The minimal polynomial of $\cos(\pi/7)$ is $8x^3-4x^2-4x+1=0$ with
three real roots. The degree is $<3$ so the roots are expressible
using radicals with the imaginary $i$. Since the roots are real,
taking only the real part, the roots are expressible using radicals of
rational numbers.

Why not? See [here][1] on Wikipedia.


  [1]: https://en.wikipedia.org/wiki/Casus_irreducibilis