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

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Answer 1 · 2018-05-09T05:11:55

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

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 ) } #