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Math Help - Using De Moivre's Theorem

  1. #1
    Member alexgeek's Avatar
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    Using De Moivre's Theorem

    Hello,
    Doing a practice paper and found this question, not sure what to do:
    Write down de Moivre's Theorem for n=5. hence show that for \sin \theta \neq 0

     \frac{\sin 5 \theta}{\sin \theta} = A cos^4 \theta + B \cos^2 \theta + C

    where A, B, C are constants to be determined.
    Deduce the limiting value of  \frac{\sin 5 \theta}{\sin \theta} as  \theta tends to zero.
    I've written de Moivre for n=5:

    (r(\cos \theta + i \sin \theta))^5  = r^5(\cos 5\theta + i \sin 5 \theta)

    But don't know where to go from there, at all.

    Thanks
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  2. #2
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    Quote Originally Posted by alexgeek View Post
    Hello,
    Doing a practice paper and found this question, not sure what to do:


    I've written de Moivre for n=5:

    (r(\cos \theta + i \sin \theta))^5 = r^5(\cos 5\theta + i \sin 5 \theta)

    But don't know where to go from there, at all.

    Thanks
    From De'moivre's theorem it follows that

    (\cos (\theta) + i \sin (\theta))^5 = \cos (5\theta) + i \sin (5 \theta).

    Now I suggest you expand the left hand side of this identity and equate the imaginary part of it to \sin (5 \theta).
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  3. #3
    Member alexgeek's Avatar
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    Finally got it, tried expanding it by hand then realised I could do it by binomial. Cheers
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