The function $f(x)=\arcsin(a\sinh(\sin(x)))$ While playing around with Desmos, I came across the function $$f(x)=\arcsin(a\sinh(\sin(x)))$$
where $a\in\mathbb{R}$ is a constant (whose value changes the function drastically; if you see the graph, you will understand).
So I came up with the following question (which I don't know how to extract an answer other than an approximation):

We have the function $f:A\to\mathbb{R}$, given by $$f(x)=\arcsin(a\sinh(\sin(x)))$$ with $a\in B=[-m,m]\subset\mathbb{R}$, and $m\in\mathbb{R}$.
What is the biggest value of $m$ such that $f$ is continuous for every $x=(2k+1)\dfrac\pi{2}$ (with $k$ an integer), and thus $A=\mathbb{R}$? Is the number $m$ rational or irrational?
(A little hint: $m\approx 0.850918$.)


The exact value of $m$ must be such that:
$$
m\sinh(1)=1
$$
and thus:
$$
m=\frac{1}{\sinh(1)}\approx 0.8509181282
$$
which I would suspect is irrational. For greater values of $m$ we will have $m\sinh(\sin(x))$ outside the domain $[-1,1]$ of $\arcsin$ which is why the graph breaks.
