How can we find the algebraic degree of $\cos\left(\frac{p\pi}{q}\right)$ for $p$, $q$ coprime integers?
I know that the algebraic degree of $e^{\frac{p\pi i}{q}}$ is $\phi(q)$, since cyclotomic polynomials are irreducible. I also know that $$\cos(x)=\frac{e^{ix}+e^{-ix}}{2}.$$ But my knowledge of abstract algebra is quite slim, and I don't know how to relate these two facts.
I've checked on Google, and I haven't been able to find anything related. Any result or reference would be appreciated.
Hint: Assume that $|q|\neq 1$. There is a quadratic polynomial $f(x)$ with coefficients in $\mathbb{K}:=\mathbb{Q}\Biggl(\cos\left(\frac{p\pi}{q}\right)\Biggr)$ such that $\exp\left(\frac{p\pi\text{i}}{q}\right)$ is a root of $f(x)$. Prove that $f(x)$ is irreducible over $\mathbb{K}$ by showing that $$\left[\mathbb{Q}\Bigg(\exp\left(\frac{p\pi\text{i}}{q}\right)\Bigg)_{\vphantom{\frac{1}{\frac{1}{\frac{1}{1}}}}}^{\vphantom{\frac{1}{\frac{1}{\frac{1}{1}}}}}:\mathbb{K}\right]>1\,.$$