Is there an angle $\alpha$ that makes both $\cos \alpha$ and $\sin\alpha$ rational?

Question

This may be a duplicate, but I couldn't find it yet.

My attempt is to write $\cos\alpha = p/q$ and $\sin\alpha = j/k$, where $p,q,j,k \in \mathbb Z$ and $|p|\le|q|,\ |j|\le|k|$, then $$ \frac{p^2}{q^2} + \frac{j^2}{k^2} = 1 \implies k^2p^2+q^2j^2 = q^2k^2 $$ and the question becomes are there any integrers that satisfy the last equation. But to procede it may require some knowledge in number theory which I lack.

Answer 1

Yes, any angle of a Pythagorean triangle satisfies this. For example, there is a 3-4-5 triangle; one of its acute angles satisfies $\cos \alpha=4/5$ and $\sin \alpha=3/5$.

Answer 2

To add to @EspeciallyLime's answer a general example, consider $m,n \in \mathbb{Z}$ which are non zero such that $$ \sin(\alpha) = \frac{m^2-n^2}{m^2 + n^2}$$ and $$\cos(\alpha) = \frac{2mn}{m^2+n^2} $$ and it is easy to check that denominators and numerators satisfy required properties.