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Thread: Trigonometry

  1. #1
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    Trigonometry

    sin2x+sin4x=0,9696+2cos3xsinx
    How much is x ?
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  2. #2
    TD!
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    I don't think you won't be able to solve this algebraicly, try a numerical approximation using a computer or (graphical) calculator.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by TD!
    I don't think you won't be able to solve this algebraicly, try a numerical approximation using a computer or (graphical) calculator.
    It can be simplified down to

    $\displaystyle
    2sin(2x)=0.9696
    $

    but since the next stage is to look up an inverse trig function the effort
    of simplifying is probably not worth it and one might as well go straight
    for a numerical solution.

    The first solution is $\displaystyle x\approx 0.2530672418$

    The only advantage of simplifiing is that it will give you all of the solutions.

    RonL
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  4. #4
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    Quote Originally Posted by totalnewbie
    sin2x+sin4x=0,9696+2cos3xsinx
    How much is x ?
    Use the sum formula to get,
    $\displaystyle \sin 2x+\sin 4x=2\sin 3x\cos x$
    Thus,
    $\displaystyle 2\sin 3x\cos x=.9696+2\cos 3x\sin x$
    Thus,
    $\displaystyle 2\sin 3x\cos x-2\cos 3x\sin x=.9696$
    Thus,
    $\displaystyle 2(\sin 3x\cos x-\cos 3x\sin x)=.9696$
    Recognzing this sum formula for sine we have,
    $\displaystyle 2\sin 2x=.9696$ (As CaptainBlack said)
    Thus,
    $\displaystyle \sin 2x=.4848$
    Thus,
    $\displaystyle x=\sin^{-1}(.4848)+\pi (2k)$
    $\displaystyle x=-\sin^{-1}(.4848)+\pi (2k+1)$
    Which can be written more elegantly as,
    $\displaystyle x=(-1)^k\sin^{-1}(.4848)+\pi k$
    Q.E.D.
    Last edited by ThePerfectHacker; Mar 14th 2006 at 09:55 AM.
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