Roots of unity with equation.

Question

$z+\frac{1}{z}=2\cos{3^\circ}$

Find $z^{2000}+\frac{1}{z^{2000}}$

I'm pretty sure it has to to do with roots of unity, but the $2\cos{3^\circ}$ is throwing me off. Some work: $$z+\frac{1}{z}=2\cos{3^\circ}$$ $$\implies z^2-(2\cos{3^\circ})z+1=0$$ If the "$2\cos{3^\circ}$" term was changed to, we could say $z$ is the complex root of $z^3=1$ and then apply roots of unity. I'm also conjecturing $z^n$ repeats every 3.

Answer 1

Use chebyshev polynomials. Answer is $-1$

Answer 2

If you have $z+1/z=2\cos t$, then $z=e^{it}$ or $e^{-it}$. In either case, $z^n+1/z^n=e^{int}+e^{-int}=2\cos nt$.