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Thread: Find the sum of the values of

  1. #1
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    Find the sum of the values of

    Find the sum of the values of x such that

    $\displaystyle \cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x, $

    where x is measured in degrees and 0< x< 360.
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  2. #2
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    Use the product to sum formula on the RHS,
    $\displaystyle 8\cos^34x\cos^3x=(2\cos4x\cos x)^3=(\cos5x+\cos3x)^3$
    Factor the LHS,
    $\displaystyle \cos^35x+\cos^33x=(\cos5x+\cos3x)(\cos^25x-\cos5x\cos3x+\cos^23x)$
    So LHS=RHS if $\displaystyle \cos5x+\cos3x=2\cos4x\cos x=0$ which happens iff $\displaystyle x=22.5+45k,90,270$ where $\displaystyle 0\leq k\leq7$.
    Now assume $\displaystyle \cos5x+\cos3x\neq0$. Then we can divide it from both sides, and expand the RHS to get
    $\displaystyle 3\cos5x\cos3x=0$
    Similarly, this has solutions $\displaystyle x=18+36k,30+60j$ where $\displaystyle 0\leq k\leq9$ and $\displaystyle 0\leq j\leq5$.
    Summing up all the solutions, we have
    $\displaystyle \sum^7_{k=0}(22.5+45k)+\sum^9_{k=0}(18+36k)+\sum^5 _{k=0}(30+60k)-90-270=3960$
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