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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    $\begingroup$ As I recall, the set of values $a\pi$ for algebraic numbers $a$ produces algebraic numbers $\sin(a\pi)$. $\endgroup$ – abiessu Mar 25 '15 at 20:20
  • $\begingroup$ ok thanks! (thats in radians right?) $\endgroup$ – stanley dodds Mar 25 '15 at 20:22
  • $\begingroup$ @abiessu, are you saying $\sin(\sqrt2\pi)$ is algebraic? $\endgroup$ – Barry Cipra Mar 25 '15 at 20:23
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    $\begingroup$ See math.stackexchange.com/questions/176889/… $\endgroup$ – Barry Cipra Mar 25 '15 at 20:24
  • $\begingroup$ @BarryCipra: I may have misunderstood other discussions on this topic; my answer is "I thought so, but now I'm second-guessing..." $\endgroup$ – abiessu Mar 25 '15 at 20:24

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