{ "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", "
$\\;\\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 }