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# Urgent help needed watch

1. By writing cos(theta) in terms of z=e^i(theta) express cos^5(theta) in terms of cos(theta), cos3(theta) and cos5(theta).

Could anyone give me some help with this?
2. cos(5t) + i sin(5t)
= e^(5it)
= [e^(it)]^5
= [cos(t) + i sin(t)]^5
= cos^5(t)
= + 5i cos^4(t) sin(t)
= - 10 cos^3(t) sin^2(t)
= - 10i cos^2(t) sin^3(t)
= + 5 cos(t) sin^4(t)
= + i sin^5(t)

The real part of the LHS equals the real part of the RHS. So

cos(5t)
= cos^5(t) - 10 cos^3(t) sin^2(t) + 5 cos(t) sin^4(t)
= cos^5(t) - 10 cos^3(t) (1 - cos^2(t)) + 5 cos(t) (1 - cos^2(t))^2
= 16 cos^5(t) - 20 cos^3(t) + 5 cos(t)

A good check on the answer is to substitute t = 0.
3. Thank you so much you're a star!!

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Updated: November 24, 2005
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