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Thread: Trigonomertric identities

  1. #1
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    Question Trigonomertric identities

    [tex]cos20°cos100°+cos100°cos140°-cos140°cos200°=\frac{-3}{4}[tex]
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  2. #2
    Super Member malaygoel's Avatar
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    Quote Originally Posted by matsci0000 View Post
    [tex]cos20°cos100°+cos100°cos140°-cos140°cos200°=\frac{-3}{4}[tex]
    $\displaystyle cos20°cos100°+cos100°cos140°-cos140°cos200°$

    =$\displaystyle cos20°cos100°+cos100°cos140°+cos140°cos20°$

    Now, see $\displaystyle
    sec\frac{\pi}{9},sec\frac{5\pi}{9},sec\frac{7\pi}{ 9}
    $ are the roots of equation

    $\displaystyle
    sec^{3}\theta+6sec^{2}\theta-8=0
    $

    see thread: http://www.mathhelpforum.com/math-he...trigo-sum.html

    hence, $\displaystyle sec\frac{\pi}{9}*sec\frac{5\pi}{9}*sec\frac{7\pi}{ 9}=8$

    =$\displaystyle cos20°cos100°+cos100°cos140°+cos140°cos20°$

    =$\displaystyle sec\frac{\pi}{9}*sec\frac{5\pi}{9}*sec\frac{7\pi}{ 9} \left(
    sec\frac{\pi}{9}+sec\frac{5\pi}{9}+sec\frac{7\pi}{ 9} \right)
    $

    =$\displaystyle \frac{-6}{8}$

    =$\displaystyle \frac{-3}{4}$
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