Sine values being rational
Can $$\sin r\pi $$ be rational if $r$ is irrational? Either a direct or existence proof is fine.
Solution 1:
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