Finding $n$ that satisfies $\left(\frac{1}{2} +i\frac{\sqrt{3}}{2}\right)^n=1 $

Question

Using De Moivre's Formula, I have to find the value of $n$ that satisfies:

$$ \left(\frac{1}{2} +i\frac{\sqrt{3}}{2}\right)^n=1$$

I found that $\cos(\frac{n\pi}{6})=1$, which means that $n$ should be equal to $12$.

Are there any other values of $n$ that I missed?

Answer 1

Hint

You have $$\frac{1}{2} +i\frac{\sqrt{3}}{2} = e^{i \frac{\pi}{3}}$$ and therefore...

Answer 2

Yes your findings are correct so far, but it is important to remember that the cos function is periodic with period $2\pi$. So $$\cos(t+2\pi) = \cos(t)$$, but we also have the rule $$\cos(-t) = \cos(t)$$These things together should help you find all solutions.