Cosines and polynomials?

Algebra Level 5

If $\cos \left( \frac {720^\circ}{7} \right)$ is a root of the sixth-degree polynomial

$ax^{6}-bx^{4}+cx^{2}-x-1$

where $a, b, c$ are positive integers, find $a+b+c$ .

Bonus: Can you find the monic third degree polynomial with rational coefficients that has $\cos (\frac {720^\circ}{7})$ as a root?