TITLE: Is there any use for $\sin(\sin x)$?
QUESTION [26 upvotes]: The convention that $\sin^2 x = (\sin x)^2$, while in general $f^2(x) = f(f(x))$, is often called illogical, but it does not lead to conflicts because nobody uses $\sin(\sin x)$.
But is this really true? Or is there a real-world application in which $\sin(\sin x)$ occurs? Or maybe something a bit more general, like $\sin(C \sin x)$ for some constant $C \neq 0$?

REPLY [14 votes]: The characteristic function for the Poisson distribution is given by
$$
e^{\lambda\left(e^{i\theta}-1\right)}=e^{\lambda\left(\cos\theta-1\right)}\left(\cos(\lambda\sin\theta)+i\sin(\lambda\sin\theta)\right)
$$

REPLY [6 votes]: Valerii Salov, in Notation for Iteration of Functions, Iteral, has written a 23 page (!) paper on how to denote the iteration of a function, specifically for the issue raised by the OP in Inevitable Dottie Number. Iterals of cosine and sine.
The notation proposed by Salov uses the first Cyrillic letter И of the Russian word Итерация for iteration, as a variation on the product symbol $\prod$, so that И$_v^n f$ is the n-fold iteration of the function $f(x)$ for initial value $x=v$.
Some examples of the use of this compact notation:

This can be typeset in MathJax as a LARGE version of Unicode character 1048:
$$\large{
{\LARGE\unicode{1048}}_0^n(x+1)=n,\
{\LARGE\unicode{1048}}_{x=1}^n ax=a^n,\
{\LARGE\unicode{1048}}_{1}^\infty \frac{1}{x+1}=\lim_{n\to\infty}
{\LARGE\unicode{1048}}_{1}^n \frac{1}{x+1}=\frac{2}{1+\sqrt{5}}
}$$