB)*cos(CA))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/great-circle", "concept/pole-of-a-circle", "concept/side", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-c5ede1faa2", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 156", "location": "Miscellaneous Propositions", "latex": "\\tan \\alpha \\tan \\alpha' = \\tan \\beta \\tan \\beta' = \\tan \\gamma \\tan \\gamma'", "name": null, "statement": "For the three cevian arcs perpendicular to the sides, the products of the tangents of their segments are equal (example 8).", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\alpha, \\alpha'", "meaning": "segments into which the perpendicular p is divided" }, { "unit": null, "symbol": "\\beta, \\beta'", "meaning": "segments of the perpendicular q" }, { "unit": null, "symbol": "\\gamma, \\gamma'", "meaning": "segments of the perpendicular r" } ], "sympy": "Eq(tan(alpha)*tan(alpha_p), tan(beta)*tan(beta_p))", "physics": false, "states": [], "concepts": [ "concept/angle-of-elevation", "concept/perpendicular", "concept/spherical-triangle", "concept/tangent-function" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-1eb32181fb", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 156", "location": "Miscellaneous Propositions", "latex": "\\frac{\\cos p}{\\cos \\alpha \\cos \\alpha'} = \\frac{\\cos q}{\\cos \\beta \\cos \\beta' } = \\frac{\\cos r}{\\cos \\gamma \\cos \\gamma'}", "name": null, "statement": "For the perpendicular arcs p, q, r of a spherical triangle, the ratios of their cosines to products of cosines of their segments are equal (example 8).", "kind": "result", "symbols": [ { "unit": null, "symbol": "p, q, r", "meaning": "arcs of the perpendiculars from the angles to the opposite sides" }, { "unit": null, "symbol": "\\alpha, \\alpha'", "meaning": "segments of p" }, { "unit": null, "symbol": "\\beta, \\beta'", "meaning": "segments of q" }, { "unit": null, "symbol": "\\gamma, \\gamma'", "meaning": "segments of r" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/angle-of-elevation", "concept/cosine", "concept/perpendicular", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-f585cb1c96", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 157", "location": "Miscellaneous Propositions", "latex": "\\frac{\\sin \\alpha}{\\sin \\alpha'} = 2 \\cos \\frac{a}{2}", "name": null, "statement": "For an arc from a vertex to the midpoint of the opposite side, the ratio of the sines of its two parts equals twice the cosine of half that side (example 9).", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\alpha, \\alpha'", "meaning": "the two parts into which the median arc is divided by its foot" }, { "unit": null, "symbol": "a", "meaning": "side bisected by the arc" } ], "sympy": "Eq(sin(alpha)/sin(alpha_p), 2*cos(a/2))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/midpoint", "concept/side", "concept/sine", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-3fabae2229", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 157", "location": "Miscellaneous Propositions", "latex": "\\cos AQ \\sin \\frac{a}{2} = \\sin \\frac{c - b}{2} \\sin \\frac{c + b}{2}", "name": null, "statement": "For the arc bisecting AB and AC meeting BC produced at Q, the cosine of AQ times the sine of half a equals a product of sines (example 10).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "AQ", "meaning": "arc from A to Q" }, { "unit": null, "symbol": "a, b, c", "meaning": "sides of the triangle" } ], "sympy": "Eq(cos(AQ)*sin(a/2), sin((c - b)/2)*sin((c + b)/2))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/great-circle", "concept/side", "concept/sine", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-5fed82367d", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 157", "location": "Miscellaneous Propositions", "latex": "\\sin AB \\sin CD \\cos P = \\sin AD \\sin BC \\cos Q = \\sin AC \\sin BD \\cos R", "name": null, "statement": "For a spherical quadrilateral with the stated intersection points, the three products of sines and cosines are equal (example 12).", "kind": "result", "symbols": [ { "unit": null, "symbol": "P, Q, R", "meaning": "intersection points of the opposite sides and of the diagonals" }, { "unit": null, "symbol": "AB, CD, AD, BC, AC, BD", "meaning": "arcs joining the vertices of the quadrilateral" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/diagonal", "concept/side", "concept/sine", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-874e33a367", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 158", "location": "Miscellaneous Propositions", "latex": "a \\cos AP + b \\cos BP = s \\cos SP", "name": null, "statement": "For fixed points A and B on a sphere and constants a and b, a fixed point S exists on AB such that the combined cosine relation holds for every P (example 19).", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, b", "meaning": "given constants" }, { "unit": null, "symbol": "s", "meaning": "constant" }, { "unit": null, "symbol": "AP, BP, SP", "meaning": "angular distances on the sphere from P to A, B and S" } ], "sympy": "Eq(a*cos(AP) + b*cos(BP), s*cos(SP))", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/cosine", "concept/locus", "concept/point", "concept/spherical-distance" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-0a1b0e48a5", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 158", "location": "Miscellaneous Propositions", "latex": "a \\cos AP + b \\cos BP + c \\cos CP + \\ldots = \\text{constant}", "name": null, "statement": "For fixed points A, B, C, ... on a sphere with given constants, the locus of P satisfying a constant weighted cosine sum is a circle (example 20).", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, b, c", "meaning": "given constants" }, { "unit": null, "symbol": "AP, BP, CP", "meaning": "angular distances from P to the fixed points" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/circle", "concept/constant", "concept/cosine", "concept/locus", "concept/spherical-distance" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-7b665ead8b", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 161", "location": "Numerical Solution of Spherical Triangles", "latex": "\\sin c=\\dfrac{\\sin a}{\\sin A}", "name": null, "statement": "In a right spherical triangle, the sine of the hypotenuse c equals the sine of side a divided by the sine of angle A; this gives two values of c, one acute and one obtuse.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "c", "meaning": "side of the spherical triangle opposite the right angle" }, { "unit": "degree", "symbol": "a", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "A", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(sin(c), sin(a)/sin(A))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/sine", "concept/spherical-triangle", "method/solving-a-right-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-1965fa6c3f", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 161", "location": "Numerical Solution of Spherical Triangles", "latex": "\\sin b = \\tan a \\cot A", "name": null, "statement": "In a right spherical triangle, the sine of side b equals the tangent of side a times the cotangent of angle A.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "b", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "a", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "A", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(sin(b), tan(a)*cot(A))", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/sine", "concept/spherical-triangle", "concept/tangent-function", "method/solving-a-right-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-affc03b8b6", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 162", "location": "Numerical Solution of Spherical Triangles", "latex": "\\tan\\tfrac12 A = \\Surd{\\left\\{\\frac{\\sin(s - b)\\sin(s - c)}{\\sin s \\sin(s - a)}\\right\\}}", "name": null, "statement": "For an oblique spherical triangle, the tangent of half angle A equals the square root of sin(s−b) sin(s−c) divided by sin s sin(s−a), where s is the semi-perimeter.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "A", "meaning": "angle of the spherical triangle" }, { "unit": "degree", "symbol": "a", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "b", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "c", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "s", "meaning": "semi-perimeter of the spherical triangle, half the sum of its sides" } ], "sympy": "Eq(tan(A/2), sqrt(sin(s - b)*sin(s - c)/(sin(s)*sin(s - a))))", "physics": false, "states": [], "concepts": [ "concept/formula", "concept/spherical-triangle", "concept/tangent-function", "method/solving-an-oblique-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-0439ff4fe9", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 164", "location": "Numerical Solution of Spherical Triangles", "latex": "\\tan\\dfrac12 (A - B) = \\dfrac{\\sin\\tfrac12 (a - b)}{\\sin\\tfrac12 (a + b)}\\cot\\tfrac12 C", "name": null, "statement": "For an oblique spherical triangle, the tangent of half the difference of angles A and B equals the ratio of sines of half the difference and half the sum of the sides a and b, times the cotangent of half of C.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "A", "meaning": "angle of the spherical triangle" }, { "unit": "degree", "symbol": "B", "meaning": "angle of the spherical triangle" }, { "unit": "degree", "symbol": "a", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "b", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "C", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(tan((A - B)/2), sin((a - b)/2)/sin((a + b)/2)*cot(C/2))", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/sine", "concept/spherical-triangle", "concept/tangent-function", "method/solving-an-oblique-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-92630403c2", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 164", "location": "Numerical Solution of Spherical Triangles", "latex": "\\tan\\tfrac12 (A + B) = \\dfrac{\\cos\\tfrac12 (a - b)}{\\cos\\tfrac12 (a + b)}\\cot\\tfrac12 C", "name": null, "statement": "For an oblique spherical triangle, the tangent of half the sum of angles A and B equals the ratio of cosines of half the difference and half the sum of sides a and b, times the cotangent of half of C.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "A", "meaning": "angle of the spherical triangle" }, { "unit": "degree", "symbol": "B", "meaning": "angle of the spherical triangle" }, { "unit": "degree", "symbol": "a", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "b", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "C", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(tan((A + B)/2), cos((a - b)/2)/cos((a + b)/2)*cot(C/2))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cotangent", "concept/spherical-triangle", "concept/tangent-function", "method/solving-an-oblique-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-a744f1059a", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 164", "location": "Numerical Solution of Spherical Triangles", "latex": "\\sin c = \\dfrac{\\sin a \\sin C}{\\sin A}", "name": null, "statement": "In an oblique spherical triangle, the sine of side c equals the sine of a times the sine of C divided by the sine of A; when sin C exceeds sin A the formula gives two values, which is the ambiguous case.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "c", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "a", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "A", "meaning": "angle of the spherical triangle" }, { "unit": "degree", "symbol": "C", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(sin(c), sin(a)*sin(C)/sin(A))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/sine", "concept/spherical-triangle", "method/solving-an-oblique-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-6804b95e2b", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 165", "location": "Numerical Solution of Spherical Triangles", "latex": "\\cos \\tfrac{1}{2} c = \\dfrac{\\cos \\tfrac{1}{2} (a + b) \\sin \\tfrac{1}{2}C}{\\cos\\tfrac{1}{2}(A+B)}", "name": null, "statement": "In an oblique spherical triangle, the cosine of half of side c equals the cosine of half the sum of a and b times the sine of half of C, divided by the cosine of half the sum of A and B; this determines c without ambiguity.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "c", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "a", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "b", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "A", "meaning": "angle of the spherical triangle" }, { "unit": "degree", "symbol": "B", "meaning": "angle of the spherical triangle" }, { "unit": "degree", "symbol": "C", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(cos(c/2), cos((a + b)/2)*sin(C/2)/cos((A + B)/2))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/cosine", "concept/sine", "concept/spherical-triangle", "method/solving-an-oblique-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-fe2bfe5ccd", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 165", "location": "Numerical Solution of Spherical Triangles", "latex": "\\tan \\theta = \\tan b \\cos C", "name": null, "statement": "The auxiliary angle theta is defined by the tangent of theta equals the tangent of b times the cosine of C; a negative cos C makes theta greater than a right angle.", "kind": "definition", "symbols": [ { "unit": "degree", "symbol": "theta", "meaning": "auxiliary angle introduced in the second method" }, { "unit": "degree", "symbol": "b", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "C", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(tan(theta), tan(b)*cos(C))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/spherical-triangle", "concept/tangent-function", "method/solving-an-oblique-spherical-triangle", "quantity/angle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-ec148c206b", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 165", "location": "Numerical Solution of Spherical Triangles", "latex": "\\cos c = \\dfrac{\\cos b \\cos (a - \\theta)}{\\cos \\theta}", "name": null, "statement": "In an oblique spherical triangle, the cosine of side c equals the cosine of b times the cosine of a minus theta, divided by the cosine of theta.