I need to find the real and imaginary part of this $Z_{n}=\left (\frac{ \sqrt{3} + i }{2}\right )^{n} + \left (\frac{ \sqrt{3} - i }{2}\right )^{n}$ I have a test tomorrow and i have some troubles understanding this kind of problems, would really appreciate some help with this
$$ Z_{n}=\left (\frac{ \sqrt{3} + i }{2}\right )^{n} + \left (\frac{ \sqrt{3} - i }{2}\right )^{n}
$$
$$ 
Z_{n}\epsilon  \mathbb{C}
$$

Let $\omega=\frac{\sqrt{3}+i}{2}$. Then $Z_n$ is twice the real part of $\omega^n$.
Now, $\omega$ is a primitive $12$-th root of unity and so $\omega = \exp(\frac{2\pi i}{12})$.
Therefore, $\omega^n=\exp(n\frac{2\pi i}{12})$, whose real part is $\cos(n\frac{2\pi}{12})=\cos(\frac{n\pi}{6})$.
