Finding sine in the real subfield of a cyclotomic field

Question

Consider the cyclotomic field $K=\mathbb{Q}(\zeta)$ where $\zeta = \exp(2\pi i/n)$ is a primitive $n$-th root of unity. I think it's pretty well known that the maximal totally real subfield of $K$ is $\mathbb{Q}(\zeta+\zeta^{-1}) = \mathbb{Q}(2\cos(2\pi/n))$. So the real subfield contains $c=\cos(2\pi/n)$.

Does $s=\sin(2\pi/n)$ exist in this subfield? If so, how might I express $s$ as a rational function of $c$?

Answer 1

It depends on $n$.

We have $\sin(2\pi/n) = \frac{\zeta -\zeta^{-1}}{2i}$.

We see that $\sin(2\pi/n)$ is in the maximal totally real subfield of $\mathbb Q(\zeta, i)$. Hence, the real subfield contains $\sin(2\pi/n)$ if and only if $4\mid n$.

Answer 2

For $n=3$ this does not hold: $K=\mathbb{Q}(\omega)$ for $\omega=-1/2+\sqrt{3}/2$. Here $\cos(2\pi/3)=-1/2$, so $\mathbb{Q}(\omega+\omega^{-1})=\mathbb{Q}$. But $\sin(2\pi/3)=\sqrt{3}/2$, which is clearly not contained there.