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "c", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "b", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "a", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "theta", "meaning": "auxiliary angle introduced in the second method" } ], "sympy": "Eq(cos(c), cos(b)*cos(a - theta)/cos(theta))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/formula", "concept/spherical-triangle", "method/solving-an-oblique-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-89542f733c", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 166", "location": "Numerical Solution of Spherical Triangles", "latex": "\\sin B = \\dfrac{\\sin b}{\\sin a}\\sin A", "name": null, "statement": "In an oblique spherical triangle, the sine of angle B equals the sine of b over the sine of a, times the sine of A.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "B", "meaning": "angle of the spherical triangle" }, { "unit": "degree", "symbol": "b", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "a", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "A", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(sin(B), sin(b)/sin(a)*sin(A))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-case", "concept/sine", "concept/spherical-triangle", "method/solving-an-oblique-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-432e2e689d", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 166", "location": "Numerical Solution of Spherical Triangles", "latex": "\\tan \\tfrac12 C = \\dfrac{\\cos \\tfrac12 (b - a)}{\\cos \\tfrac12 (b + a)}\\cot \\tfrac12 (B + A)", "name": "Napier's analogies", "statement": "Napier's analogy: the tangent of half of C equals the ratio of cosines of half the difference and half the sum of sides b and a, times the cotangent of half the sum of B and A.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "C", "meaning": "angle of the spherical triangle" }, { "unit": "degree", "symbol": "b", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "a", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "B", "meaning": "angle of the spherical triangle" }, { "unit": "degree", "symbol": "A", "meaning": "angle of the spherical triangle" } ], "sympy": "Eq(tan(C/2), cos((b - a)/2)/cos((b + a)/2)*cot((B + A)/2))", "physics": false, "states": [ "theorem/napier-s-analogies" ], "concepts": [ "concept/cosine", "concept/cotangent", "concept/spherical-triangle", "concept/tangent-function", "method/solving-an-oblique-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/eq-69c3063014", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "scan 166", "location": "Numerical Solution of Spherical Triangles", "latex": "\\tan \\tfrac12 c = \\dfrac{\\cos \\tfrac12 (B + A)}{\\cos \\tfrac12 (B - A)}\\tan \\tfrac12 (b + a)", "name": "Napier's analogies", "statement": "Napier's analogy: the tangent of half of side c equals the ratio of cosines of half the sum and half the difference of angles B and A, times the tangent of half the sum of b and a.", "kind": "formula", "symbols": [ { "unit": "degree", "symbol": "c", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "B", "meaning": "angle of the spherical triangle" }, { "unit": "degree", "symbol": "A", "meaning": "angle of the spherical triangle" }, { "unit": "degree", "symbol": "b", "meaning": "side of the spherical triangle" }, { "unit": "degree", "symbol": "a", "meaning": "side of the spherical triangle" } ], "sympy": "Eq(tan(c/2), cos((B + A)/2)/cos((B - A)/2)*tan((b + a)/2))", "physics": false, "states": [ "theorem/napier-s-analogies" ], "concepts": [ "concept/cosine", "concept/spherical-triangle", "concept/tangent-function", "method/solving-an-oblique-spherical-triangle" ] } ], "exercise_sets": [ { "id": "todhunter-spherical-trigonometry-1886/ex-v", "set": "V", "page": "053", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles", "practices": [ "concept/cosecant", "concept/cosine", "concept/cotangent", "concept/great-circle", "concept/half-angle", "concept/quadrant", "concept/right-spherical-triangle", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle", "concept/tangent-function", "theorem/right-spherical-triangle-relations" ] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv", "set": "IV", "page": "040", "chapter": "todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle", "practices": [ "concept/great-circle", "concept/polar-triangle", "concept/quadrant", "concept/side", "concept/spherical-angle", "concept/spherical-triangle", "theorem/cosine-formula-for-a-side-of-a-spherical-triangle", "theorem/cosine-formula-for-an-angle-of-a-spherical-triangle", "theorem/half-angle-formulae-for-a-spherical-triangle", "theorem/law-of-sines" ] }, { "id": "todhunter-spherical-trigonometry-1886/ex-ix", "set": "IX", "page": "093", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae", "practices": [ "concept/approximation", "concept/area", "concept/chord", "concept/chordal-triangle", "concept/cosine", "concept/cotangent", "concept/diameter", "concept/equilateral-triangle", "concept/great-circle", "concept/inscribed-circle", "concept/plane-triangle", "concept/pole-of-a-circle", "concept/quadrant", "concept/sine", "concept/small-circle", "concept/spherical-triangle", "quantity/spherical-excess", "theorem/legendre-s-theorem" ] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vi", "set": "VI", "page": "068", "chapter": "todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles", "practices": [ "concept/ambiguous-case", "concept/isosceles-spherical-triangle", "concept/quadrant", "concept/spherical-triangle", "method/solving-a-triangle", "theorem/half-angle-formulae-for-a-spherical-triangle", "theorem/law-of-sines" ] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xi", "set": "XI", "page": "105", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle", "practices": [ "concept/cosine", "concept/cotangent", "concept/quadrant", "concept/side", "concept/sine", "concept/small-variation", "concept/spherical-angle", "concept/spherical-triangle", "concept/tangent-function", "theorem/area-of-a-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv", "set": "XV", "page": "155", "chapter": "todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions", "practices": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vii", "set": "VII", "page": "076", "chapter": "todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles", "practices": [ "concept/circumscribed-circle", "concept/excircle", "concept/inscribed-circle", "concept/polar-triangle", "concept/small-circle", "concept/spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xii", "set": "XII", "page": "122", "chapter": "todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry", "practices": [ "concept/area", "concept/cosine", "concept/great-circle", "concept/nine-points-circle", "concept/perpendicular", "concept/plane-angle", "concept/plane-triangle", "concept/plane-trigonometry", "concept/side", "concept/sine", "concept/spherical-angle", "concept/spherical-triangle", "concept/spherical-trigonometry", "theorem/delambre-s-analogies", "theorem/napier-s-analogies" ] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xvi", "set": "XVI", "page": "168", "chapter": "todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles", "practices": [ "method/numerical-solution-of-a-spherical-triangle", "method/solving-a-right-spherical-triangle", "method/solving-an-oblique-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii", "set": "VIII", "page": "083", "chapter": "todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess", "practices": [ "concept/great-circle", "concept/spherical-polygon", "concept/spherical-triangle", "quantity/spherical-excess", "theorem/area-of-a-sphere", "theorem/area-of-a-spherical-polygon", "theorem/area-of-a-spherical-triangle" ] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii", "set": "XIII", "page": "133", "chapter": "todhunter-spherical-trigonometry-1886/ch-polyhedrons", "practices": [] } ], "problems": [ { "id": "todhunter-spherical-trigonometry-1886/ex-iv/1", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 1, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 1", "problem_latex": "If $A = a$, shew that $B$ and $b$ are equal or supplemental, as\nalso $C$ and $c$.", "markdown": "If $A = a$, shew that $B$ and $b$ are equal or supplemental, as also $C$ and $c$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/10", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 10, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 10", "problem_latex": "$AB$, $CD$ are quadrants on the surface of a sphere intersecting\nat $E$, the extremities being joined by great circles: shew\nthat\n\\[\n\\cos AEC = \\cos AC \\cos BD - \\cos BC \\cos AD.\n\\]", "markdown": "$AB$, $CD$ are quadrants on the surface of a sphere intersecting at $E$, the extremities being joined by great circles: shew that AEC = AC BD - BC AD.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/11", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 11, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 11", "problem_latex": "If $b + c = \\pi$, shew that $\\sin 2B + \\sin 2C = 0$.", "markdown": "If $b + c = \\pi$, shew that $\\sin 2B + \\sin 2C = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.const", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/12", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 12, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 12", "problem_latex": "If $DE$ be an arc of a great circle bisecting the sides $AB$,\n$AC$ of a spherical triangle at $D$ and $E$, $P$ a pole of $DE$, and $PB$,\n$PD$, $PE$, $PC$ be joined by arcs of great circles, shew that the angle\n$BPC =$ twice the angle $DPE$.", "markdown": "If $DE$ be an arc of a great circle bisecting the sides $AB$, $AC$ of a spherical triangle at $D$ and $E$, $P$ a pole of $DE$, and $PB$, $PD$, $PE$, $PC$ be joined by arcs of great circles, shew that the angle $BPC =$ twice the angle $DPE$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/13", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 13, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 13", "problem_latex": "In a spherical triangle shew that\n\\[\n\\sin b \\sin c + \\cos b \\cos c \\cos A =\n\\sin B \\sin C - \\cos B \\cos C \\cos a.\n\\]", "markdown": "In a spherical triangle shew that b c + b c A = B C - B C a.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/14", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 14, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 14", "problem_latex": "If $D$ be any point in the side $BC$ of a triangle, shew that\n\\[\n\\cos AD \\sin BC = \\cos AB \\sin DC + \\cos AC \\sin BD.\n\\]", "markdown": "If $D$ be any point in the side $BC$ of a triangle, shew that AD BC = AB DC + AC BD.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/15", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 15, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 15", "problem_latex": "In a spherical triangle shew that $\\theta$, $\\phi$, $\\psi$ be the lengths\nof arcs of great circles drawn from $A$, $B$, $C$ perpendicular to the\nopposite sides,\n\\begin{gather*}\n \\sin a \\sin \\theta =\n \\sin b \\sin \\phi =\n \\sin c \\sin \\psi \\\\\n= \\surd(1 - \\cos^2 a - \\cos^2 b - \\cos^2 c + 2 \\cos a \\cos b \\cos c).\n\\end{gather*}", "markdown": "In a spherical triangle shew that $\\theta$, $\\phi$, $\\psi$ be the lengths of arcs of great circles drawn from $A$, $B$, $C$ perpendicular to the opposite sides, gather* a = b = c = (1 - ^2 a - ^2 b - ^2 c + 2 a b c). gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/16", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 16, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 16", "problem_latex": "In a spherical triangle, if $\\theta$, $\\phi$, $\\psi$ be the arcs bisecting the\nangles $A$, $B$, $C$ respectively and terminated by the opposite sides,\nshew that\n\\[\n \\cot\\theta\\cos\\dfrac A2 +\n \\cot\\phi \\cos\\dfrac B2 +\n \\cot\\psi \\cos\\dfrac C2 =\n \\cot a + \\cot b + \\cot c.\n\\]", "markdown": "In a spherical triangle, if $\\theta$, $\\phi$, $\\psi$ be the arcs bisecting the angles $A$, $B$, $C$ respectively and terminated by the opposite sides, shew that A2 + B2 + C2 = a + b + c.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/17", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 17, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 17", "problem_latex": "Two ports are in the same parallel of latitude, their common\nlatitude being $l$ and their difference of longitude $2\\lambda$: shew\nthat the saving of distance in sailing from one to the other on the\ngreat circle, instead of sailing due East or West, is\n\\[\n 2r\\{\\lambda \\cos l - \\sin^{-1}(\\sin \\lambda \\cos l)\\},\n\\]\n$\\lambda$ being expressed in circular measure, and $r$ being the radius of\nthe Earth.", "markdown": "Two ports are in the same parallel of latitude, their common latitude being $l$ and their difference of longitude $2\\lambda$: shew that the saving of distance in sailing from one to the other on the great circle, instead of sailing due East or West, is 2rl - ^-1(l), $\\lambda$ being expressed in circular measure, and $r$ being the radius of the Earth.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/18", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 18, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 18", "problem_latex": "If a ship be proceeding uniformly along a great circle and\nthe observed latitudes be $l_1$, $l_2$, $l_3$, at equal intervals of time, in\neach of which the distance traversed is $s$, shew that\n\\[\n s = r\\cos^{-1}\n \\frac{\\sin\\tfrac 12(l_1 + l_3)\\cos\\tfrac 12(l_1 - l_3)}\n {\\sin l_2},\n\\]\n$r$ denoting the Earth's radius: and shew that the change of longitude\nmay also be found in terms of the three latitudes.", "markdown": "If a ship be proceeding uniformly along a great circle and the observed latitudes be $l_1$, $l_2$, $l_3$, at equal intervals of time, in each of which the distance traversed is $s$, shew that s = r^-1 12(l_1 + l_3)12(l_1 - l_3) l_2, $r$ denoting the Earth’s radius: and shew that the change of longitude may also be found in terms of the three latitudes.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/2", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 2, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 2", "problem_latex": "If one angle of a triangle be equal to the sum of the other\ntwo, the greatest side is double of the distance of its middle point\nfrom the opposite angle.", "markdown": "If one angle of a triangle be equal to the sum of the other two, the greatest side is double of the distance of its middle point from the opposite angle.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/3", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 3, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 3", "problem_latex": "When does the polar triangle coincide with the primitive\ntriangle?", "markdown": "When does the polar triangle coincide with the primitive triangle?