#!/usr/bin/env python # coding: utf-8 # # Basic trigonometry # # > Marcos Duarte # > Laboratory of Biomechanics and Motor Control ([http://demotu.org/](http://demotu.org/)) # > Federal University of ABC, Brazil # 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. # These proportionality constants are the values of $\sin\theta$, $\cos\theta$, and $\tan\theta$. # Here is a geometric representation of the main [trigonometric functions](http://en.wikipedia.org/wiki/Trigonometric_function) of an angle $\theta$: #
#

# ## Radian # # An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian: #
#

# ## Common trigonometric values # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

$\;\theta\;(^o)$	$\;\theta\;(rad)$	$\;\sin \theta\;$	$\;\cos \theta\;$	$\;\tan \theta\;$
$0^o$	$0$	$0$	$1$	$0$
$30^o$	$\pi/6$	$1/2$	$\sqrt{3}/2$	$1\sqrt{3}$
$45^o$	$\pi/4$	$\sqrt{2}/2$	$\sqrt{2}/2$	$1$
$60^o$	$\pi/3$	$\sqrt{3}/2$	$1/2$	$\sqrt{3}$
$90^o$	$\pi/2$	$1$	$0$	$\infty$

# ## Trigonometric identities # # Some of the main trigonometric identities are (see a [complete list at Wikipedia](http://en.wikipedia.org/wiki/List_of_trigonometric_identities)): # # $$ \sin^2{\alpha} + \cos^2{\alpha} = 1 $$ # # $$ \sin(2\alpha) = 2\sin{\alpha} \cos{\alpha} $$ # # $$ \cos(2\alpha) = \cos^2{\alpha} - \sin^2{\alpha} $$ # # $$ \sin(\alpha \pm \beta) = \sin{\alpha} \cos{\beta} \pm \cos{\alpha} \sin{\beta} $$ # # $$ \cos(\alpha \pm \beta) = \cos{\alpha} \cos{\beta} \mp \sin{\alpha} \cos{\beta} $$ # ## References # # - [Trigonometric functions [Wikipedia]](http://en.wikipedia.org/wiki/Trigonometric_function). # - [Trigonometry [S.O.S. Mathematics]](http://www.sosmath.com/trig/trig.html). # In[ ]: