Write an equivalent expression for sin x/2 using compound angle formulas?

May 9, 2018

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

Explanation:

The half angle formula for sine comes from the double angle formula for cosine:

$\cos \left(a + b\right) = \cos a \cos b - \sin a \sin b$

$\cos \left(2 a\right) = \cos \left(a + a\right) = {\cos}^{2} a - {\sin}^{2} a$

${\cos}^{2} a + {\sin}^{2} a = 1$

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

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

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

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

Let $a = \frac{x}{2} , x = 2 a$

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