How to find the principal argument of $z^4$, given $z$?

Question

I am having trouble with a homework question.

Let $ z= \cos\left(\frac{3}{4}\pi\right)+i \sin\left(\frac{3}{4}\pi\right)$.

What is the principal argument of $z^4$ in radians?

Is it undefined? If not then I am lost...

Answer 1

Use de Moivre's formula, with

$$\cos\frac34\pi+i\sin\frac34\pi=:\text{cis}\frac34\pi=:e^{\frac34\pi i}\;\;:$$

$$z^4=\left(\text{cis}\frac34\pi\right)^4=\text{cis} \,3\pi$$