# trigonometric expressions as algebraic numbers

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.

• As 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
• ok thanks! (thats in radians right?) – stanley dodds Mar 25 '15 at 20:22
• @abiessu, are you saying $\sin(\sqrt2\pi)$ is algebraic? – Barry Cipra Mar 25 '15 at 20:23
• – Barry Cipra Mar 25 '15 at 20:24
• @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