which trigonometric expressions give results which are rational or expressible as surds? is there a complete set? are there infinite? for example, the well known $sin(30)=1/2$, and a range of others like $sin(45)=\sqrt2/2$, I know of, but how many are there? it seems like multiples of 30, 18, and other fractions of 360 degrees give results as surds more often.
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1As I recall, the set of values $a\pi$ for algebraic numbers $a$ produces algebraic numbers $\sin(a\pi)$. – abiessu Mar 25 '15 at 20:20
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ok thanks! (thats in radians right?) – stanley dodds Mar 25 '15 at 20:22
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@abiessu, are you saying $\sin(\sqrt2\pi)$ is algebraic? – Barry Cipra Mar 25 '15 at 20:23
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1See http://math.stackexchange.com/questions/176889/for-which-angles-we-know-the-sin-value-algebraically-exact – Barry Cipra Mar 25 '15 at 20:24
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@BarryCipra: I may have misunderstood other discussions on this topic; my answer is "I thought so, but now I'm second-guessing..." – abiessu Mar 25 '15 at 20:24
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@abiessu, to quote a comment by Robert Israel from my previous comment's link, "If $x/\pi$ is an algebraic irrational, $\sin(x)$ is transcendental by the Gelfond-Schneider theorem." – Barry Cipra Mar 25 '15 at 20:28
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@BarryCipra: so the correct answer is to replace $a$ as a rational instead of $a$ as algebraic in my initial comment... – abiessu Mar 25 '15 at 21:03