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/4", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 4, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 4", "problem_latex": "If $D$ be the middle point of $AB$, shew that\n\\[\n\\cos AC + \\cos BC = 2 \\cos \\tfrac{1}{2} AB \\cos CD.\n\\]", "markdown": "If $D$ be the middle point of $AB$, shew that AC + BC = 2 12 AB CD.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/5", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 5, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 5", "problem_latex": "If two angles of a spherical triangle be respectively equal\nto the sides opposite to them, shew that the remaining side is the\nsupplement of the remaining angle; or else that the triangle has\ntwo quadrants and two right angles, and then the remaining side\nis equal to the remaining angle.", "markdown": "If two angles of a spherical triangle be respectively equal to the sides opposite to them, shew that the remaining side is the supplement of the remaining angle; or else that the triangle has two quadrants and two right angles, and then the remaining side is equal to the remaining angle.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/6", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 6, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 6", "problem_latex": "In an equilateral triangle, shew that $2 \\cos \\dfrac a2 \\sin \\dfrac A2 = 1$.", "markdown": "In an equilateral triangle, shew that $2 \\cos \\dfrac a2 \\sin \\dfrac A2 = 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/7", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 7, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 7", "problem_latex": "In an equilateral triangle, shew that $\\tan^2 \\dfrac a2 = 1 - 2 \\cos A$;\\\\\nhence deduce the limits between which the sides and the angles of\nan equilateral triangle are restricted.", "markdown": "In an equilateral triangle, shew that $\\tan^2 \\dfrac a2 = 1 - 2 \\cos A$; hence deduce the limits between which the sides and the angles of an equilateral triangle are restricted.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/8", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 8, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 8", "problem_latex": "In an equilateral triangle, shew that $\\sec A = 1 + \\sec a$.", "markdown": "In an equilateral triangle, shew that $\\sec A = 1 + \\sec a$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-iv/9", "set": "todhunter-spherical-trigonometry-1886/ex-iv", "number": 9, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "040", "location": "Exercise IV, problem 9", "problem_latex": "If the three sides of a spherical triangle be halved and\na new triangle formed, the angle $\\theta$ between the new sides $\\dfrac b2$ and $\\dfrac c2$\nis given by $\\cos \\theta = \\cos A + \\tfrac 12 \\tan \\dfrac b2 \\tan \\dfrac c2 \\sin^2 \\theta$.", "markdown": "If the three sides of a spherical triangle be halved and a new triangle formed, the angle $\\theta$ between the new sides $\\dfrac b2$ and $\\dfrac c2$ is given by $\\cos \\theta = \\cos A + \\tfrac 12 \\tan \\dfrac b2 \\tan \\dfrac c2 \\sin^2 \\theta$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-ix/1", "set": "todhunter-spherical-trigonometry-1886/ex-ix", "number": 1, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "093", "location": "Exercise IX, problem 1", "problem_latex": "If the sides of a spherical triangle $AB$, $AC$ be produced to\n$B'$, $C'$, so that $BB'$, $CC'$ are the semi-supplements of $AB$, $AC$\nrespectively, shew that the arc $B'C'$ will subtend an angle at the\ncentre of the sphere equal to the angle between the chords of $AB$\nand $AC$.", "markdown": "If the sides of a spherical triangle $AB$, $AC$ be produced to $B'$, $C'$, so that $BB'$, $CC'$ are the semi-supplements of $AB$, $AC$ respectively, shew that the arc $B'C'$ will subtend an angle at the centre of the sphere equal to the angle between the chords of $AB$ and $AC$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-ix/10", "set": "todhunter-spherical-trigonometry-1886/ex-ix", "number": 10, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "093", "location": "Exercise IX, problem 10", "problem_latex": "From Arts.~110 and 111, shew that approximately\n\\[\n\\log\\beta = \\log\\alpha + \\log\\sin B - \\log\\sin A + \\frac{S}{3r^2}(\\cot A-\\cot B).\n\\]", "markdown": "From Arts. 110 and 111, shew that approximately = + B - A + S3r^2(A-B).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-ix/11", "set": "todhunter-spherical-trigonometry-1886/ex-ix", "number": 11, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "093", "location": "Exercise IX, problem 11", "problem_latex": "By continuing the approximation in Art.~106 so as to\ninclude the terms involving $r^4$, shew that approximately\n\\[\n\\cos A = \\cos A' - \\frac{\\beta\\gamma\\sin^2 A'}{6r^2}\n + \\frac{\\beta\\gamma(\\alpha^2-3\\beta^2-3\\gamma^2)\\sin^2 A'}{180r^4}\\,.\n\\]", "markdown": "By continuing the approximation in Art. 106 so as to include the terms involving $r^4$, shew that approximately A = A’ - ^2 A’6r^2 + (^2-3^2-3^2)^2 A’180r^4 .", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-ix/12", "set": "todhunter-spherical-trigonometry-1886/ex-ix", "number": 12, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "093", "location": "Exercise IX, problem 12", "problem_latex": "From the preceding result shew that if $A = A' + \\theta$ then\napproximately\n\\[\n\\theta = \\frac{\\beta\\gamma\\sin A'}{6r^2}\n \\left( 1+\\frac{7\\beta^2 + 7\\gamma^2 + \\alpha^2}{120 r^2} \\right)\\,.\n\\]", "markdown": "From the preceding result shew that if $A = A' + \\theta$ then approximately = A’6r^2 ( 1+7^2 + 7^2 + ^2120 r^2 ) .", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-ix/2", "set": "todhunter-spherical-trigonometry-1886/ex-ix", "number": 2, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "093", "location": "Exercise IX, problem 2", "problem_latex": "Deduce Legendre's Theorem from the formula\n\\[\n\\tan^2\\frac{A}{2}\n= \\frac{\\sin\\tfrac12(a+b-c) \\sin\\tfrac12(c+a-b) }\n {\\sin\\tfrac12(b+c-a) \\sin\\tfrac12(a+b+c) }\\,.\n\\]", "markdown": "Deduce Legendre’s Theorem from the formula ^2A2 = 12(a+b-c) 12(c+a-b) 12(b+c-a) 12(a+b+c)  .", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-ix/3", "set": "todhunter-spherical-trigonometry-1886/ex-ix", "number": 3, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "093", "location": "Exercise IX, problem 3", "problem_latex": "Four points $A$, $B$, $C$, $D$ on the surface of a sphere are\njoined by arcs of great circles, and $E$, $F$ are the middle points\nof the arcs $AC$, $BD$: shew that\n\\[\n\\cos AB + \\cos BC + \\cos CD + \\cos DA = 4 \\cos AE \\cos BF \\cos FE.\n\\]", "markdown": "Four points $A$, $B$, $C$, $D$ on the surface of a sphere are joined by arcs of great circles, and $E$, $F$ are the middle points of the arcs $AC$, $BD$: shew that AB + BC + CD + DA = 4 AE BF FE.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-ix/4", "set": "todhunter-spherical-trigonometry-1886/ex-ix", "number": 4, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "093", "location": "Exercise IX, problem 4", "problem_latex": "If a quadrilateral $ABCD$ be inscribed in a small circle on\na sphere so that two opposite angles $A$ and $C$ may be at opposite\nextremities of a diameter, the sum of the cosines of the sides is\nconstant.", "markdown": "If a quadrilateral $ABCD$ be inscribed in a small circle on a sphere so that two opposite angles $A$ and $C$ may be at opposite extremities of a diameter, the sum of the cosines of the sides is constant.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-ix/5", "set": "todhunter-spherical-trigonometry-1886/ex-ix", "number": 5, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "093", "location": "Exercise IX, problem 5", "problem_latex": "In a spherical triangle if $A = B = 2C$, shew that\n\\[\n\\cos a\\cos\\frac{a}{2} = \\cos\\left( c+\\frac{a}{2}\\right).\n\\]", "markdown": "In a spherical triangle if $A = B = 2C$, shew that aa2 = ( c+a2).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-ix/6", "set": "todhunter-spherical-trigonometry-1886/ex-ix", "number": 6, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "093", "location": "Exercise IX, problem 6", "problem_latex": "$ABC$ is a spherical triangle each of whose sides is a quadrant;\n$P$ is any point within the triangle: shew that\n\\[\n\\cos PA \\cos PB \\cos PC + \\cot BPC \\cot CPA \\cot APB = 0,\n\\]\nand \\hfill$\n\\tan ABP \\tan BCP \\tan CAP = 1.\n$\\hfill\\phantom{and}", "markdown": "$ABC$ is a spherical triangle each of whose sides is a quadrant; $P$ is any point within the triangle: shew that PA PB PC + BPC CPA APB = 0, and $ \\tan ABP \\tan BCP \\tan CAP = 1. $and", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-ix/7", "set": "todhunter-spherical-trigonometry-1886/ex-ix", "number": 7, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "093", "location": "Exercise IX, problem 7", "problem_latex": "If $O$ be the middle point of an equilateral triangle $ABC$,\nand $P$ any point on the surface of the sphere, then\n\\begin{gather*}\n\\tfrac{1}{4} (\\tan PO \\tan OA)^2 (\\cos PA + \\cos PB + \\cos PC)^2 = \\\\\n\\cos^2 PA + \\cos^2 PB + \\cos^2 PC - \\cos PA \\cos PB - \\cos PB \\cos PC - \\cos PC \\cos PA.\n\\end{gather*}", "markdown": "If $O$ be the middle point of an equilateral triangle $ABC$, and $P$ any point on the surface of the sphere, then gather* 14 (PO OA)^2 (PA + PB + PC)^2 = ^2 PA + ^2 PB + ^2 PC - PA PB - PB PC - PC PA. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-ix/8", "set": "todhunter-spherical-trigonometry-1886/ex-ix", "number": 8, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "093", "location": "Exercise IX, problem 8", "problem_latex": "If $ABC$ be a triangle having each side a quadrant, $O$ the\npole of the inscribed circle, $P$ any point on the sphere, then\n\\[\n(\\cos PA + \\cos PB + \\cos PC)^2 = 3\\cos^2 PO.\n\\]", "markdown": "If $ABC$ be a triangle having each side a quadrant, $O$ the pole of the inscribed circle, $P$ any point on the sphere, then (PA + PB + PC)^2 = 3^2 PO.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-ix/9", "set": "todhunter-spherical-trigonometry-1886/ex-ix", "number": 9, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "093", "location": "Exercise IX, problem 9", "problem_latex": "From each of three points on the surface of a sphere arcs\nare drawn on the surface to three other points situated on a great\ncircle of the sphere, and their cosines are $a$, $b$, $c$; $a'$, $b'$, $c'$; $a''$, $b''$, $c''$.\nShew that $ab''c' + a'bc'' + a''b'c = ab'c'' + a'b''c + a''bc'$.", "markdown": "From each of three points on the surface of a sphere arcs are drawn on the surface to three other points situated on a great circle of the sphere, and their cosines are $a$, $b$, $c$; $a'$, $b'$, $c'$; $a''$, $b''$, $c''$. Shew that $ab''c' + a'bc'' + a''b'c = ab'c'' + a'b''c + a''bc'$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/1", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 1, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 1", "problem_latex": "$\\operatorname{Sin}^2\\dfrac{c}{2} =\n \\sin^2 \\dfrac{a}{2}\\, \\cos^2 \\dfrac{b}{2} +\n \\cos^2 \\dfrac{a}{2}\\, \\sin^2 \\dfrac{b}{2}$.", "markdown": "$\\operatorname{Sin}^2\\dfrac{c}{2} = \\sin^2 \\dfrac{a}{2}\\, \\cos^2 \\dfrac{b}{2} + \\cos^2 \\dfrac{a}{2}\\, \\sin^2 \\dfrac{b}{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/10", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 10, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 10", "problem_latex": "$OAA_1$ is a spherical triangle right-angled at $A_1$ and acute-angled\nat $A$; the arc $A_1A_2$ of a great circle is drawn perpendicular\nto $OA$, then $A_2A_3$ is drawn perpendicular to $OA_1$, and so on: shew\nthat $A_nA_{n+1}$ vanishes when $n$ becomes infinite; and find the value\nof $\\cos AA_1 \\cos A_1A_2 \\cos A_2A_3\\ldots\\ldots$ to infinity.", "markdown": "$OAA_1$ is a spherical triangle right-angled at $A_1$ and acute-angled at $A$; the arc $A_1A_2$ of a great circle is drawn perpendicular to $OA$, then $A_2A_3$ is drawn perpendicular to $OA_1$, and so on: shew that $A_nA_{n+1}$ vanishes when $n$ becomes infinite; and find the value of $\\cos AA_1 \\cos A_1A_2 \\cos A_2A_3\\ldots\\ldots$ to infinity.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig", "other:infinite_product" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/11", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 11, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 11", "problem_latex": "$ABC$ is a right-angled spherical triangle, $A$ not being the\nright angle: shew that if $A = a$, then $c$ and $b$ are quadrants.", "markdown": "$ABC$ is a right-angled spherical triangle, $A$ not being the right angle: shew that if $A = a$, then $c$ and $b$ are quadrants.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/12", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 12, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 12", "problem_latex": "If $\\delta$ be the length of the arc drawn from $C$ perpendicular\nto $AB$ in \\textit{any} triangle, shew that\n\\[\n\\cos\\delta = \\operatorname{cosec} c\\, (\\cos^2 a + \\cos^2 b - 2 \\cos a\\, \\cos b\\, \\cos c)^{\\tfrac{1}{2}}.\n\\]", "markdown": "If $\\delta$ be the length of the arc drawn from $C$ perpendicular to $AB$ in *any* triangle, shew that = cosec c  (^2 a + ^2 b - 2 a  b  c)^12.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/13", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 13, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 13", "problem_latex": "$ABC$ is a great circle of a sphere; $AA'$, $BB'$, $CC'$, are arcs\nof great circles drawn at right angles to $ABC$ and reckoned positive\n%-----File: 055.png------------------------------------------------\nwhen they lie on the same side of it: shew that the condition\nof $A'$, $B'$, $C'$ lying in a great circle is\n\\[\n\\tan AA'\\, \\sin BC + \\tan BB'\\, \\sin CA + \\tan CC'\\, \\sin AB = 0.\n\\]", "markdown": "$ABC$ is a great circle of a sphere; $AA'$, $BB'$, $CC'$, are arcs of great circles drawn at right angles to $ABC$ and reckoned positive %-----File: 055.png------------------------------------------------ when they lie on the same side of it: shew that the condition of $A'$, $B'$, $C'$ lying in a great circle is AA’  BC + BB’  CA + CC’  AB = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/14", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 14, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 14", "problem_latex": "Perpendiculars are drawn from the angles $A$, $B$, $C$ of any\ntriangle meeting the opposite sides at $D$, $E$, $F$ respectively: shew\nthat\n\\[\n\\tan BD\\, \\tan CE\\, \\tan AF = \\tan DC\\, \\tan EA\\, \\tan FB.