Is $\arcsin(1/4) / \pi$ rational? An approximation given by a calculator seem to suggest that it isn't, but I found no proof. Thanks in advance!

2$\begingroup$ See the essentially identical question math.stackexchange.com/q/3617176/442 already at math.stackexchange.com . $\endgroup$– Gerald EdgarJul 23, 2020 at 14:14

$\begingroup$ yes, seems very relevant, I'll look. Thanks a lot! $\endgroup$– ikpJul 23, 2020 at 14:21

$\begingroup$ "An approximation given by a calculator seem to suggest that it isn't [rational]": how come you can see (ir)rationality from a finite decimal representation? (If I got what you meant right.)(This applies to the MSE post as well.) $\endgroup$– Loïc TeyssierJul 23, 2020 at 16:56

$\begingroup$ Of course, "seem to suggest" is not very precise. I meant that I couldn't see any pattern immerging from the finitely many digits. $\endgroup$– ikpJul 23, 2020 at 17:48
3 Answers
This is a partial case of the classical result.
Yes, $\arcsin(\frac14)/\pi$ is irrational.
Suppose $\arcsin(\frac14)/\pi = m/n$, where $m$ and $n$ are integers.
Then $\sin(n \arcsin(\frac14))=\sin(m \pi)=0$.
We analyze this usng the formulas from Browmich as cited on Mathworld:
$$\frac{\sin(n\arcsin(x))}{n}=x\frac{(n^21^2)x^3}{3!} + \frac{(n^21^2)(n^23^2)x^5}{5!} + \cdots$$ $$\frac{\sin(n\arcsin(x))}{n \cos(\arcsin(x))}=x\frac{(n^22^2)x^3}{3!} + \frac{(n^22^2)(n^24^2)x^5}{5!} + \cdots$$ for $n$ odd or even respectively.
So the righthand sides must be 0 for $x=\frac14$.
However, when we multiply the terms on the righthand sides by $2^nn$ (if $n$ is odd) or $2^{n2}n$ (if $n$ is even), we find that all the terms are integers, except that the last nonzero term is $\pm\frac12$.
So the righthand side can't be 0, the lefthand side can't be 0, and $\arcsin(\frac14)/\pi$ must be irrational.

1$\begingroup$ Can you assume WLOG that $n$ is even, to simplify this somewhat? I don't think you use the fact that $m/n$ is in lowest terms anywhere, so you can replace $m$ and $n$ by $2m$ and $2n$, respectively, so that you don't have to split into cases later. $\endgroup$ Jul 23, 2020 at 15:42
Let $\theta= \arcsin(1/4)$. Assume $\theta$ is a rational multiple of $\pi$.
Then, there exists some $n$ such that $\sin(n\theta)=0$. This gives $\cos(n \theta)= \pm 1$.
Set $z=\cos(\theta)+i \sin(\theta)$, then $z^n= \pm 1$ and $\frac{1}{z^n}=\pm 1$.
This gives that $z$ and $\frac{1}{z}$ are algebraic integers, and hence so is $$2 i\sin(\theta)=z \frac{1}{z}$$
Therefore, $\frac{i}{2}$ is an algebraic integer, which is a contradiction.

2$\begingroup$ Nice proof! It's essentially the same as Matt F's, but replaces the explicit formulas with closed ringness of algebraic integers. Pretty neat. $\endgroup$ Jul 23, 2020 at 16:54