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Trig Function Power Problem

  1. Nov 17, 2011 #1
    1. The problem statement, all variables and given/known data
    Find constants a, b and c such that
    sin(∅)^5=asin∅+bsin3∅+csin5∅



    2. Relevant equations



    3. The attempt at a solution
    I expressed sin in its complex form (if thats what its called) and put it to the power 5, and then multiplied each bracket out one at a time, and got the correct answers, as checked by wolfram, but I was wondering if there is a quicker way to find the bracket to the power 5 other than just multiplying each bracket out 5 times with its self? This is probably a simple question but I never seem to be able to get the simple stuff. Any help is greatly appreciated. Thanks.

    Sorry for the wording; I don't know how to use latex, and it's really ugly just typed out.
     
  2. jcsd
  3. Nov 18, 2011 #2

    fzero

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    Using the exponential form of sin is probably the fastest way to work this out. Expanding [itex](e^{i\phi} - e^{-i\phi})^5[/itex] might be a bit simpler using the binomial theorem: http://en.wikipedia.org/wiki/Binomial_theorem

    An alternative to using complex numbers is to use the identity for [itex]\sin(A+B)[/itex] to write [itex]\sin 3\phi[/itex] and [itex]\sin 5\phi[/itex] in terms of [itex]\sin\phi[/itex] and [itex]\cos\phi[/itex]. Then use [itex]\cos^2\phi = 1-\sin^2\phi[/itex] to write the whole thing as a polynomial in [itex]\sin\phi[/itex].
     
  4. Nov 18, 2011 #3
    Ah, thanks for clearing that up. My tutor used the binomial theorem, but didn't mention it, probably because he's been doing it for so long. Thanks for your help :)
     
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