Is there a way to characterise the set of all angles $0°<\phi<360°$ such that $\phi$ can be classically trisected? (That is, the trisection can only be done with a finite sequence of straight lines and circles.)
Of course, it is well-known that $90°$ can be trisected. (In fact, this question arose after I had shown my students a trisection of the right angle.) Now some other angles are trivially trisectable (e.g., $180°,270°, 360°$), but apart from these are there more interesting angles? Since there are infinitely many of them the only way to systematically characterise the set of all trisectable angles is to use algebra. From the trigonometric identity $$\cos\phi=4\cos^3\frac\phi 3-3\cos\frac\phi 3,$$ we see that the problem is essentially cubic. So in other words is there some criterion on $\cos\phi$ that tells us whether the cubic in $\cos\frac\phi 3$ is reducible to a lower degree polynomial over the extension of $\mathrm Q$ by $\cos\phi$? Clearly, it cannot simply be about the rationality or otherwise of $\cos\phi,$ since it is well-known, for example, that $60°$ cannot be trisected.
So is there a way to tell exactly which angles can be trisected? Or rather is there a provably complete list of all trisectable angles somewhere?
Thank you.