For integers $n \neq 0$ is $\sin n$ irrational or transcendental?

Question

For integers $n \neq 0$ is $\sin n$ irrational or transcendental?

This arose from another question. I would hypothesize yes and yes, possibly with proof for irrationality existing and but not for the more difficult property of transcendentality.

Any ideas?

Answer 1

If $n\in\mathbb{N}\setminus\{0\}$, $\sin(n)$ is trascendental (hence irrational) by the Lindemann-Weierstrass theorem, since $2i\sin(n)=e^{in}-e^{-in}$.