If $|z| = 1, z \in \mathbb{C}$, what can you say about the real and imaginary parts of $z^i$

Question

If $|z| = 1, z \in \mathbb{C}$, what can you say about the real and imaginary parts of $z^i$?

I am not sure how to approach this question.

I have let $z = re^{i\theta}$ and then using $|z| = 1$, I find $$|re^{i\theta}| = 1\Rightarrow |r| |e^{i\theta}| = 1$$ so $ |r| = 1$.

Then $z^i = r^i e^{-\theta}$, but I don't see how I can proceed using what I know...

Answer 1

You may put it as follows. $$ z^i=e^{i\log z}=e^{i\log\left(re^{i\theta}\right)}=e^{i\left(\log r+i\theta\right)}=e^{i\log r-\theta}. $$ Provided that $\left|z\right|=1$, we have $r=1$. Thus $$ z^i=e^{i\log r-\theta}=e^{-\theta}. $$