TITLE: Sine values being rational
QUESTION [8 upvotes]: Can $$\sin r\pi $$ be rational if $r$ is irrational? Either a direct or existence proof is fine.

REPLY [12 votes]: As J. M. said, Niven's theorem does it.  There is some $r$ such that $\sin (r\pi) = \frac{1}{3}$  As $\sin (r\pi)$ is rational and not $0, \pm1, \pm \frac{1}{2}$, $r$ is not rational