# 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.
• By L-W theorem, $e^1$, $e^{-1}$, and $e^0$ are linearly independent over the algebraic numbers, so in particular $\frac{e^1+e^{-1}}{2}=\sinh 1$ cannot be rational. Jan 6 '19 at 23:15