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Math Help - Sums of cos(nx) and sin(nx)

  1. #1
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    Sums of cos(nx) and sin(nx)

    Find simple formulas for: 1 + cos(x) + cos(2x) + ... + cos(nx)
    and: sin(x) + sin(2x) + ... + sin(nx).
    Remember that: 1 + a + a^2 + ... + a^n = (a^(n+1) - 1) / (a - 1)
    and: (cosx + i*sinx)^n = cos(nx) + i*sin(nx).


    So far, I showed that cos(nx) = (cosx + i*sinx)^n - i*sin(nx) and plugged that in for the first sum to get:
    1 + cos(x) + cos(2x) + ... + cos(nx)
    = 1 + (cosx + i*sinx) - i*sin(x) + (cosx + i*sinx)^2 - i*sin(2x) + ... + (cosx + i*sinx)^n - i*sin(nx)
    = ((cosx + i*sinx)^(n+1) - 1)/(cosx + i*sinx - 1) - (i*sin(x) + i*sin(2x) + ... + i*sin(nx))

    I'm not sure where to proceed from here to simplify it any further.
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  2. #2
    MHF Contributor chisigma's Avatar
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    The sum of this 'geometric series'...

    \displaystyle \sum_{k=0}^{n} e^{i k x}= \frac{1-e^{i n x}}{1-e^{i x}} (1)

    ... allows You to write...

    \displaystyle \sum_{k=0}^{n} \cos k x= \Re \{\frac{1-e^{i n x}}{1-e^{i x}}\} (2)

    \displaystyle \sum_{k=0}^{n} \sin k x = \Im \{\frac{1-e^{i n x}}{1-e^{i x}}\} (3)

    Kind regards

    \chi \sigma
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