# How do you prove sin(pi+x) = - sinx? Double Angle Question: 1.) Express sin x/2 in terms of cos x Thank you so much!?

May 2, 2015

I) $\sin \left(x + \Pi\right) = - \sin x$. There are 2 ways to prove:
a. By the trig unit circle,.
Example: x = Pi/3 (Quadrant I). And $\left(x + \pi\right)$ is in Quadrant III.
Then $\sin \left(x + \pi\right) = - \sin x$.
b. By trig identity sin (a + b) = sin a.cos b + sin a. sin b
sin (x + pi) = sin x cos pi + sin pi.cos x Because cos pi = -1 and sin pi = 0. then sin (x + Pi) = -sin x#

2) $\cos x = 1 - 2 {\sin}^{2} \left(\frac{x}{2}\right)$
${\sin}^{2} \left(\frac{x}{2}\right) = \frac{1 - \cos x}{2}$

$\sin \left(\frac{x}{2}\right) = + \left(\mathmr{and} -\right) s q r \left[\frac{\left(1 - \cos x\right)}{2}\right]$