Trig Function Power Problem

  • #1

Homework Statement


Find constants a, b and c such that
sin(∅)^5=asin∅+bsin3∅+csin5∅



Homework Equations





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.
 

Answers and Replies

  • #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].
 
  • #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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