\n\\]", "markdown": "Perpendiculars are drawn from the angles $A$, $B$, $C$ of any triangle meeting the opposite sides at $D$, $E$, $F$ respectively: shew that BD  CE  AF = DC  EA  FB.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/15", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 15, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 15", "problem_latex": "$Ox$, $Oy$ are two great circles of a sphere at right angles to\neach other, $P$ is any point in $AB$ another great circle. $OC = p$ is\nthe arc perpendicular to $AB$ from $O$, making the angle $COx = a$\nwith $Ox$. $PM$, $PN$ are arcs perpendicular to $Ox$, $Oy$ respectively:\nshew that if $OM = x$ and $ON = y$,\n\\[\n\\cos a\\, \\tan x + \\sin a\\, \\tan y = \\tan p.\n\\]", "markdown": "$Ox$, $Oy$ are two great circles of a sphere at right angles to each other, $P$ is any point in $AB$ another great circle. $OC = p$ is the arc perpendicular to $AB$ from $O$, making the angle $COx = a$ with $Ox$. $PM$, $PN$ are arcs perpendicular to $Ox$, $Oy$ respectively: shew that if $OM = x$ and $ON = y$, a  x + a  y = p.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/16", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 16, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 16", "problem_latex": "The position of a point on a sphere, with reference to two\ngreat circles at right angles to each other as axes, is determined\nby the portions $\\theta$, $\\phi$ of these circles cut off by great circles through\nthe point, and through two points on the axes, each $\\dfrac{\\pi}{2}$ from their\npoint of intersection: shew that if the three points ($\\theta$, $\\phi$), ($\\theta'$, $\\phi'$),\n($\\theta''$, $\\phi''$) lie on the same great circle\n\\begin{gather*}\n \\tan \\phi\\, (\\tan \\theta' - \\tan \\theta'')\n+ \\tan \\phi'\\, (\\tan \\theta'' - \\tan \\theta)\\\\\n+ \\tan \\phi''\\, (\\tan \\theta - \\tan \\theta') = 0.\n\\end{gather*}", "markdown": "The position of a point on a sphere, with reference to two great circles at right angles to each other as axes, is determined by the portions $\\theta$, $\\phi$ of these circles cut off by great circles through the point, and through two points on the axes, each $\\dfrac{\\pi}{2}$ from their point of intersection: shew that if the three points ($\\theta$, $\\phi$), ($\\theta'$, $\\phi'$), ($\\theta''$, $\\phi''$) lie on the same great circle gather* (’ - ”) + ’  (” - ) + ”  (- ’) = 0. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/17", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 17, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 17", "problem_latex": "If a point on a sphere be referred to two great circles at\nright angles to each other as axes, by means of the portions of\nthese axes cut off by great circles drawn through the point and\ntwo points on the axes each $90^\\circ$ from their intersection, shew that\nthe equation to a great circle is\n\\[\n\\tan \\theta\\, \\cot \\alpha + \\tan \\phi\\, \\cot \\beta = 1.\n\\]", "markdown": "If a point on a sphere be referred to two great circles at right angles to each other as axes, by means of the portions of these axes cut off by great circles drawn through the point and two points on the axes each $90^\\circ$ from their intersection, shew that the equation to a great circle is +   = 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/18", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 18, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 18", "problem_latex": "In a spherical triangle, if $A = \\dfrac{\\pi}{5}$, $B = \\dfrac{\\pi}{3}$, and, $C = \\dfrac{\\pi}{2}$, shew that\\, $a + b + c = \\dfrac{\\pi}{2}$.", "markdown": "In a spherical triangle, if $A = \\dfrac{\\pi}{5}$, $B = \\dfrac{\\pi}{3}$, and, $C = \\dfrac{\\pi}{2}$, shew that  $a + b + c = \\dfrac{\\pi}{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/2", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 2, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 2", "problem_latex": "$\\operatorname{Tan}\\tfrac{1}{2}(c + a)\\,\n \\tan \\tfrac{1}{2}(c - a) =\n \\tan^2 \\dfrac{b}{2}$.", "markdown": "$\\operatorname{Tan}\\tfrac{1}{2}(c + a)\\, \\tan \\tfrac{1}{2}(c - a) = \\tan^2 \\dfrac{b}{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/3", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 3, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 3", "problem_latex": "$\\operatorname{Sin}(c - b) = \\tan^2 \\dfrac{A}{2}\\,\\sin(c + b)$.", "markdown": "$\\operatorname{Sin}(c - b) = \\tan^2 \\dfrac{A}{2}\\,\\sin(c + b)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/4", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 4, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 4", "problem_latex": "$\\operatorname{Sin} a\\, \\tan \\tfrac{1}{2}A\n - \\sin b\\, \\tan \\tfrac{1}{2}B = \\sin (a - b)$.", "markdown": "$\\operatorname{Sin} a\\, \\tan \\tfrac{1}{2}A - \\sin b\\, \\tan \\tfrac{1}{2}B = \\sin (a - b)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/5a", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 5, "part": "a", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 5a", "problem_latex": " && \\operatorname{Sin} (c - a)\n&= \\sin b\\, \\cos a\\, \\tan \\tfrac{1}{2}B, &&\\\\\n&& \\operatorname{Sin} (c - a)\n&= \\tan b\\, \\cos c\\, \\tan \\tfrac{1}{2}B. &&", "markdown": "&& Sin (c - a) &= b  a  12B, && && Sin (c - a) &= b  c  12B. &&", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/5b", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 5, "part": "b", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 5b", "problem_latex": " && \\operatorname{Sin} (c - a)\n&= \\sin b\\, \\cos a\\, \\tan \\tfrac{1}{2}B, &&\\\\\n&& \\operatorname{Sin} (c - a)\n&= \\tan b\\, \\cos c\\, \\tan \\tfrac{1}{2}B. &&", "markdown": "&& Sin (c - a) &= b  a  12B, && && Sin (c - a) &= b  c  12B. &&", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/6", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 6, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 6", "problem_latex": "If $ABC$ be a spherical triangle, right-angled at $C$, and\n$\\cos A = \\cos^2 a$, shew that if $A$ be not a right angle $b + c = \\tfrac{1}{2}\\pi$ or\n$\\dfrac{3}{2}\\pi$, according as $b$ and $c$ are both less or both greater than $\\dfrac{\\pi}{2}$.", "markdown": "If $ABC$ be a spherical triangle, right-angled at $C$, and $\\cos A = \\cos^2 a$, shew that if $A$ be not a right angle $b + c = \\tfrac{1}{2}\\pi$ or $\\dfrac{3}{2}\\pi$, according as $b$ and $c$ are both less or both greater than $\\dfrac{\\pi}{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/7", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 7, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 7", "problem_latex": "If $\\alpha$, $\\beta$ be the arcs drawn from the right angle respectively\nperpendicular to and bisecting the hypotenuse $c$, shew that\n\\[\n\\sin^2 \\dfrac{c}{2}\\,(1 + \\sin^2\\alpha) = \\sin^2\\beta.\n\\]", "markdown": "If $\\alpha$, $\\beta$ be the arcs drawn from the right angle respectively perpendicular to and bisecting the hypotenuse $c$, shew that ^2 c2 (1 + ^2) = ^2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/8", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 8, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 8", "problem_latex": "In a triangle, if $C$ be a right angle and $D$ the middle point\nof $AB$, shew that\n\\[\n4\\cos^2\\dfrac{c}{2}\\, \\sin^2 CD = \\sin^2 a + \\sin^2 b.\n\\]", "markdown": "In a triangle, if $C$ be a right angle and $D$ the middle point of $AB$, shew that 4^2c2  ^2 CD = ^2 a + ^2 b.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-v/9", "set": "todhunter-spherical-trigonometry-1886/ex-v", "number": 9, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "053", "location": "Exercise V, problem 9", "problem_latex": "In a right-angled triangle, if $\\delta$ be the length of the arc\ndrawn from $C$ perpendicular to the hypotenuse $AB$, shew that\n\\[\n\\cot\\delta = \\surd{(\\cot^2a + \\cot^2b)}.\n\\]", "markdown": "In a right-angled triangle, if $\\delta$ be the length of the arc drawn from $C$ perpendicular to the hypotenuse $AB$, shew that = (^2a + ^2b).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vi/1", "set": "todhunter-spherical-trigonometry-1886/ex-vi", "number": 1, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "068", "location": "Exercise VI, problem 1", "problem_latex": "The sides of a triangle are $105^\\circ$, $90^\\circ$, and $75^\\circ$ respectively:\nfind the sines of all the angles.", "markdown": "The sides of a triangle are $105^\\circ$, $90^\\circ$, and $75^\\circ$ respectively: find the sines of all the angles.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vi/10", "set": "todhunter-spherical-trigonometry-1886/ex-vi", "number": 10, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "068", "location": "Exercise VI, problem 10", "problem_latex": "If $c_1$, $c_2$ be the two values of the third side when $A$, $a$, $b$\nare given and the triangle is ambiguous, shew that\n\\[\n \\tan \\dfrac{c_1}{2} \\tan \\dfrac{c_2}{2}\n= \\tan \\tfrac{1}{2} (b - a) \\tan \\tfrac{1}{2} (b + a).\n\\]", "markdown": "If $c_1$, $c_2$ be the two values of the third side when $A$, $a$, $b$ are given and the triangle is ambiguous, shew that c_12 c_22 = 12 (b - a) 12 (b + a).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.trig", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vi/2a", "set": "todhunter-spherical-trigonometry-1886/ex-vi", "number": 2, "part": "a", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "068", "location": "Exercise VI, problem 2a", "problem_latex": "Shew that $\\tan \\tfrac{1}{2} A \\tan \\tfrac{1}{2} B= \\dfrac{\\sin(s-c)}{\\sin s}$. Solve a triangle\nwhen a side, an adjacent angle, and the sum of the other two\nsides are given.", "markdown": "Shew that $\\tan \\tfrac{1}{2} A \\tan \\tfrac{1}{2} B= \\dfrac{\\sin(s-c)}{\\sin s}$. Solve a triangle when a side, an adjacent angle, and the sum of the other two sides are given.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.trig", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vi/2b", "set": "todhunter-spherical-trigonometry-1886/ex-vi", "number": 2, "part": "b", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "068", "location": "Exercise VI, problem 2b", "problem_latex": "Shew that $\\tan \\tfrac{1}{2} A \\tan \\tfrac{1}{2} B= \\dfrac{\\sin(s-c)}{\\sin s}$. Solve a triangle\nwhen a side, an adjacent angle, and the sum of the other two\nsides are given.", "markdown": "Shew that $\\tan \\tfrac{1}{2} A \\tan \\tfrac{1}{2} B= \\dfrac{\\sin(s-c)}{\\sin s}$. Solve a triangle when a side, an adjacent angle, and the sum of the other two sides are given.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vi/3", "set": "todhunter-spherical-trigonometry-1886/ex-vi", "number": 3, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "068", "location": "Exercise VI, problem 3", "problem_latex": "Solve a triangle having given a side, an adjacent angle,\nand the sum of the other two angles.", "markdown": "Solve a triangle having given a side, an adjacent angle, and the sum of the other two angles.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vi/4", "set": "todhunter-spherical-trigonometry-1886/ex-vi", "number": 4, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "068", "location": "Exercise VI, problem 4", "problem_latex": "A triangle has the sum of two sides equal to a semicircumference:\nfind the arc joining the vertex with the middle of\nthe base.", "markdown": "A triangle has the sum of two sides equal to a semicircumference: find the arc joining the vertex with the middle of the base.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vi/5a", "set": "todhunter-spherical-trigonometry-1886/ex-vi", "number": 5, "part": "a", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "068", "location": "Exercise VI, problem 5a", "problem_latex": "If $a$, $b$, $c$ are known, $c$ being a \\textit{quadrant}, determine the\nangles: shew also that if $\\delta$ be the perpendicular on $c$ from the\nopposite angle, $\\cos^2 \\delta = \\cos^2 a + \\cos^2 b$.", "markdown": "If $a$, $b$, $c$ are known, $c$ being a *quadrant*, determine the angles: shew also that if $\\delta$ be the perpendicular on $c$ from the opposite angle, $\\cos^2 \\delta = \\cos^2 a + \\cos^2 b$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vi/5b", "set": "todhunter-spherical-trigonometry-1886/ex-vi", "number": 5, "part": "b", "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "068", "location": "Exercise VI, problem 5b", "problem_latex": "If $a$, $b$, $c$ are known, $c$ being a \\textit{quadrant}, determine the\nangles: shew also that if $\\delta$ be the perpendicular on $c$ from the\nopposite angle, $\\cos^2 \\delta = \\cos^2 a + \\cos^2 b$.", "markdown": "If $a$, $b$, $c$ are known, $c$ being a *quadrant*, determine the angles: shew also that if $\\delta$ be the perpendicular on $c$ from the opposite angle, $\\cos^2 \\delta = \\cos^2 a + \\cos^2 b$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.trig", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vi/6", "set": "todhunter-spherical-trigonometry-1886/ex-vi", "number": 6, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "068", "location": "Exercise VI, problem 6", "problem_latex": "If one side of a spherical triangle be divided into four\nequal parts, and $\\theta_1$, $\\theta_2$, $\\theta_3$, $\\theta_4$, be the angles subtended at the opposite\nangle by the parts taken in order, shew that\n\\[\n\\sin(\\theta_1 + \\theta_2) \\sin \\theta_2 \\sin\\theta_4 = \\sin(\\theta_3 + \\theta_4) \\sin\\theta_1 \\sin\\theta_3.\n\\]", "markdown": "If one side of a spherical triangle be divided into four equal parts, and $\\theta_1$, $\\theta_2$, $\\theta_3$, $\\theta_4$, be the angles subtended at the opposite angle by the parts taken in order, shew that (_1 + _2) _2 _4 = (_3 + _4) _1 _3.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.trig", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vi/7", "set": "todhunter-spherical-trigonometry-1886/ex-vi", "number": 7, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "068", "location": "Exercise VI, problem 7", "problem_latex": "In a spherical triangle if $A = B = 2C$, shew that\n\\[\n8 \\sin\\left(a + \\dfrac{c}{2}\\right) \\sin^2 \\dfrac{c}{2} \\cos \\dfrac{c}{2} = \\sin^3 a.\n\\]", "markdown": "In a spherical triangle if $A = B = 2C$, shew that 8 (a + c2) ^2 c2 c2 = ^3 a.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.trig", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vi/8", "set": "todhunter-spherical-trigonometry-1886/ex-vi", "number": 8, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "068", "location": "Exercise VI, problem 8", "problem_latex": "In a spherical triangle if $A = B = 2C$, shew that\n\\[\n8 \\sin^2 \\dfrac{C}{2} \\left(\\cos s + \\sin \\dfrac{C}{2}\\right)\n\\dfrac{\\cos\\dfrac{c}{2}}{\\cos a} = 1.