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I'm aware of the existence of trig formulas such as double angle and half angle formulas. I was curious are there any for exponential angles.

What I mean is: If I were presented with the expression $\sin(\theta^2)$, and if I know $\sin(\theta)$, is there a formula that allows me to find it?

Blue
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violetorigin
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    Not as far as I know (beyond stuff like Taylor series). The reason being, well ... in what sense would it ever naturally arise to speak of the square of an angle? Products of angles (or even just their measures) is weird, but adding and subtracting them is a bit more natural. – PrincessEev Apr 21 '22 at 19:58
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    You can easily check $\sin\pi^2\ne\sin0^2$, so $\sin\theta$ doesn't in general determine $\sin\theta^2$. – J.G. Apr 21 '22 at 20:00
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    Loosely speaking, a formula in $\sin\theta$ (and $\cos\theta$, $\tan\theta$, etc) is periodic in $\theta$, but $\sin(\theta^2)$ is not. (There is no constant $k\neq0$ such that $\sin((\theta+k)^2)=\sin(\theta^2)$ for all $\theta$.) – Blue Apr 21 '22 at 20:25

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