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

1 Answer
May 9, 2018

sin (x/2) = pm sqrt{ 1/2 (1 - cos x ) }

Explanation:

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

cos(a+b) = cos a cos b - sin a sin b

cos(2a) = cos(a+a) = cos ^2 a - sin^2 a

cos^2 a + sin ^2 a = 1

cos(2a) = (1 - sin ^2 a )- sin ^2 a = 1 - 2 sin ^2 a

2 sin ^2 a = 1 - cos 2a

sin ^2 a = 1/2 (1 - cos 2a )

sin a = pm sqrt{ 1/2 (1 - cos 2a ) }

Let a=x/2, x=2a

sin (x/2) = pm sqrt{ 1/2 (1 - cos x ) }