\n\\]", "markdown": "In a spherical triangle if $A = B = 2C$, shew that 8 ^2 C2 (s + C2) c2a = 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.trig", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vi/9", "set": "todhunter-spherical-trigonometry-1886/ex-vi", "number": 9, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "068", "location": "Exercise VI, problem 9", "problem_latex": "If the equal sides of an isosceles triangle $ABC$ be bisected\nby an arc $DE$, and $BC$ be the base, shew that\n\\[\n\\sin \\dfrac{DE}{2} = \\tfrac{1}{2} \\sin \\dfrac{BC}{2} \\sec \\dfrac{AC}{2}.\n\\]", "markdown": "If the equal sides of an isosceles triangle $ABC$ be bisected by an arc $DE$, and $BC$ be the base, shew that DE2 = 12 BC2 AC2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.trig", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vii/1", "set": "todhunter-spherical-trigonometry-1886/ex-vii", "number": 1, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "076", "location": "Exercise VII, problem 1", "problem_latex": "$\\operatorname{Tan} r_1 \\tan r_2 \\tan r_3 = \\tan r \\sin^2 s$.", "markdown": "$\\operatorname{Tan} r_1 \\tan r_2 \\tan r_3 = \\tan r \\sin^2 s$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vii/10", "set": "todhunter-spherical-trigonometry-1886/ex-vii", "number": 10, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "076", "location": "Exercise VII, problem 10", "problem_latex": "If three small circles be inscribed in a spherical triangle\nhaving each of its angles $120^\\circ$, so that each touches the other two\nas well as two sides of the triangle, shew that the radius of each\nof the small circles $= 30^\\circ$, and that the centres of the three small\ncircles coincide with the angular points of the polar triangle.", "markdown": "If three small circles be inscribed in a spherical triangle having each of its angles $120^\\circ$, so that each touches the other two as well as two sides of the triangle, shew that the radius of each of the small circles $= 30^\\circ$, and that the centres of the three small circles coincide with the angular points of the polar triangle.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vii/2", "set": "todhunter-spherical-trigonometry-1886/ex-vii", "number": 2, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "076", "location": "Exercise VII, problem 2", "problem_latex": "$\\operatorname{Tan} R + \\cot r = \\tan R_1 + \\cot r_1 = \\tan R_2 + \\cot r_2$\\\\\n\\rightline{$= \\tan R_3 + \\cot r_3 = \\tfrac{1}{2} (\\cot r + \\cot r_1 + \\cot r_2 + \\cot r_3)$.}", "markdown": "$\\operatorname{Tan} R + \\cot r = \\tan R_1 + \\cot r_1 = \\tan R_2 + \\cot r_2$ $= \\tan R_3 + \\cot r_3 = \\tfrac{1}{2} (\\cot r + \\cot r_1 + \\cot r_2 + \\cot r_3)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vii/3", "set": "todhunter-spherical-trigonometry-1886/ex-vii", "number": 3, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "076", "location": "Exercise VII, problem 3", "problem_latex": "$\\operatorname{Tan}^2 R + \\tan^2 R_1 + \\tan^2 R_2 + \\tan^2 R_3$\\\\\n\\rightline{$ = \\cot^2 r + \\cot^2 r_1 + \\cot^2 r_2 + \\cot^2 r_3$.}", "markdown": "$\\operatorname{Tan}^2 R + \\tan^2 R_1 + \\tan^2 R_2 + \\tan^2 R_3$ $ = \\cot^2 r + \\cot^2 r_1 + \\cot^2 r_2 + \\cot^2 r_3$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vii/4", "set": "todhunter-spherical-trigonometry-1886/ex-vii", "number": 4, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "076", "location": "Exercise VII, problem 4", "problem_latex": "$\\dfrac{\\operatorname{Tan} r_1 + \\tan r_2 + \\tan r_3 - \\tan r}\n {\\cot r_1 + \\cot r_2 + \\cot r_3 - \\cot r}\n= \\tfrac{1}{2} (1 + \\cos a + \\cos b + \\cos c)$.", "markdown": "$\\dfrac{\\operatorname{Tan} r_1 + \\tan r_2 + \\tan r_3 - \\tan r} {\\cot r_1 + \\cot r_2 + \\cot r_3 - \\cot r} = \\tfrac{1}{2} (1 + \\cos a + \\cos b + \\cos c)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vii/5", "set": "todhunter-spherical-trigonometry-1886/ex-vii", "number": 5, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "076", "location": "Exercise VII, problem 5", "problem_latex": "$\\operatorname{Cosec}^2 r\n= \\cot (s - a) \\cot (s - b)\n+ \\cot (s - b) \\cot (s - c)\n+ \\cot (s - c) (s - a)$.", "markdown": "$\\operatorname{Cosec}^2 r = \\cot (s - a) \\cot (s - b) + \\cot (s - b) \\cot (s - c) + \\cot (s - c) (s - a)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vii/6", "set": "todhunter-spherical-trigonometry-1886/ex-vii", "number": 6, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "076", "location": "Exercise VII, problem 6", "problem_latex": "$\\operatorname{Cosec}^2 r_1\n= \\cot (s - b) \\cot (s - c)\n- \\cot s \\cot (s - b)\n- \\cot s \\cot (s - c)$.", "markdown": "$\\operatorname{Cosec}^2 r_1 = \\cot (s - b) \\cot (s - c) - \\cot s \\cot (s - b) - \\cot s \\cot (s - c)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vii/7", "set": "todhunter-spherical-trigonometry-1886/ex-vii", "number": 7, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "076", "location": "Exercise VII, problem 7", "problem_latex": "$\\operatorname{Tan} R_1 \\tan R_2 \\tan R_3 = \\tan R \\sec^2 S$.", "markdown": "$\\operatorname{Tan} R_1 \\tan R_2 \\tan R_3 = \\tan R \\sec^2 S$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vii/8", "set": "todhunter-spherical-trigonometry-1886/ex-vii", "number": 8, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "076", "location": "Exercise VII, problem 8", "problem_latex": "Shew that in an equilateral triangle $\\tan R = 2\\tan r$.", "markdown": "Shew that in an equilateral triangle $\\tan R = 2\\tan r$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-vii/9", "set": "todhunter-spherical-trigonometry-1886/ex-vii", "number": 9, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "076", "location": "Exercise VII, problem 9", "problem_latex": "If $ABC$ be an equilateral spherical triangle, $P$ the pole of\nthe circle circumscribing it, $Q$ any point on the sphere, shew that\n\\[\n \\cos QA + \\cos QB + \\cos QC = 3\\cos PA \\cos PQ.\n\\]", "markdown": "If $ABC$ be an equilateral spherical triangle, $P$ the pole of the circle circumscribing it, $Q$ any point on the sphere, shew that QA + QB + QC = 3PA PQ.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/1", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 1, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 1", "problem_latex": "Find the angles and sides of an equilateral triangle whose\narea is one-fourth of that of the sphere on which it is described.", "markdown": "Find the angles and sides of an equilateral triangle whose area is one-fourth of that of the sphere on which it is described.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.const", "core.solve.num", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/10", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 10, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 10", "problem_latex": "If the angles of a spherical triangle be together equal to\nfour right angles\n\\[\n \\cos^2\\tfrac{1}{2}a + \\cos^2\\tfrac{1}{2}b + \\cos^2\\tfrac{1}{2}c = 1.\n\\]", "markdown": "If the angles of a spherical triangle be together equal to four right angles ^212a + ^212b + ^212c = 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/11", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 11, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 11", "problem_latex": "If $r_1$, $r_2$, $r_3$ be the radii of three small circles of a\nsphere of radius $r$ which touch one another at $P$, $Q$, $R$, and\n$A$, $B$, $C$ be the angles of the spherical triangle formed by joining\ntheir centres,\n\\[\n\\text{area }PQR = (A\\cos r_1 + B\\cos r_2 + C\\cos r_3 - \\pi)r^2.\n\\]", "markdown": "If $r_1$, $r_2$, $r_3$ be the radii of three small circles of a sphere of radius $r$ which touch one another at $P$, $Q$, $R$, and $A$, $B$, $C$ be the angles of the spherical triangle formed by joining their centres, areaPQR = (Ar_1 + Br_2 + Cr_3 - )r^2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.const", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/12", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 12, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 12", "problem_latex": "Shew that\n\\[\n\\sin s\n= \\frac{\\Bigl\\{\\sin\\frac{1}{2}E \\sin(A-\\frac{1}{2}E)\n \\sin(B-\\frac{1}{2}E) \\sin(C-\\frac{1}{2}E)\n \\Bigr\\}^{\\frac{1}{2}} }\n {2\\sin\\frac{1}{2}A \\sin\\frac{1}{2}B \\sin\\frac{1}{2}C } \\,.\n\\]", "markdown": "Shew that s = 12E (A-12E) (B-12E) (C-12E) ^12 212A 12B 12C  .", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/13", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 13, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 13", "problem_latex": "Given two sides of a spherical triangle, determine when\nthe area is a maximum.", "markdown": "Given two sides of a spherical triangle, determine when the area is a maximum.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/14", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 14, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 14", "problem_latex": "Find the area of a regular polygon of a given number of\nsides formed by arcs of great circles on the surface of a sphere;\nand hence deduce that, if $\\alpha$ be the angular radius of a small\ncircle, its area is to that of the whole surface of the sphere as\n$\\operatorname{versin}\\alpha$ is to 2.", "markdown": "Find the area of a regular polygon of a given number of sides formed by arcs of great circles on the surface of a sphere; and hence deduce that, if $\\alpha$ be the angular radius of a small circle, its area is to that of the whole surface of the sphere as $\\operatorname{versin}\\alpha$ is to 2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.const", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/15", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 15, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 15", "problem_latex": "$A$, $B$, $C$ are the angular points of a spherical triangle;\n$A'$, $B'$, $C'$ are the middle points of the respectively opposite sides.\nIf $E$ be the spherical excess of the triangle, shew that\n\\[\n \\cos\\tfrac{1}{2}E\n= \\frac{\\cos A'B'}{\\cos \\frac{1}{2}c}\n= \\frac{\\cos B'C'}{\\cos \\frac{1}{2}a}\n= \\frac{\\cos C'A'}{\\cos \\frac{1}{2}b}\\,.\n\\]", "markdown": "$A$, $B$, $C$ are the angular points of a spherical triangle; $A'$, $B'$, $C'$ are the middle points of the respectively opposite sides. If $E$ be the spherical excess of the triangle, shew that 12E = A’B’12c = B’C’12a = C’A’12b .", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/16", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 16, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 16", "problem_latex": "If one of the arcs of great circles which join the middle\npoints of the sides of a spherical triangle be a quadrant, shew that\nthe other two are also quadrants.", "markdown": "If one of the arcs of great circles which join the middle points of the sides of a spherical triangle be a quadrant, shew that the other two are also quadrants.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/2", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 2, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 2", "problem_latex": "Find the surface of an equilateral and equiangular spherical\npolygon of $n$ sides, and determine the value of each of the\nangles when the surface equals half the surface of the sphere.", "markdown": "Find the surface of an equilateral and equiangular spherical polygon of $n$ sides, and determine the value of each of the angles when the surface equals half the surface of the sphere.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.solve.num", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/3", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 3, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 3", "problem_latex": "If $a=b=\\dfrac{\\pi}{3}$, and $c=\\dfrac{\\pi}{2}$, shew that $E=\\cos^{-1}\\dfrac{7}{9}$.", "markdown": "If $a=b=\\dfrac{\\pi}{3}$, and $c=\\dfrac{\\pi}{2}$, shew that $E=\\cos^{-1}\\dfrac{7}{9}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/4", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 4, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 4", "problem_latex": "If the angle $C$ of a spherical triangle be a right angle,\nshew that\n\\[\n\\sin \\tfrac{1}{2} E= \\sin \\tfrac{1}{2} a \\sin \\tfrac{1}{2} b \\sec \\tfrac{1}{2} c, \\quad\n\\cos \\tfrac{1}{2} E= \\cos \\tfrac{1}{2} a \\cos \\tfrac{1}{2} b \\sec \\tfrac{1}{2} c.\n\\]", "markdown": "If the angle $C$ of a spherical triangle be a right angle, shew that 12 E= 12 a 12 b 12 c, 12 E= 12 a 12 b 12 c.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/5", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 5, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 5", "problem_latex": "If the angle $C$ be a right angle, shew that\n\\[\n\\frac{\\sin^2 c}{\\cos c}\\cos E= \\frac{\\sin^2 a}{\\cos a}+\\frac{\\sin^2 b}{\\cos b}.\n\\]", "markdown": "If the angle $C$ be a right angle, shew that ^2 ccE= ^2 aa+^2 bb.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/6", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 6, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 6", "problem_latex": "If $a=b$ and $C=\\dfrac{\\pi}{2}$, shew that $\\tan E=\\dfrac{\\sin^2 a}{2\\cos a}$.", "markdown": "If $a=b$ and $C=\\dfrac{\\pi}{2}$, shew that $\\tan E=\\dfrac{\\sin^2 a}{2\\cos a}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/7", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 7, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 7", "problem_latex": "The sum of the angles in a right-angled triangle is less\nthan four right angles.", "markdown": "The sum of the angles in a right-angled triangle is less than four right angles.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/8", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 8, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 8", "problem_latex": "Draw through a given point in the side of a spherical\ntriangle an arc of a great circle cutting off a given part of the\ntriangle.", "markdown": "Draw through a given point in the side of a spherical triangle an arc of a great circle cutting off a given part of the triangle.