For which integer valued $\alpha=\angle BAC$ is it possible to find a right-angled triangle ABC with at least 2 integer valued sides?

Question

Besides of a=b where $\alpha=45°$ I wouldn't know how to tackle this problem.

Answer 1

This reduces to finding rational multiples of $\pi$ for which $\sin(\theta)$, $\cos(\theta)$, or $\tan(\theta)$ are rational.

The argument in this paper proves that the only such values are the obvious ones

i.e. the answer is $30^{\circ}$, $60^{\circ}$ and $45^{\circ}$.