By definition $ z \in\mathbb{C}$ is a n-th root of unity iff $z^n = 1$. My assignment is to
(iv) List all n-th roots of unity in their polar form. You may use that there are only $n$ Elements with z $\in\mathbb{C}$ so that $z^n = 1$.
What I started out with is this:
$ z^n = 1 = 1 + 0i = 1(cos(0 + 2\pi k) + i \ sin(0+2\pi k))$ for $k \in\mathbb{Z}$.
$z = 1^{1/n} (cos(2\pi k) + i \ sin(2\pi k))^{1/n} $
However I don't see how to go further without knowing the De Moivre Theorem or Euler's Formula. If it helps, before in this assignment I have shown that $ z \in S^1$ , $\bar z = z^{-1}$ and that the set of n-th roots of unity is a subgroup of $S^1$.
You shouldn't need De Moivre's theorem, just the geometric view of multiplication of complex numbers. $(re^{i\theta})^n = r^ne^{in\theta}$. You should be able to find $n$ numbers whose $n$th power is $1\cdot e^{i(0)}$.