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-viii/9", "set": "todhunter-spherical-trigonometry-1886/ex-viii", "number": 9, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "083", "location": "Exercise VIII, problem 9", "problem_latex": "In a spherical triangle if $\\cos C=-\\tan\\dfrac{a}{2}\\tan\\dfrac{b}{2}$, then\n$C=A+B$.", "markdown": "In a spherical triangle if $\\cos C=-\\tan\\dfrac{a}{2}\\tan\\dfrac{b}{2}$, then $C=A+B$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xi/1", "set": "todhunter-spherical-trigonometry-1886/ex-xi", "number": 1, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "105", "location": "Exercise XI, problem 1", "problem_latex": "In a spherical triangle, if $C$ and $c$ remain constant while\n$a$ and $b$ receive the small increments $\\delta a$ and $\\delta b$ respectively, shew\nthat\n\\[\n \\frac{\\delta a}{\\surd{(1 - n^2 \\sin^2 a)}} +\n \\frac{\\delta b}{\\surd{(1 - n^2 \\sin^2 b)}} = 0 \\text{ where }\nn = \\frac{\\sin C}{\\sin c}\\,.\n\\]", "markdown": "In a spherical triangle, if $C$ and $c$ remain constant while $a$ and $b$ receive the small increments $\\delta a$ and $\\delta b$ respectively, shew that a(1 - n^2 ^2 a) + b(1 - n^2 ^2 b) = 0 where n = Cc .", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xi/2", "set": "todhunter-spherical-trigonometry-1886/ex-xi", "number": 2, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "105", "location": "Exercise XI, problem 2", "problem_latex": "If $C$ and $c$ remain constant, and a small change be made\nin $a$, find the consequent changes in the other parts of the triangle.\nFind also the change in the area.", "markdown": "If $C$ and $c$ remain constant, and a small change be made in $a$, find the consequent changes in the other parts of the triangle. Find also the change in the area.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xi/3", "set": "todhunter-spherical-trigonometry-1886/ex-xi", "number": 3, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "105", "location": "Exercise XI, problem 3", "problem_latex": "Supposing $A$ and $c$ to remain constant, prove the following\nequations, connecting the small variations of pairs of the other\nelements:\n\\begin{gather*}\n \\sin C \\delta b = \\sin a \\delta B,\\quad\n \\delta b \\sin C = -\\delta C \\tan a,\\quad\n \\delta a \\tan C = \\delta B \\sin a,\n\\\\\n \\delta a \\tan C = -\\delta C \\tan a,\\quad\n \\delta b \\cos C = \\delta a,\\quad\n \\delta B \\cos a = -\\delta C.\n\\end{gather*}", "markdown": "Supposing $A$ and $c$ to remain constant, prove the following equations, connecting the small variations of pairs of the other elements: gather* C b = a B, b C = -C a, a C = B a, a C = -C a, b C = a, B a = -C. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xi/4", "set": "todhunter-spherical-trigonometry-1886/ex-xi", "number": 4, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "105", "location": "Exercise XI, problem 4", "problem_latex": "Supposing $b$ and $c$ to remain constant, prove the following\nequations connecting the small variations of pairs of the other\nelements:\n\\begin{align*}\n& \\delta B \\tan C = \\delta C \\tan B, &\n& \\delta a \\cot C = -\\delta B \\sin a,\n\\\\\n& \\delta a = \\delta A \\sin c \\sin B, &\n& \\delta A \\sin B \\cos C = -\\delta B \\sin A.\n\\end{align*}", "markdown": "Supposing $b$ and $c$ to remain constant, prove the following equations connecting the small variations of pairs of the other elements: align* & B C = C B, & & a C = -B a, & a = A c B, & & A B C = -B A. align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xi/5", "set": "todhunter-spherical-trigonometry-1886/ex-xi", "number": 5, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "105", "location": "Exercise XI, problem 5", "problem_latex": "Supposing $B$ and $C$ to remain constant, prove the following\nequations connecting the small variations of pairs of the\nother elements:\n\\begin{align*}\n \\delta b \\tan c &= \\delta c \\tan b, &\\quad\n \\delta A \\cot c &= \\delta b \\sin A,\n\\\\\n \\delta A &= \\delta a \\sin b \\sin C, &\\quad\n \\delta a \\sin B \\cos c &= \\delta b \\sin A.\n\\end{align*}", "markdown": "Supposing $B$ and $C$ to remain constant, prove the following equations connecting the small variations of pairs of the other elements: align* b c &= c b, & A c &= b A, A &= a b C, & a B c &= b A. align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xi/6", "set": "todhunter-spherical-trigonometry-1886/ex-xi", "number": 6, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "105", "location": "Exercise XI, problem 6", "problem_latex": "If $A$ and $C$ are constant, and $b$ be increased by a small\nquantity, shew that $a$ will be increased or diminished according as\n$c$ is less or greater than a quadrant.", "markdown": "If $A$ and $C$ are constant, and $b$ be increased by a small quantity, shew that $a$ will be increased or diminished according as $c$ is less or greater than a quadrant.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xii/1", "set": "todhunter-spherical-trigonometry-1886/ex-xii", "number": 1, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "122", "location": "Exercise XII, problem 1", "problem_latex": "From the formula $\\sin\\dfrac{a}{2}=\\Surd{\\left\\{\\dfrac{-\\cos S\\cos(S-A)}{\\sin B\\sin C}\\right\\}}$ deduce the expression for the area of a plane triangle, namely\n$\\dfrac{a^2\\sin B\\sin C}{2\\sin A}$, when the radius of the sphere is indefinitely increased.", "markdown": "From the formula $\\sin\\dfrac{a}{2}=\\Surd{\\left\\{\\dfrac{-\\cos S\\cos(S-A)}{\\sin B\\sin C}\\right\\}}$ deduce the expression for the area of a plane triangle, namely $\\dfrac{a^2\\sin B\\sin C}{2\\sin A}$, when the radius of the sphere is indefinitely increased.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xii/10", "set": "todhunter-spherical-trigonometry-1886/ex-xii", "number": 10, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "122", "location": "Exercise XII, problem 10", "problem_latex": "Shew that the points determined in Examples 8 and 9,\nand the point $N$ of Art.\\ 146 are on a great circle.\n\nState the corresponding theorem in Plane Geometry.", "markdown": "Shew that the points determined in Examples 8 and 9, and the point $N$ of Art. 146 are on a great circle. State the corresponding theorem in Plane Geometry.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xii/11", "set": "todhunter-spherical-trigonometry-1886/ex-xii", "number": 11, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "122", "location": "Exercise XII, problem 11", "problem_latex": "If one angle of a spherical triangle remains constant while\nthe adjacent sides are increased, shew that the area and the sum\nof the angles are increased.", "markdown": "If one angle of a spherical triangle remains constant while the adjacent sides are increased, shew that the area and the sum of the angles are increased.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xii/12", "set": "todhunter-spherical-trigonometry-1886/ex-xii", "number": 12, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "122", "location": "Exercise XII, problem 12", "problem_latex": "If the arcs bisecting two angles of a spherical triangle and\nterminated at the opposite sides are equal, the bisected angles will\nbe equal provided their sum be less than $180^\\circ$.", "markdown": "If the arcs bisecting two angles of a spherical triangle and terminated at the opposite sides are equal, the bisected angles will be equal provided their sum be less than $180^\\circ$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xii/2", "set": "todhunter-spherical-trigonometry-1886/ex-xii", "number": 2, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "122", "location": "Exercise XII, problem 2", "problem_latex": "Two triangles $ABC$, $abc$, spherical or plane, equal in all\nrespects, differ slightly in position: shew that\n\\[\n\\cos ABb\\cos BCc\\cos CAa+\\cos ACc\\cos CBb\\cos BAa=0.\n\\]", "markdown": "Two triangles $ABC$, $abc$, spherical or plane, equal in all respects, differ slightly in position: shew that ABbBCcCAa+ACcCBbBAa=0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xii/3", "set": "todhunter-spherical-trigonometry-1886/ex-xii", "number": 3, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "122", "location": "Exercise XII, problem 3", "problem_latex": "Deduce formul\\ae\\ in Plane Trigonometry from Napier's\nAnalogies.", "markdown": "Deduce formul in Plane Trigonometry from Napier’s Analogies.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xii/4", "set": "todhunter-spherical-trigonometry-1886/ex-xii", "number": 4, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "122", "location": "Exercise XII, problem 4", "problem_latex": "Deduce formul\\ae\\ in Plane Trigonometry from Delambre's\nAnalogies.", "markdown": "Deduce formul in Plane Trigonometry from Delambre’s Analogies.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xii/5", "set": "todhunter-spherical-trigonometry-1886/ex-xii", "number": 5, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "122", "location": "Exercise XII, problem 5", "problem_latex": "From the formula\n$\\cos\\dfrac{c}{2}\\cos\\dfrac{A+B}{2}\n=\\sin\\dfrac{C}{2}\\cos\\dfrac{a+b}{2}$ deduce\nthe area of a plane triangle in terms of the sides and one of the\nangles.", "markdown": "From the formula $\\cos\\dfrac{c}{2}\\cos\\dfrac{A+B}{2} =\\sin\\dfrac{C}{2}\\cos\\dfrac{a+b}{2}$ deduce the area of a plane triangle in terms of the sides and one of the angles.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "cas.trig", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xii/6", "set": "todhunter-spherical-trigonometry-1886/ex-xii", "number": 6, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "122", "location": "Exercise XII, problem 6", "problem_latex": "What result is obtained from Example 7 to Chapter VI.,\nby supposing the radius of the sphere infinite?", "markdown": "What result is obtained from Example 7 to Chapter VI., by supposing the radius of the sphere infinite?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xii/7", "set": "todhunter-spherical-trigonometry-1886/ex-xii", "number": 7, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "122", "location": "Exercise XII, problem 7", "problem_latex": "From the angle $C$ of a spherical triangle a perpendicular is\ndrawn to the arc which joins the middle points of the sides $a$ and\n$b$: shew that this perpendicular makes an angle $S-B$ with the\nside $a$, and an angle $S-A$ with the side $b$.", "markdown": "From the angle $C$ of a spherical triangle a perpendicular is drawn to the arc which joins the middle points of the sides $a$ and $b$: shew that this perpendicular makes an angle $S-B$ with the side $a$, and an angle $S-A$ with the side $b$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xii/8", "set": "todhunter-spherical-trigonometry-1886/ex-xii", "number": 8, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "122", "location": "Exercise XII, problem 8", "problem_latex": "From each angle of a spherical triangle a perpendicular is\ndrawn to the arc which joins the middle points of the adjacent\nsides. Shew that these perpendiculars meet at a point; and that\n%-----File: 123.png------------------------------------------------\nif $x$, $y$, $z$ are the perpendiculars from this point on the sides $a$, $b$, $c$\nrespectively,\n\\[\n \\frac{\\sin x}{\\sin(S-B)\\sin(S-C)}\n= \\frac{\\sin y}{\\sin(S-C)\\sin(S-A)}\n= \\frac{\\sin z}{\\sin(S-A)\\sin(S-B)}.\n\\]", "markdown": "From each angle of a spherical triangle a perpendicular is drawn to the arc which joins the middle points of the adjacent sides. Shew that these perpendiculars meet at a point; and that %-----File: 123.png------------------------------------------------ if $x$, $y$, $z$ are the perpendiculars from this point on the sides $a$, $b$, $c$ respectively, x(S-B)(S-C) = y(S-C)(S-A) = z(S-A)(S-B).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xii/9", "set": "todhunter-spherical-trigonometry-1886/ex-xii", "number": 9, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "122", "location": "Exercise XII, problem 9", "problem_latex": "Through each angle of a spherical triangle an arc is drawn\nso as to make the same angle with one side which the perpendicular\non the base makes with the other side. Shew that these\narcs meet at a point; and that if $x$, $y$, $z$ are the perpendiculars\nfrom this point on the sides $a$, $b$, $c$ respectively,\n\\[\n\\frac{\\sin x}{\\cos A}=\\frac{\\sin y}{\\cos B}=\\frac{\\sin z}{\\cos C}.\n\\]", "markdown": "Through each angle of a spherical triangle an arc is drawn so as to make the same angle with one side which the perpendicular on the base makes with the other side. Shew that these arcs meet at a point; and that if $x$, $y$, $z$ are the perpendiculars from this point on the sides $a$, $b$, $c$ respectively, xA=yB=zC.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/1", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 1, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 1", "problem_latex": "If $I$ denote the inclination of two adjacent faces of a\nregular polyhedron, shew that $\\cos I=\\tfrac{1}{3}$ in the tetrahedron, $=0$\nin the cube, $=-\\tfrac{1}{3}$ in the octahedron, $=-\\tfrac{1}{5}\\surd{5}$ in the dodecahedron,\nand $=-\\tfrac{1}{3}\\surd{5}$ in the icosahedron.", "markdown": "If $I$ denote the inclination of two adjacent faces of a regular polyhedron, shew that $\\cos I=\\tfrac{1}{3}$ in the tetrahedron, $=0$ in the cube, $=-\\tfrac{1}{3}$ in the octahedron, $=-\\tfrac{1}{5}\\surd{5}$ in the dodecahedron, and $=-\\tfrac{1}{3}\\surd{5}$ in the icosahedron.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/10", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 10, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 10", "problem_latex": "The sum of the squares of the four diagonals of a parallelepiped\nis equal to four times the sum of the squares of the\nedges.", "markdown": "The sum of the squares of the four diagonals of a parallelepiped is equal to four times the sum of the squares of the edges.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:solid_geometry" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/11", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 11, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 11", "problem_latex": "If with all the angular points of any parallelepiped as\ncentres equal spheres be described, the sum of the intercepted\nportions of the parallelepiped will be equal in volume to one of\nthe spheres.", "markdown": "If with all the angular points of any parallelepiped as centres equal spheres be described, the sum of the intercepted portions of the parallelepiped will be equal in volume to one of the spheres.