Re: trig identity help

From: Dennis May (dennis_at_NOkernelSPAMthread.e7even.com)
Date: 12/05/04 

Date: Sun, 05 Dec 2004 19:09:32 +0000

tsmith wrote:
> Hi,
>
> In my physics book, for superposition of waves it uses the identity
>
> sin(a) + sin(b) = 2cos( (a-b) / 2 )sin( (a+b) / 2 )
>
> I looked in my calculus book's trig review appendix and they do not prove
> this one.
>
> So how to derive it?

sin (A+B) = sin A cos B + cos A sin B (1)
sin (A-B) = sin A cos B - cos A sin B

Add:

sin (A+B) + sin (A-B) = 2 sin A cos B

Put A+B = a, A-B = b
then A = (a+b)/2, B = (a-b)/2
Hence

sin a + sin b = 2 sin ((a+b)/2) cos ((a-b)/2)

To derive (1) use

exp(ix) = cos x + i sin x

so

sin x = Im exp(ix) (for real x)

Thus for real x, y

sin (x+y) = Im (exp(ix)exp(iy))
           = Im((cos x + i sin x)(cos y + i sin y))
           = cos x sin y + sin x cos y

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