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Math Help - 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

     \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
    Junior Member
    Joined
    Nov 2007
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    Use the product to sum formula on the RHS,
    8\cos^34x\cos^3x=(2\cos4x\cos x)^3=(\cos5x+\cos3x)^3
    Factor the LHS,
    \cos^35x+\cos^33x=(\cos5x+\cos3x)(\cos^25x-\cos5x\cos3x+\cos^23x)
    So LHS=RHS if \cos5x+\cos3x=2\cos4x\cos x=0 which happens iff x=22.5+45k,90,270 where 0\leq k\leq7.
    Now assume \cos5x+\cos3x\neq0. Then we can divide it from both sides, and expand the RHS to get
    3\cos5x\cos3x=0
    Similarly, this has solutions x=18+36k,30+60j where 0\leq k\leq9 and 0\leq j\leq5.
    Summing up all the solutions, we have
    \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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