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Math Help - DeMoivre's Theorem

  1. #1
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    DeMoivre's Theorem

    I was asked to find both sin4() and cos4() in terms of sin() and cos()
    using DeMoivre's Theorem.


    I couldnt figure out how to type the actual symbol for theta on here so i just used ()....

    I have the Theorem in front of me and understand it (pretty much), but am struggling getting past the question above.
    I need some help please! Its been driving me crazy....
    Last edited by wick; December 15th 2008 at 10:31 AM.
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  2. #2
    MHF Contributor
    Joined
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    France
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    Hi

    (\cos \theta + i \,\sin \theta)^4 = \cos(4 \theta) + i \sin (4 \theta)

    Developing the left member
    \cos^4 \theta + 4i \cos^3 \theta \,\sin \theta - 6 \cos^2 \theta \sin^2 \theta - 4i \cos \theta \sin^3 \theta + \sin^4 \theta = \cos(4 \theta) + i \sin (4 \theta)

    Separating real and imaginary parts
    \cos(4 \theta) = \cos^4 \theta - 6 \cos^2 \theta \sin^2 \theta + \sin^4 \theta

    \sin (4 \theta) = 4 \cos^3 \theta \,\sin \theta - 4 \cos \theta \sin^3 \theta

    Using \sin^2 \theta = 1 - \cos^2 \theta

    \cos(4 \theta) = 8\cos^4 \theta - 8 \cos^2 \theta + 1
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