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.
Here $p$ and $q$ are co-prime integers. Is it necessarily an algebraic number?
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