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Math Help - Find general solution of the equation?

  1. #1
    Super Member fardeen_gen's Avatar
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    Find general solution of the equation?

    Find general solution of the equation:
    \sin \frac{2x + 1}{x} + \sin \frac{2x + 1}{3x} - 3\cos^2 \frac{2x + 1}{3x} = 0

    Answer:
    Spoiler:
    \frac{2}{3(2n - 1)\pi - 4}

    I am unable to solve it. Anyone?
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  2. #2
    MHF Contributor red_dog's Avatar
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    Use \sin a+\sin b=2\sin\frac{a+b}{2}\cos\frac{a-b}{2}

    2\sin 2\left(\frac{2x+1}{3x}\right)\cos\frac{2x+1}{3x}-3\cos^2\frac{2x+1}{3x}=0

    Now use \sin 2a=2\sin a\cos a

    2\sin\frac{2x+1}{3x}\cos^2\frac{2x+1}{3x}-3\cos^2\frac{2x+1}{3x}=0

    \cos^2\frac{2x+1}{3x}\left(2\sin\frac{2x+1}{3x}-3\right)=0

    Then, \cos\frac{2x+1}{3x}=0\Rightarrow\frac{2x+1}{3x}=(2  k+1)\frac{\pi}{2} and find x.

    If 2\sin\frac{2x+1}{3x}-3=0\Rightarrow\sin\frac{2x+1}{3x}=\frac{3}{2}>1 and it has no solution.
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