What is the value of $\operatorname{Arg}(\frac{z}{\bar{z}})$?

Question

As part of an exercise, I was asked to plot this condition on the complex plane:

$$-\frac{\pi}{2} \leq \operatorname{Arg}\left(\frac{z}{\bar{z}}\right) \leq\frac{\pi}{2}$$

My question is:

What is the value of $\displaystyle\operatorname{Arg}\left(\frac{z}{\bar{z}}\right)$?

Answer 1

$$z=re^{i\theta}, \bar{z} = re^{-i\theta} \implies\frac{z}{\bar{z}} = e^{2i\theta} $$

$\arg (\frac{z}{\overline{z}}) = 2\theta = 2 \arg (z) $

Answer 2

$$\frac{a+bi}{a-bi}=\frac{a^2+2abi-b^2}{a^2+b^2}$$

$$\arg\left(\frac{a+bi}{a-bi}\right)=\arctan\left(\frac{2ab}{a^2-b^2}\right)$$

As normally $\arctan(b/a)$ is the $\arg(z)$