Prove that this equation is true.

sin22a-sin2a=sin3a x sina

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- May 4th 2008, 09:30 AMblerttatrigonometric identity
Prove that this equation is true.

sin22a-sin2a=sin3a x sina - May 4th 2008, 09:48 AMReckoner
I really don't understand your question. Do you mean

$\displaystyle \sin(22a)-\sin(2a)=\sin(3a)\sin a$?

$\displaystyle \sin^2(2a)-\sin^2a=\sin^3a\sin a$?

$\displaystyle \sin^{22}a-\sin^2a=\sin^3a\sin a$?

$\displaystyle \sin^2(2a)-\sin(2a)=\sin(3a)\sin a$?

Please be more careful with your notation. - May 4th 2008, 01:27 PMblertta
- May 4th 2008, 01:42 PMReckoner
Sorry. This equation isn't true for all $\displaystyle a$, so I thought you must have meant something else. Anyway, if we let $\displaystyle a=\frac{\pi}3$, we get

$\displaystyle \sin(22a)-\sin(2a)=-\frac{\sqrt{3}}2$

but

$\displaystyle \sin(3a)\sin a = 0$.

Was there supposed to be a restriction on $\displaystyle a$? - May 5th 2008, 03:23 AMblertta
- May 5th 2008, 03:42 AMtopsquark
Then you want to solve the equation for a.

My calculator comes up with 18 solutions. I would not be surprised if there were actually 22 possibles (thus leaving 4 complex solutions.) Generally an equation with $\displaystyle sin(na)$ in it will reduce to an nth degree polynomial so you would be stuck solving a 22nd degree polynomial equation, which is impossible to do in general. You are going to be stuck doing numerical approximations.

However notice that there is at least one nice solution: a = 0.

-Dan