$S = \{z ∈ C | z^6 = 1\}$. List the elements of S as complex numbers in polar form and in Cartesian form.

Question

I'm stuck on this question: I need to list the elements of S in cartesian and polar form where $S = \{z ∈ C | z^6 = 1\}$

What I did so far is write $z^6=r^6(\cos6θ+i\sin6θ)=1$ which also equals to $r^6e^{i6θ}$.

For $z^6$ to be $=1$, $r$ must be $= 1$ and $i\sin6θ$ needs to be equal to $0$ and $\cos6θ =1$ (or the other way round) so $θ = 2πn/6$. This is where I got up to, not sure if I'm right though. How can I list the elements?

Really appreciate your time and help. Thank you!

Answer 1

There are $6$ roots given by

$$z_n=\cos (\frac {n\pi}{3})+i\sin (\frac {n\pi}{3}) $$ $$=e^{i\frac {n\pi}{3}} $$

with $$n\in\{0,1,2,3,4,5\} $$

$$z_0=1$$ $$z_1=\frac {1}{2}+i\frac {\sqrt {3}}{2} $$ $$z_2=-\frac {1}{2}+i\frac {\sqrt {3}}{2} $$ $$z_3=-1$$ $$z_4=-\frac {1}{2}-i\frac {\sqrt {3}}{2} $$ $$z_5=\frac {1}{2}-i\frac {\sqrt {3}}{2} $$

Answer 2

First of all, $\theta<2\pi$. Since if not, then we have repetitive roots. (For example $\theta=0$ and $\theta=2\pi$ are the same)

$r^6e^{i6θ}=1=1e^{2k\pi}$

$\to r=1, \theta=2k\pi/6$ for $\theta < 2\pi$

As a result, $z=re^{i\theta}=\cos{(k\pi/3)}+i\sin{(k\pi/3)}$ for $k=0,1,...,5$