In solving $\displaystyle z^n = 1 $ we get the distinct solutions $\displaystyle z = e^{i(2k \pi/n)} $. We get this as follows:

Let $\displaystyle z = re^{i \theta} $. Then $\displaystyle (re^{i \theta})^{n} = 1 $ or $\displaystyle r^{n}e^{i n \theta} = 1e^{i0} $. Thus $\displaystyle r = 1 $ and $\displaystyle \theta = 2 k \pi/n $.

How do we draw the solutions? Why do we write $\displaystyle \omega_{n} = \cos \frac{2 \pi}{n} + i \sin \frac{2 \pi}{n} $? Like the three cube roots of $\displaystyle 1 $ form an equilateral triangle inscribed in the unit circle. But does it matter how we draw it?