Possible trig identity?

Question

Is there a trigonometric identity for $\sin(ab)$? Thanks in advance! I can't find it anywhere. Bothering me a lot.

For that matter, what about $\sin(a^{-1})$?

Both of these for cosine, too, but if you can get any one of these 4, that would be great.

This might help get the creative juices flowing!

There is none, unless e.g. $\sin (ab)=\frac{1}{\csc (ab)}$ or $\sin (ab)=\pm\sqrt{1-\cos^2 (ab)}$. — Américo Tavares, Mar 20 '14 at 00:47
My first instinct is to say "no" for the first question since using it on $\sin(a)$ would mean there would be a bunch of $\sin(1)$'s floating around — Chris Brooks, Mar 20 '14 at 00:47

score 0 · Answer 1 · answered Mar 20 '14 at 01:41

I don't know the answer, and this isn't really a complete answer, it is more of a comment:

Perhaps use the fact that $sin(ab)$ is the imaginary part of $e^{iab}=(e^{ia})^b = (\cos a + i \sin a)^b$, and then use some binomial expansion formula. For example, taking $b=2$ recovers $\sin(2a)=2\cos(a)\sin(a)$, and similar expressions for other positive integers $b$ (I think chebyshev polynomials arise). Of course, $b$ need not be an integer, but there is still a binomial expansion for real numbers $b$, and maybe something can still be said, I don't know.

Possible trig identity?

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