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.const", "other:solid_geometry" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/12", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 12, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 12", "problem_latex": "A regular octahedron is inscribed in a cube so that the\ncorners of the octahedron are at the centres of the faces of the\ncube: shew that the volume of the cube is six times that of the\noctahedron.", "markdown": "A regular octahedron is inscribed in a cube so that the corners of the octahedron are at the centres of the faces of the cube: shew that the volume of the cube is six times that of the octahedron.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:solid_geometry" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/13", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 13, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 13", "problem_latex": "It is not possible to fill any given space with a number\nof regular polyhedrons of the same kind, except cubes; but this\nmay be done by means of tetrahedrons and octahedrons which\nhave equal faces, by using twice as many of the former as of\nthe latter.", "markdown": "It is not possible to fill any given space with a number of regular polyhedrons of the same kind, except cubes; but this may be done by means of tetrahedrons and octahedrons which have equal faces, by using twice as many of the former as of the latter.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:solid_geometry" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/14", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 14, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 14", "problem_latex": "A spherical triangle is formed on the surface of a sphere\nof radius $\\rho$; its angular points are joined, forming thus a pyramid\nwith the straight lines joining them with the centre: shew that\nthe volume of the pyramid is\n\\[\n \\tfrac{1}{3}\\rho^3\\surd{(\\tan r\\tan{r_1}\\tan{r_2}\\tan{r_3})},\n\\]\nwhere $r$, $r_1$, $r_2$, $r_3$ are the radii of the inscribed and escribed circles\nof the triangle.", "markdown": "A spherical triangle is formed on the surface of a sphere of radius $\\rho$; its angular points are joined, forming thus a pyramid with the straight lines joining them with the centre: shew that the volume of the pyramid is 13^3(rr_1r_2r_3), where $r$, $r_1$, $r_2$, $r_3$ are the radii of the inscribed and escribed circles of the triangle.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/15", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 15, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 15", "problem_latex": "The angular points of a regular tetrahedron inscribed\nin a sphere of radius $r$ being taken as poles, four equal small\ncircles of the sphere are described, so that each circle touches\nthe other three. Shew that the area of the surface bounded by\neach circle is $2\\pi r^2\\left( 1 - \\dfrac{1}{\\surd{3}} \\right)$.", "markdown": "The angular points of a regular tetrahedron inscribed in a sphere of radius $r$ being taken as poles, four equal small circles of the sphere are described, so that each circle touches the other three. Shew that the area of the surface bounded by each circle is $2\\pi r^2\\left( 1 - \\dfrac{1}{\\surd{3}} \\right)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.const", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/16", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 16, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 16", "problem_latex": "If $O$ be any point within a spherical triangle $ABC$, the\nproduct of the sines of any two sides and the sine of the included\nangle\n\\begin{multline*}\n=\\sin{AO}\\sin{BO}\\sin{CO} \\biggl\\{\\cot{AO}\\sin{BOC} \\\\\n+\\cot{BO}\\sin{COA}+\\cot{CO}\\sin{AOB} \\biggr\\}.\n\\end{multline*}", "markdown": "If $O$ be any point within a spherical triangle $ABC$, the product of the sines of any two sides and the sine of the included angle multline* =AOBOCO AOBOC +BOCOA+COAOB . multline*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/2", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 2, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 2", "problem_latex": "With the notation of Art.\\ 153, shew that the radius of\nthe sphere which touches one face of a regular polyhedron and all\nthe adjacent faces produced is $\\tfrac{1}{2}a\\cot{\\dfrac{\\pi}{m}}\\cot{\\tfrac{1}{2}}I$.", "markdown": "With the notation of Art. 153, shew that the radius of the sphere which touches one face of a regular polyhedron and all the adjacent faces produced is $\\tfrac{1}{2}a\\cot{\\dfrac{\\pi}{m}}\\cot{\\tfrac{1}{2}}I$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.const", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/3", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 3, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 3", "problem_latex": "A sphere touches one face of a regular tetrahedron and\nthe other three faces produced: find its radius.", "markdown": "A sphere touches one face of a regular tetrahedron and the other three faces produced: find its radius.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/4", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 4, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 4", "problem_latex": "If $a$ and $b$ are the radii of the spheres inscribed in and\ndescribed about a regular tetrahedron, shew that $b=3a$.", "markdown": "If $a$ and $b$ are the radii of the spheres inscribed in and described about a regular tetrahedron, shew that $b=3a$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/5", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 5, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 5", "problem_latex": "If $a$ is the radius of a sphere inscribed in a regular tetrahedron,\nand $R$ the radius of the sphere which touches the edges,\nshew that $R^2=3a^2$.", "markdown": "If $a$ is the radius of a sphere inscribed in a regular tetrahedron, and $R$ the radius of the sphere which touches the edges, shew that $R^2=3a^2$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/6", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 6, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 6", "problem_latex": "If $a$ is the radius of a sphere inscribed in a regular tetrahedron,\nand $R'$ the radius of the sphere which touches one face and\nthe others produced, shew that $R'=2a$.", "markdown": "If $a$ is the radius of a sphere inscribed in a regular tetrahedron, and $R'$ the radius of the sphere which touches one face and the others produced, shew that $R'=2a$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/7", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 7, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 7", "problem_latex": "If a cube and an octahedron be described about a given\nsphere, the sphere described about these polyhedrons will be the\nsame; and conversely.", "markdown": "If a cube and an octahedron be described about a given sphere, the sphere described about these polyhedrons will be the same; and conversely.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:solid_geometry" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/8", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 8, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 8", "problem_latex": "If a dodecahedron and an icosahedron be described about\na given sphere, the sphere described about these polyhedrons will\nbe the same; and conversely.", "markdown": "If a dodecahedron and an icosahedron be described about a given sphere, the sphere described about these polyhedrons will be the same; and conversely.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:solid_geometry" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xiii/9", "set": "todhunter-spherical-trigonometry-1886/ex-xiii", "number": 9, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "133", "location": "Exercise XIII, problem 9", "problem_latex": "A regular tetrahedron and a regular octahedron are inscribed\nin the same sphere: compare the radii of the spheres\nwhich can be inscribed in the two solids.", "markdown": "A regular tetrahedron and a regular octahedron are inscribed in the same sphere: compare the radii of the spheres which can be inscribed in the two solids.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/1", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 1, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 1", "problem_latex": "Find the locus of the vertices of all right-angled spherical\ntriangles having the same hypotenuse; and from the equation\nobtained, prove that the locus is a circle when the radius of the\nsphere is infinite.", "markdown": "Find the locus of the vertices of all right-angled spherical triangles having the same hypotenuse; and from the equation obtained, prove that the locus is a circle when the radius of the sphere is infinite.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/10", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 10, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 10", "problem_latex": "The arc of a great circle bisecting the sides $AB$, $AC$ of a\nspherical triangle cuts $BC$ produced at $Q$: shew that\n\\[\n\\cos AQ \\sin \\frac{a}{2} = \\sin \\frac{c - b}{2} \\sin \\frac{c + b}{2}.\n\\]", "markdown": "The arc of a great circle bisecting the sides $AB$, $AC$ of a spherical triangle cuts $BC$ produced at $Q$: shew that AQ a2 = c - b2 c + b2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/11", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 11, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 11", "problem_latex": "If $ABCD$ be a spherical quadrilateral, and the opposite\nsides $AB$, $CD$ when produced meet at $E$, and $AD$, $BC$ meet at $F$,\nthe ratio of the sines of the arcs drawn from $E$ at right angles to\nthe diagonals of the quadrilateral is the same as the ratio of those\nfrom $F$.", "markdown": "If $ABCD$ be a spherical quadrilateral, and the opposite sides $AB$, $CD$ when produced meet at $E$, and $AD$, $BC$ meet at $F$, the ratio of the sines of the arcs drawn from $E$ at right angles to the diagonals of the quadrilateral is the same as the ratio of those from $F$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/12", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 12, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 12", "problem_latex": "If $ABCD$ be a spherical quadrilateral whose sides $AB$,\n$DC$ are produced to meet at $P$, and $AD$, $BC$ at $Q$, and whose\ndiagonals $AC$, $BD$ intersect at $R$, then\n\\[\n\\sin AB \\sin CD \\cos P = \\sin AD \\sin BC \\cos Q = \\sin AC \\sin BD \\cos R.\n\\]", "markdown": "If $ABCD$ be a spherical quadrilateral whose sides $AB$, $DC$ are produced to meet at $P$, and $AD$, $BC$ at $Q$, and whose diagonals $AC$, $BD$ intersect at $R$, then AB CD P = AD BC Q = AC BD R.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/13", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 13, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 13", "problem_latex": "If $A'$ be the angle of the chordal triangle which corresponds\nto the angle $A$ of a spherical triangle, shew that\n\\[\n\\cos A' = \\sin (S - A) \\cos \\frac{a}{2}.\n\\]", "markdown": "If $A'$ be the angle of the chordal triangle which corresponds to the angle $A$ of a spherical triangle, shew that A’ = (S - A) a2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/14", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 14, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 14", "problem_latex": "If the tangent of the radius of the circle described about a\nspherical triangle is equal to twice the tangent of the radius of the\ncircle inscribed in the triangle, the triangle is equilateral.", "markdown": "If the tangent of the radius of the circle described about a spherical triangle is equal to twice the tangent of the radius of the circle inscribed in the triangle, the triangle is equilateral.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/15", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 15, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 15", "problem_latex": "The arc $AP$ of a circle of the same radius as the sphere\nis equal to the greater of two sides of a spherical triangle, and\nthe arc $AQ$ taken in the same direction is equal to the less; the\nsine $PM$ of $AP$ is divided at $E$, so that $\\dfrac{EM}{PM} =$ the natural cosine\n%-----File: 158.png------------------------------------------------\nof the angle included by the two sides, and $EZ$ is drawn parallel\nto the tangent to the circle at $Q$. Shew that the remaining side\nof the spherical triangle is equal to the arc $QPZ$.", "markdown": "The arc $AP$ of a circle of the same radius as the sphere is equal to the greater of two sides of a spherical triangle, and the arc $AQ$ taken in the same direction is equal to the less; the sine $PM$ of $AP$ is divided at $E$, so that $\\dfrac{EM}{PM} =$ the natural cosine %-----File: 158.png------------------------------------------------ of the angle included by the two sides, and $EZ$ is drawn parallel to the tangent to the circle at $Q$. Shew that the remaining side of the spherical triangle is equal to the arc $QPZ$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:construction" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/16", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 16, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 16", "problem_latex": "If through any point $P$ within a spherical triangle $ABC$\ngreat circles be drawn from the angular points $A$, $B$, $C$ to meet\nthe opposite sides at $a$, $b$, $c$ respectively, prove that\n\\[\n\\dfrac{\\sin Pa \\cos PA}{\\sin Aa} +\n\\dfrac{\\sin Pb \\cos PB}{\\sin Bb} +\n\\dfrac{\\sin Pc \\cos PC}{\\sin Cc} = 1.\n\\]", "markdown": "If through any point $P$ within a spherical triangle $ABC$ great circles be drawn from the angular points $A$, $B$, $C$ to meet the opposite sides at $a$, $b$, $c$ respectively, prove that Pa PAAa + Pb PBBb + Pc PCCc = 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/17", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 17, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 17", "problem_latex": "$A$ and $B$ are two places on the Earth's surface on the\nsame side of the equator, $A$ being further from the equator\nthan $B$. If the bearing of $A$ from $B$ be more nearly due East\nthan it is from any other place in the same latitude as $B$, find\nthe bearing of $B$ from $A$.", "markdown": "$A$ and $B$ are two places on the Earth’s surface on the same side of the equator, $A$ being further from the equator than $B$. If the bearing of $A$ from $B$ be more nearly due East than it is from any other place in the same latitude as $B$, find the bearing of $B$ from $A$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/18", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 18, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 18", "problem_latex": "From the result given in example 18 of Chapter~V.