An angle x is rational when measured in degrees. sin(x) is can be written using radicals. What are the conditions on x? If nested square roots are allowed?

What I know so far: 
If sin(x) can be expressed in radicals then so can $\cos{x}=\sqrt{1-\sin^{2}{x}}$, and $\tan{x}=\sin{x}/\cos{x}$. As can sin{x/2} and sin{x/3}, because I can use trig identities to get them in terms of sin{x} and the resultant quadratics and cubics can be solved.