When are $\alpha$ and $\cos\alpha$ both rational?

Question

I've seen a bunch of related questions, but none lead me to a solution to this problem:

Is there any "easy" characterisation of $\alpha\in\mathbb Q$ such that $\cos\alpha\in\mathbb Q$?

It feels like there should be only "few" such numbers.
Are there only finitely many?

Answer 1

As requested in the comments:

It follows from Lindemann-Weierstrass that $\alpha=0$ is the only example. To see that:

Suppose $\cos(\alpha)$ is algebraic. Then, of course, $\sin(\alpha)$ is also algebraic. It follows that $e^{i\alpha}$ is algebraic. But Lindemann-Weierstrass tells us that, unless $\alpha=0$, this implies that $\alpha$ is transcendental. In particular, it can not be rational (unless it is $0$).