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Math Help - \tan{2\alpha}=24/7

  1. #1
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    \tan{2\alpha}=24/7

    How can I show that \displaystyle\tan{2\alpha}=\frac{24}{7}\equiv\tan{  \alpha}=\frac{3}{4}\mbox{?}
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  2. #2
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    Quote Originally Posted by dwsmith View Post
    How can I show that \displaystyle\tan{2\alpha}=\frac{24}{7}\equiv\tan{  \alpha}=\frac{3}{4}\mbox{?}
    Apply the usual double angle formula, substitute \tan{\alpha}=\frac{3}{4} and simplify.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Apply the usual double angle formula, substitute \tan{\alpha}=\frac{3}{4} and simplify.
    \displaystyle\tan{2\alpha}=\frac{2\tan{\alpha}}{1-\tan^2{\alpha}}=\frac{24}{7}\Rightarrow 7\tan{\alpha}=12-12\tan^2{\alpha}

    \displaystyle\Rightarrow 12\tan^2{\alpha}+7\tan{\alpha}-12=(3\tan{\alpha}+4)(4\tan{\alpha}-3)=0

    \displaystyle\tan{\alpha}=\frac{3}{4}, \ \frac{-4}{3}

    These are both 3,4,5 right triangles.

    This comes from PDE so it doesn't matter which one I would pick for the overall solution of the PDE.
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