Theorem: TRIG.PERIOD The period for the sine, cosine, and tangent functions.

$\sin(x + 2\pi)= \sin(x)$ for all $x$.
$\cos(x + 2\pi)= \cos(x)$ for all $x$.
$\tan(x + \pi)= \tan(x)$ for all $x \ne \frac {(2k+1)\pi}2$.

The justification for periodicity makes sense by considering the mapping diagrams for the unit circle definitions of these functions.