Efficient Method for Finding Constants in Trig Function Power Problem

[Pandabasher]

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 that's 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.

 

[fzero]

Using the exponential form of sin is probably the fastest way to work this out. Expanding $(e^{i\phi} - e^{-i\phi})^5$ 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 $\sin(A+B)$ to write $\sin 3\phi$ and $\sin 5\phi$ in terms of $\sin\phi$ and $\cos\phi$. Then use $\cos^2\phi = 1-\sin^2\phi$ to write the whole thing as a polynomial in $\sin\phi$.

 

[Pandabasher]

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 :)