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Math Help - Complex Algebra (De Moivre?)

  1. #1
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    Complex Algebra (De Moivre?)

    Hello,

    Would you use De Moivres theorem here and expand it? Not to sure!

    Thanks.

    By writing sin theta in terms of z = e ^(i theta), express sin^6 theta as a sum of
    terms of the form k cos(h theta)
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  2. #2
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    You don't need to write it in terms of \displaystyle e^{i\theta}.

    If \displaystyle z = \cos{\theta} + i\sin{\theta}, then

    \displaystyle z^6 = (\cos{\theta} + i\sin{\theta})^6.

    Expand this using the Binomial Theorem and write in terms of real and imaginary parts.


    You should also know that \displaystyle z^6 = \cos{6\theta} + i\sin{6\theta} by DeMoivre's Theorem.

    So once you have written the binomial expansion in terms of its real and imaginary parts, you can then equate real and imaginary parts and solve the problem.
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    Quote Originally Posted by MattWT View Post
    Hello,

    Would you use De Moivres theorem here and expand it? Not to sure!

    Thanks.
    By writing sin theta in terms of z = e ^(i theta), express sin^6 theta as a sum of
    terms of the form k cos(h theta)
    I would write \sin\theta = \frac1{2i}(z-z^{-1}), where z=e^{i\theta}. Then \sin^6\theta = \bigl(\frac1{2i}\bigr)^6(z-z^{-1})^6. Expand that by the binomial theorem, then pair off terms from opposite ends of the expansion and use facts like z^6+z^{-6} = 2\cos6\theta (by DeMoivre's Theorem).
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