**Question amended following response from lulu.**

Am I misunderstanding Niven's Theorem? I understand it to mean that only angles $0$, $\pi/6$, and $\pi/2$ of the angles between $0$ and $\pi$ radians that are rational multiples of $\pi$ can have rational sines, but there seem to be loads of rational sines - eg $1/4$ and $5/13$ with angles $0.25268026$ and $0.39479112$ radians according to my calculator. Can angles that are irrational multiples of $\pi$ have rational trig ratios?

Thank you for your response, lulu. You are right, in that according to Niven's Theorem, the rational angles are 0,/6,/2 for rational sines. I was thinking of the cosines, which is why I originally wrote /3 instead of /6. I have now amended that. When I wrote "rational angles" I meant rational multiples of radians. I am not so much bothered about the angles as I am about the cosines of the angles in this range and I take it from Niven's Theorem that the **cosines** must be irrational for all angles that are rational multiples of except for 0,/3,/2. Since there are clearly many rational cosines between 0 and 1, I was asking if these are necessarily cosines of angles that are products of and z, where z is some irrational number.