{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Basic trigonometry\n", "\n", "> Marcos Duarte \n", "> Laboratory of Biomechanics and Motor Control ([http://demotu.org/](http://demotu.org/)) \n", "> Federal University of ABC, Brazil" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If two right triangles (triangles with an angle of $90^o$ ($\\pi/2$ radians)) have equal acute angles, they are similar, so their side lengths are proportional. \n", "These proportionality constants are the values of $\\sin\\theta$, $\\cos\\theta$, and $\\tan\\theta$. \n", "Here is a geometric representation of the main [trigonometric functions](http://en.wikipedia.org/wiki/Trigonometric_function) of an angle $\\theta$: \n", "
\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Radian\n", "\n", "An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian: \n", "
\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Common trigonometric values\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "
$\\;\\theta\\;(^o)$$\\;\\theta\\;(rad)$$\\;\\sin \\theta\\;$$\\;\\cos \\theta\\;$$\\;\\tan \\theta\\;$
$0^o$$0$$0$$1$$0$
$30^o$$\\pi/6$$1/2$$\\sqrt{3}/2$$1\\sqrt{3}$
$45^o$$\\pi/4$$\\sqrt{2}/2$$\\sqrt{2}/2$$1$
$60^o$$\\pi/3$$\\sqrt{3}/2$$1/2$$\\sqrt{3}$
$90^o$$\\pi/2$$1$$0$$\\infty$
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Trigonometric identities\n", "\n", "Some of the main trigonometric identities are (see a [complete list at Wikipedia](http://en.wikipedia.org/wiki/List_of_trigonometric_identities)):\n", "\n", "$$\\sin^2{\\alpha} + \\cos^2{\\alpha} = 1$$\n", "\n", "$$\\sin(2\\alpha) = 2\\sin{\\alpha} \\cos{\\alpha}$$\n", "\n", "$$\\cos(2\\alpha) = \\cos^2{\\alpha} - \\sin^2{\\alpha}$$\n", "\n", "$$\\sin(\\alpha \\pm \\beta) = \\sin{\\alpha} \\cos{\\beta} \\pm \\cos{\\alpha} \\sin{\\beta}$$\n", "\n", "$$\\cos(\\alpha \\pm \\beta) = \\cos{\\alpha} \\cos{\\beta} \\mp \\sin{\\alpha} \\cos{\\beta}$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## References\n", "\n", "- [Trigonometric functions [Wikipedia]](http://en.wikipedia.org/wiki/Trigonometric_function).\n", "- [Trigonometry [S.O.S. Mathematics]](http://www.sosmath.com/trig/trig.html)." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" }, "varInspector": { "cols": { "lenName": 16, "lenType": 16, "lenVar": 40 }, "kernels_config": { "python": { "delete_cmd_postfix": "", "delete_cmd_prefix": "del ", "library": "var_list.py", "varRefreshCmd": "print(var_dic_list())" }, "r": { "delete_cmd_postfix": ") ", "delete_cmd_prefix": "rm(", "library": "var_list.r", "varRefreshCmd": "cat(var_dic_list()) " } }, "types_to_exclude": [ "module", "function", "builtin_function_or_method", "instance", "_Feature" ], "window_display": false } }, "nbformat": 4, "nbformat_minor": 1 }