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.

  • 3
    $\begingroup$ 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. $\endgroup$
    – vadim123
    Jan 6 '19 at 23:15

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