$\sin(2\pi q)$ is algebraic over $\mathbb{Q}$

Question

Let $q$ be an element of $\mathbb{Q}$ (rational numbers). How can I prove that $\sin (2\pi q)$ is algebraic over $\mathbb{Q}$ for any $q$? I am trying the method: Euler formula: $e^{i\theta} =\cos \theta+i\sin \theta$ so $2\sin\theta = ie^{i\theta}-ie^{-i\theta}$. $i$ is root of $x^2+1$. So if I prove that $e^{i\theta}$ and $e^{-i\theta}$ is algebraic over $\mathbb{Q}$f or $\theta=2\pi q$, it will be ok.

Answer 1

If $q=\frac mn$, with $m,n\in\mathbb Z$ and $n>0$, then$$e^{2\pi iq}=e^{2\pi i\frac mn},$$which is algebraic, since it is a root of $x^n-1$.