Can $\sin r\pi $ be rational if $r$ is irrational? Either a direct or existence proof is fine.
Sine values being rational
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$\begingroup$
number-theory
trigonometry
irrational-numbers
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3I've created a new Wikipedia article titled [Niven's theorem](http://en.wikipedia.org/wiki/Niven%27s_theorem). Two-and-a-half hours after I created it, I entered "Niven's theorem" into Google and that Wikipedia article was on the first page of results. So contribute to it if you can. Besides contributions _within_ the article, there's the matter of which other articles ought to link to it. I've created three such links; if you think of others that should be there, you can add those too. – 2011-12-02
1 Answers
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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