Sketch this subset in the complex plane

Question

I'm trying to plot this in the complex plane:

$$C = \{z \in\mathbb C | z \neq 0,\arg(z^2) \in \left[0, \pi/4\right)\}$$

My work so far: Let $z = re^{(i\theta)}$ $z^2 = r^2(\cos(2\theta) + i\sin(2\theta))$

I know how to plot in the complex plane, but I'm not really sure how to specifically plot this function. Thanks in advance for your help!

Answer 1

You want $2\theta $ between zero and $\pi/4$ thus $\theta$ is between zero and $\pi/8$

That is the part of the complex plane between the two rays $\theta =0$ and $\theta =\pi/8$

Answer 2

Having $z^2=e^{i\theta}$, must be $z=e^{i\theta/2}$ So,

$$C=\{z\in\mathbb C|\arg(z)\in\left[0,\pi/8\right)\}$$

It's the angle of value $\pi/8$, one side along $\mathbb R^+$, the other along the ray of complex numbers with $\pi/8$ and (consequently) vertex in $0$.