Rational value of sine

Question

Given $x$, $\sin(x) \in \mathbb{Q}$, where $x$ is in degrees, we want to find all $x$ in the range $(0,90)$.

One trivial solution is $x=30$.

Answer 1

If $x$ is a rational multiple of $\pi$ then $e^{ix}$ is a root of unity, and so an algebraic integer. So $2\sin x=-ie^{ix}+ie^{-ix}$ is an algebraic integer. If $\sin x$ is rational too, $2\sin x$ is rational and an algebraic integer, so is an integer. As $|\sin x|\le1$ then $\sin x\in\{-1,-\frac12,0,\frac12,1\}$.

Alas then, the only solutions to your problem are the obvious ones...