galois theory - When is sin(r pi) expressible in radicals for r rational?

Perhaps this question will not be considered appropriate for MO - so be it. But hear me out before you dismiss it as completely elementary.

As the question suggests, I would like to know when $sin(ppi/q)$ can be expressed in radicals (in the way that $sin(pi/4) = sqrt{2}/2$ and $sin(pi/3) = sqrt{3}/2$ can). Let $alpha = sin(x)$, and consider the field extension $mathbb{Q}[alpha]$. Using $(cos(x) + isin(x))^k = cos(kx) + i sin(kx)$ together with the binomial formula and the Pythagorean identity relating sine and cosine, we can see that $sin(kx)$ lies in a solvable extension of $mathbb{Q}[alpha]$. Thus $sin(ppi/q )$ is expressible in radicals if $sin(pi/q)$ is.

To handle $sin(pi/q)$, we start by using the same trick (which most people also learn in an elementary trig class). Write $-1 = (cos(pi/q) + isin(pi/q))^q$, use the binomial theorem to expand, compare imaginary parts, and express the right-hand-side in terms of sine using the Pythagorean identity. This gives an explicit equation for any $q$ one of whose solutions is $sin(pi/q)$. This equation is not a polynomial in $sin(pi/q)$ since it involves terms of the form $sqrt{1 - sin^2(pi/q)}$, but it is enough to prove that $sin(pi/q)$ is algebraic.

So I am curious about the number theoretic properties of this equation. What can be said about the Galois group of its "splitting field" over $mathbb{Q}$? Can we at least determine when it is solvable? Note that if the prime factors of $q$ are $p_1, ldots p_k$ and we can express each $sin(pi/p_j)$ in radicals, then the same is true for $sin(pi/q)$. So it suffices to consider the case where $q$ is prime. That's about all the progress I have made.

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