TITLE: Is sin(x) necessarily irrational where x is rational?
QUESTION [39 upvotes]: My friend and I were discussing this and we couldn't figure out how to prove it one way or another.
The only rational values I can figure out for $\sin(x)$ (or $\cos(x)$, etc...) come about when $x$ is some product of a fraction of $\pi$.
Is $\sin(x) $ (or other trigonometric function) necessarily irrational if $ x $ is rational?
Edit:
Excluding the trivial solution of 0.

REPLY [39 votes]: If $\sin x$ is rational (or even just algebraic), then $\cos x=\pm \sqrt{1-\sin^2 x}$ is algebraic. Therefore $e^{ix}=\cos x+i\sin x$ is algebraic, so by the Lindemann-Weierstrass theorem, $x$ cannot have been nonzero algebraic -- in particular not nonzero rational.