Rational angle: $\pi$ times some rational number
When I ask wolframalpha for the sine of any such number, it seems to be able to represent it as a solution to a polynomial. It seems plausible that this is always true. Is there a theorem about this, or a source I can read more about? Or is there an easy way to see why this must always be true?
See: Chebyshev polynomials. If $a=\cos x$ is algebraic then $\sin x$ also, because it is in $\mathbb Q(a)(\sqrt{1-a^2})$ and it is algebraic extension(because it is degree $2$) of algebraic extension of $\mathbb Q$ $\mathbb Q(a)$. And $a$ is root of $T_n(X)-1=0$, when $nx$ is multiple of $2\pi$ so algebraic.
Much more easy and elegant way. Let $n$ be positive integer that $nx\in 2\pi\mathbb Z$ for rational angle $x$. Then $1=e^{inx}=(e^{ix})^n$ so $e^{ix}$ is algebraic, and $\sin x=\frac1{2i}\left(e^{ix}-\frac{1}{e^{ix}}\right)$ is also algebraic.
