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May 24th, 2009, 06:55 PM   #1
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Proof of a law of cosine/sine

Can you help me prove sin 3A = 3 sin A cos^2 A - sin^3 A

or

(cos A + i sin A)^3 = cos 3A + i sin 3A?

Last edited by skipjack; February 28th, 2018 at 01:25 PM.
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May 25th, 2009, 03:35 AM   #2
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Re: Proof of a law of cosine/sine

Quote:
Originally Posted by aquamarine4443
Can you help me prove sin 3A = 3 sin A cos^2 A - sin^3 A?
Recall $\displaystyle \cos^2{A} = 1 -\sin^2{A}$.

Have you learned the power reduction formula $\displaystyle \sin^3{A}=\frac{3\sin{A}-\sin{3A}}{4}$?

Quote:
Originally Posted by aquamarine4443
Can you help me prove (cos A + i sin A)^3 = cos 3A + i sin 3A?
This is an example of De Moivre's formula. (In case you didn't know).

Last edited by skipjack; February 28th, 2018 at 01:23 PM.
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May 25th, 2009, 07:31 AM   #3
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Re: Proof of a law of cosine/sine

Depending on your classroom procedure, you can do the first by expansion formula [sin(A+B) = sinAcosB + cosAsinB] applied a couple of times. The second can be done by algebraic expansion, then using in part the first result.
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