Basic trigonometry

Marcos Duarte
Laboratory of Biomechanics and Motor Control (
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 of an angle $\theta$:

Main trigonometric functions
Figure. Main trigonometric functions (image from Wikipedia).


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

Figure. Definition of the radian (image from Wikipedia).

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):

$$ \sin^2{\alpha} + \cos^2{\alpha} = 1 $$$$ \sin(2\alpha) = 2\sin{\alpha} \cos{\alpha} $$$$ \cos(2\alpha) = \cos^2{\alpha} - \sin^2{\alpha} $$$$ \sin(\alpha \pm \beta) = \sin{\alpha} \cos{\beta} \pm \cos{\alpha} \sin{\beta} $$$$ \cos(\alpha \pm \beta) = \cos{\alpha} \cos{\beta} \mp \sin{\alpha} \cos{\beta} $$
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