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Math Help - trig identity

  1. #1
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    trig identity

    prove:
    sinx + sin(2x) + sin(3x) = sin(2x)(1+2cosx)

    thanks!
    Last edited by phthiriasis; June 14th 2008 at 04:58 PM.
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    Quote Originally Posted by phthiriasis View Post
    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.
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