# trig identity

• May 29th 2008, 10:14 PM
phthiriasis
trig identity
prove:
sinx + sin(2x) + sin(3x) = sin(2x)(1+2cosx)

thanks!
• May 29th 2008, 11:09 PM
Reckoner
Quote:

Originally Posted by phthiriasis
prove:
sinx + sin(2x) + sin(3x) = sin(2x)(1+2cosx)

thanks!

Here's one way to do it: the key step is realizing that $\sin3x = \sin(2x + x)$ and applying the sum and difference formula:

$\sin x + \sin 2x + \sin 3x$

$=\sin x + 2\sin x\cos x + \sin(2x + x)$

$=\sin x + 2\sin x\cos x + \sin 2x\cos x + \sin x\cos 2x$

$=\sin x + 2\sin x\cos x + 2\sin x\cos x\cos x + \sin x(2\cos^2x - 1)$

$=\sin x + 2\sin x\cos x + 2\sin x\cos^2 x + 2\sin x\cos^2x - \sin x$

$=2\sin x\cos x + 4\sin x\cos^2 x$

Now, factor.