Prove formulas for shifting sine and cosine by π

by RoRi

July 31, 2015

Prove

$\sin (x + \pi) = - \sin x \qquad \text{and} \qquad \cos (x+\pi) = -\cos x$

for all $x \in \mathbb{R}$ .

Proof. First,

$\sin (x+ \pi) = \sin \left( x + \frac{\pi}{2} + \frac{\pi}{2} \right) = \cos \left( x + \frac{\pi}{2} \right) = - \sin x.$

And then,

$\cos (x+ \pi) = \cos \left( x + \frac{\pi}{2} + \frac{\pi}{2}\right) = - \sin \left( x + \frac{\pi}{2} \right) = - \cos x. \qquad \blacksquare$

Stumbling Robot