Question

How do you express $\sin \frac{x}{2}$ in terms of $\cos x$ using the double angle identity?

Answer 1

Express $\sin \left(x/2\right)$ in terms of $\cos x$.

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

Explanation:

By applying the trig identity: $\cos 2a=1-{\sin }^{2}a$ we get

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

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

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

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