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DeMoivre's Theorem

  1. Mar 14, 2017 #1
    1. The problem statement, all variables and given/known data
    Can somebody explain to me how the first two steps are performed?

    3. The attempt at a solution
    I have no idea how to start the question. I tried using an equation for sin^6 x derived by (cos x + i sinx)^6 = cos 6x+isin 6x but the solution becomes way too hard.

    Attached Files:

  2. jcsd
  3. Mar 14, 2017 #2


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    If [itex]z = e^{i\theta}[/itex] then [tex]
    \cos n\theta = \frac{z^n + z^{-n}}{2} \\
    \sin n\theta = \frac{z^n - z^{-n}}{2i}.[/tex] Hence [tex]
    \sin^6 \theta = \frac{(z - z^{-1})^6}{(2i)^6} = -\frac{(z - z^{-1})^6}{64}.[/tex]
  4. Mar 14, 2017 #3
    Are these equation correct?
    [tex] \sin^n \theta = \frac{(z-z^{-1})^n}{(2i)^n} [/tex] and [tex]\cos^n \theta = \frac{(z+z^{-1})^n}{2^n}[/tex]
    Last edited: Mar 14, 2017
  5. Mar 14, 2017 #4
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