\\ infer\nthe possibility of a regular dodecahedron.", "markdown": "From the result given in example 18 of Chapter V. infer the possibility of a regular dodecahedron.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/19", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 19, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 19", "problem_latex": "$A$ and $B$ are fixed points on the surface of a sphere, and\n$P$ is any point on the surface. If $a$ and $b$ are given constants,\nshew that a fixed point $S$ can always be found, in $AB$ or $AB$ produced,\nsuch that\n\\[\na \\cos AP + b \\cos BP = s \\cos SP,\n\\]\nwhere $s$ is a constant.", "markdown": "$A$ and $B$ are fixed points on the surface of a sphere, and $P$ is any point on the surface. If $a$ and $b$ are given constants, shew that a fixed point $S$ can always be found, in $AB$ or $AB$ produced, such that a AP + b BP = s SP, where $s$ is a constant.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/2", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 2, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 2", "problem_latex": "$AB$ is an arc of a great circle on the surface of a sphere, $C$\nits middle point: shew that the locus of the point $P$, such that\nthe angle $APC =$ the angle $BPC$, consists of two great circles at\nright angles to one another. Explain this when the triangle\nbecomes plane.", "markdown": "$AB$ is an arc of a great circle on the surface of a sphere, $C$ its middle point: shew that the locus of the point $P$, such that the angle $APC =$ the angle $BPC$, consists of two great circles at right angles to one another. Explain this when the triangle becomes plane.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/20", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 20, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 20", "problem_latex": "$A$, $B$, $C$,\\ldots are fixed points on the surface of a sphere;\n$a$, $b$, $c$,\\ldots are given constants. If $P$ be a point on the surface of\nthe sphere, such that\n\\[\na \\cos AP + b \\cos BP + c \\cos CP + \\ldots = \\text{constant},\n\\]\nshew that the locus of $P$ is a circle.", "markdown": "$A$, $B$, $C$,…are fixed points on the surface of a sphere; $a$, $b$, $c$,…are given constants. If $P$ be a point on the surface of the sphere, such that a AP + b BP + c CP + …= constant, shew that the locus of $P$ is a circle.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/3", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 3, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 3", "problem_latex": "On a given arc of a sphere, spherical triangles of equal\narea are described: shew that the locus of the angular point\nopposite to the given arc is defined by the equation\n\\begin{multline*}\n \\tan^{-1} \\left\\{\\frac{\\tan (\\alpha + \\phi)}{\\sin \\theta}\\right\\}\n+ \\tan^{-1} \\left\\{\\frac{\\tan (\\alpha - \\phi)}{\\sin \\theta}\\right\\} \\\\\n{}\n+ \\tan^{-1} \\left\\{\\frac{\\tan \\theta}{\\sin (\\alpha + \\phi)}\\right\\}\n+ \\tan^{-1} \\left\\{\\frac{\\tan \\theta}{\\sin (\\alpha - \\phi)}\\right\\}\n= \\beta,\n\\end{multline*}\nwhere $2\\alpha$ is the length of the given arc, $\\theta$ the arc of the great\ncircle drawn from any point $P$ in the locus perpendicular to the\ngiven arc, $\\phi$ the inclination of the great circle on which $\\theta$ is\n%-----File: 156.png------------------------------------------------\nmeasured to the great circle bisecting the given arc at right\nangles, and $\\beta$ a constant.", "markdown": "On a given arc of a sphere, spherical triangles of equal area are described: shew that the locus of the angular point opposite to the given arc is defined by the equation multline* ^-1 (+ ) + ^-1 (- ) + ^-1 (+ ) + ^-1 (- ) = , multline* where $2\\alpha$ is the length of the given arc, $\\theta$ the arc of the great circle drawn from any point $P$ in the locus perpendicular to the given arc, $\\phi$ the inclination of the great circle on which $\\theta$ is %-----File: 156.png------------------------------------------------ measured to the great circle bisecting the given arc at right angles, and $\\beta$ a constant.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/4", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 4, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 4", "problem_latex": "In any spherical triangle\n\\[\n \\tan c = \\frac{\\cot A \\cot a + \\cot B \\cot b}\n {\\cot a \\cot b - \\cos A \\cos B}.\n\\]", "markdown": "In any spherical triangle c = A a + B b a b - A B.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/5", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 5, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 5", "problem_latex": "If $\\theta$, $\\phi$, $\\psi$ denote the distances from the angles $A$, $B$, $C$\nrespectively of the point of intersection of arcs bisecting the\nangles of the spherical triangle $ABC,$ shew that\n\\[\n \\cos \\theta \\sin(b - c)\n+ \\cos \\phi \\sin(c - a)\n+ \\cos \\psi \\sin(a - b) = 0.\n\\]", "markdown": "If $\\theta$, $\\phi$, $\\psi$ denote the distances from the angles $A$, $B$, $C$ respectively of the point of intersection of arcs bisecting the angles of the spherical triangle $ABC,$ shew that (b - c) + (c - a) + (a - b) = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/6", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 6, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 6", "problem_latex": "If $A'$, $B'$, $C'$ be the poles of the sides $BC$, $CA$, $AB$ of a\nspherical triangle $ABC$, shew that the great circles $AA'$, $BB'$, $CC'$\nmeet at a point $P$, such that\n\\[\n \\cos PA \\cos BC = \\cos PB \\cos CA = \\cos PC \\cos AB.\n\\]", "markdown": "If $A'$, $B'$, $C'$ be the poles of the sides $BC$, $CA$, $AB$ of a spherical triangle $ABC$, shew that the great circles $AA'$, $BB'$, $CC'$ meet at a point $P$, such that PA BC = PB CA = PC AB.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/7", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 7, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 7", "problem_latex": "If $O$ be the point of intersection of arcs $AD$, $BE$, $CF$\ndrawn from the angles of a triangle perpendicular to the opposite\nsides and meeting them at $D$, $E$, $F$ respectively, shew that\n\\[\n \\frac{\\tan AD}{\\tan OD}, \\qquad\n \\frac{\\tan BE}{\\tan OE}, \\qquad\n \\frac{\\tan CF}{\\tan OF}\n\\]\nare respectively equal to\n\\[\n 1+\\frac{\\cos A}{\\cos B \\cos C}, \\quad\n 1+\\frac{\\cos B}{\\cos A \\cos C}, \\quad\n 1+\\frac{\\cos C}{\\cos A \\cos B}.\n\\]", "markdown": "If $O$ be the point of intersection of arcs $AD$, $BE$, $CF$ drawn from the angles of a triangle perpendicular to the opposite sides and meeting them at $D$, $E$, $F$ respectively, shew that ADOD, BEOE, CFOF are respectively equal to 1+AB C, 1+BA C, 1+CA B.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/8", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 8, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 8", "problem_latex": "If $p$, $q$, $r$ be the arcs of great circles drawn from the\nangles of a triangle perpendicular to the opposite sides, $(\\alpha, \\alpha')$,\n$(\\beta, \\beta')$, $(\\gamma, \\gamma')$ the segments into which these arcs are divided,\nshew that\n\\begin{flalign*}\n&&& \\tan \\alpha \\tan \\alpha'\n = \\tan \\beta \\tan \\beta'\n = \\tan \\gamma \\tan \\gamma'; &&\n\\\\[2ex]\n&\\text{and }&\\multispan{2}{\\hfill$\\displaystyle\n \\frac{\\cos p}{\\cos \\alpha \\cos \\alpha'}\n= \\frac{\\cos q}{\\cos \\beta \\cos \\beta' }\n= \\frac{\\cos r}{\\cos \\gamma \\cos \\gamma'}.\n$\\hfill} &\\phantom{and }&\n\\end{flalign*}", "markdown": "If $p$, $q$, $r$ be the arcs of great circles drawn from the angles of a triangle perpendicular to the opposite sides, $(\\alpha, \\alpha')$, $(\\beta, \\beta')$, $(\\gamma, \\gamma')$ the segments into which these arcs are divided, shew that flalign* &&& ’ = ’ = ’; && [2ex] &and&2$\\displaystyle \\frac{\\cos p}{\\cos \\alpha \\cos \\alpha'} = \\frac{\\cos q}{\\cos \\beta \\cos \\beta' } = \\frac{\\cos r}{\\cos \\gamma \\cos \\gamma'}. $ &and & flalign*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xv/9", "set": "todhunter-spherical-trigonometry-1886/ex-xv", "number": 9, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "155", "location": "Exercise XV, problem 9", "problem_latex": "In a spherical triangle if arcs be drawn from the angles to\nthe middle points of the opposite sides, and if $\\alpha$, $\\alpha'$ be the two\nparts of the one which bisects the side $a$,\nshew that\n\\[\n\\frac{\\sin \\alpha}{\\sin \\alpha'} = 2 \\cos \\frac{a}{2}.\n\\]", "markdown": "In a spherical triangle if arcs be drawn from the angles to the middle points of the opposite sides, and if $\\alpha$, $\\alpha'$ be the two parts of the one which bisects the side $a$, shew that ’ = 2 a2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:geometric_proof" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xvi/1", "set": "todhunter-spherical-trigonometry-1886/ex-xvi", "number": 1, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "168", "location": "Exercise XVI, problem 1", "problem_latex": "Given $b$\n& = 137^\\circ\\, 3'\\, 48'',\\\nA = 147^\\circ\\, 2'\\, 54'',\\\nC = 90^\\circ.", "markdown": "Given $b$ & = 137^  3’  48”, A = 147^  2’  54”, C = 90^.", "answer_latex": [ "\\textit{Results.}\\quad c\n& = 47^\\circ\\, 57'\\, 15'',\\\na = 156^\\circ\\, 10'\\, 34'',\\\nB = 113^\\circ\\, 28'." ], "answer_markdown": [ "*Results.* c & = 47^  57’  15”, a = 156^  10’  34”, B = 113^  28’." ], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xvi/2", "set": "todhunter-spherical-trigonometry-1886/ex-xvi", "number": 2, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "168", "location": "Exercise XVI, problem 2", "problem_latex": "Given $c$\n& = 61^\\circ\\, 4'\\, 56'',\\\na = 40^\\circ\\, 31'\\, 20'',\\\nC = 90^\\circ.", "markdown": "Given $c$ & = 61^  4’  56”, a = 40^  31’  20”, C = 90^.", "answer_latex": [ "\\textit{Results.}\\quad b\n& = 50^\\circ\\, 30'\\, 29'',\\\nB = 61^\\circ\\, 50'\\, 28'',\\\nA = 47^\\circ\\, 54'\\, 21''." ], "answer_markdown": [ "*Results.* b & = 50^  30’  29”, B = 61^  50’  28”, A = 47^  54’  21”." ], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xvi/3", "set": "todhunter-spherical-trigonometry-1886/ex-xvi", "number": 3, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "168", "location": "Exercise XVI, problem 3", "problem_latex": "Given $A$\n& = 36^\\circ,\\\nB = 60^\\circ,\\\nC = 90^\\circ.", "markdown": "Given $A$ & = 36^, B = 60^, C = 90^.", "answer_latex": [ "\\textit{Results.}\\quad a\n& = 20^\\circ\\, 54'\\, 18''.5,\\\nb = 31^\\circ\\, 43'\\, 3'',\\\nc = 37^\\circ\\, 21'\\, 38''.5." ], "answer_markdown": [ "*Results.* a & = 20^  54’  18”.5, b = 31^  43’  3”, c = 37^  21’  38”.5." ], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xvi/4", "set": "todhunter-spherical-trigonometry-1886/ex-xvi", "number": 4, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "168", "location": "Exercise XVI, problem 4", "problem_latex": "Given $a$\n& = 59^\\circ\\, 28'\\, 27'',\\\nA = 66^\\circ\\, 7'\\, 20'',\\\nC = 90^\\circ.", "markdown": "Given $a$ & = 59^  28’  27”, A = 66^  7’  20”, C = 90^.", "answer_latex": [ "\\textit{Results.}\\quad c\n& = 70^\\circ\\, 23'\\, 42'',\\\nb = 48^\\circ\\, 39'\\, 16'',\\\nB = 52^\\circ\\, 50'\\, 20'',\n&&\\\n\\indent\\rlap{~or,}\\phantom{\\textit{Results.}\\ {}}c\n& = 109^\\circ\\, 36'\\, 18'',\\\nb = 131^\\circ\\, 20'\\, 44'',\\\nB = 127^\\circ\\, 9'\\, 40''." ], "answer_markdown": [ "*Results.* c & = 70^  23’  42”, b = 48^  39’  16”, B = 52^  50’  20”, && or,*Results.* c & = 109^  36’  18”, b = 131^  20’  44”, B = 127^  9’  40”." ], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xvi/5", "set": "todhunter-spherical-trigonometry-1886/ex-xvi", "number": 5, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "168", "location": "Exercise XVI, problem 5", "problem_latex": "Given $c$\n& = 90^\\circ,\\\na = 138^\\circ\\, 4',\\\nb = 109^\\circ\\, 41'.", "markdown": "Given $c$ & = 90^, a = 138^  4’, b = 109^  41’.", "answer_latex": [ "\\textit{Results.}\\quad C\n& = 113^\\circ\\, 28'\\, 2'',\\\nA = 142^\\circ\\, 11'\\, 38'',\\\nB = 120^\\circ\\, 15'\\, 57''." ], "answer_markdown": [ "*Results.* C & = 113^  28’  2”, A = 142^  11’  38”, B = 120^  15’  57”." ], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xvi/6", "set": "todhunter-spherical-trigonometry-1886/ex-xvi", "number": 6, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "168", "location": "Exercise XVI, problem 6", "problem_latex": "Given $c$\n& = 90^\\circ,\\\nA = 131^\\circ\\, 30',\\\nB = 120^\\circ\\, 32'.", "markdown": "Given $c$ & = 90^, A = 131^  30’, B = 120^  32’.", "answer_latex": [ "\\textit{Results.}\\quad C\n& = 109^\\circ\\, 40'\\, 20'',\\\na = 127^\\circ\\, 17'\\, 51'',\\\nb = 113^\\circ\\, 49'\\, 31''." ], "answer_markdown": [ "*Results.* C & = 109^  40’  20”, a = 127^  17’  51”, b = 113^  49’  31”." ], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xvi/7", "set": "todhunter-spherical-trigonometry-1886/ex-xvi", "number": 7, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "168", "location": "Exercise XVI, problem 7", "problem_latex": "Given $a$\n& = 76^\\circ\\, 35'\\, 36'',\\\nb = 50^\\circ\\, 10'\\, 30'',\\\nc = 40^\\circ\\, 0'\\, 10''.", "markdown": "Given $a$ & = 76^  35’  36”, b = 50^  10’  30”, c = 40^  0’  10”.", "answer_latex": [ "\\textit{Results.}\\quad A\n& = 121^\\circ\\, 36'\\, 20'',\\\nB = 42^\\circ\\, 15'\\, 13'',\\\nC = 34^\\circ\\, 15'\\, 3''." ], "answer_markdown": [ "*Results.* A & = 121^  36’  20”, B = 42^  15’  13”, C = 34^  15’  3”." ], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "todhunter-spherical-trigonometry-1886/ex-xvi/8", "set": "todhunter-spherical-trigonometry-1886/ex-xvi", "number": 8, "part": null, "book": "todhunter-spherical-trigonometry-1886", "edition": "Macmillan and Co., London, 5th ed., 1886", "page": "168", "location": "Exercise XVI, problem 8", "problem_latex": "Given $A$\n&= 129^\\circ\\, 5'\\, 28'',\\\nB= 142^\\circ\\, 12'\\, 42'',\\\nC= 105^\\circ\\, 8'\\, 10''.", "markdown": "Given $A$ &= 129^  5’  28”, B= 142^  12’  42”, C= 105^  8’  10”.", "answer_latex": [ "\\textit{Results.}\\quad a\n& = 135^\\circ\\, 49'\\, 20'',\\\nb = 144^\\circ\\, 37'\\, 15'',\\\nc = 60^\\circ\\, 4'\\, 54''." ], "answer_markdown": [ "*Results.* a & = 135^  49’  20”, b = 144^  37’  15”, c = 60^  4’  54”." ], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "core.units" ], "expectation": null, "keys": [] } ], "errata_statuses": [ "transcriber_marked", "candidate", "probable", "confirmed", "dismissed", "note" ], "errata": [] }