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    Trigonometric constants

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    1) Express (cosx)^n in terms of sums of coskx
    2) Express (sinx)^n in terms of sums of coskx or sinkx
    3) Express cosnx in terms of sums of (cosx)^k
    4) Express sinnx in terms of sums of (sinx)^k
    5) Derive exact trigonometric identities for sin(k*pi) for k being:
    1/12, 1/10,1/8,1/6,1/5,1/4,3/10,1/3,3/8,2/5,5/12

    © BrainMass Inc. brainmass.com November 30, 2021, 1:05 am ad1c9bdddf
    https://brainmass.com/math/trigonometry/powers-sine-cosine-exact-trigonometric-constants-82284

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    Powers of Sine and Cosine

    1) It can be useful to express in terms of sums of :
    First for even powers:

    Conclude
    Or

    There is a corollary from this, integrating the left hand side is a lot harder than integrating the left hand side:

    Conclude

    2) Now for odd powers:

    Conclude
    Or

    There is a corollary from this, integrating the left hand side is a lot harder than integrating the left hand side:

    Conclude ...

    Solution Summary

    This provides several examples of working with exact trigonometric constants. The expert derives the exact trigonometric identities for the functions.

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