Question

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

Answer 1

Expressing $\sin \frac{x}{2}$ in terms of $\cos x$:

By using the trigonometry identity

$\cos 2a=1-2{\sin }^{2}a$,

$$\begin{gathered} \Rightarrow \cos x=1-2{\sin }^2\left(\frac{x}{2}\right) \\ \Rightarrow 2{\sin }^2\left(\frac{x}{2}\right)=1-\cos x \\ \Rightarrow {\sin }^2\left(\frac{x}{2}\right)=\frac{1-\cos x}{2} \\ \Rightarrow \sin \left(\frac{x}{2}\right)=\pm \sqrt{\frac{1-\cos x}{2}} \end{gathered}$$

Hence, $\sin \left(\frac{x}{2}\right)=\pm \sqrt{\frac{1-\cos x}{2}}$ is the required value.