Depicting complex numbers

Question

Sorry for a stupid question but how would you depict $0\leq arg(z)\leq \frac{7\pi}{3}$ on the complex plane? Is this a circle or is it going to be the the same as when you depict $0\leq arg(z)\leq \frac{\pi}{3}$?

(Suppose $|z|=1$)

Answer 1

Firstly, as a geometrical angle, it is clear that whole circle rotations do not change the point, so angles differing by an integer multiple of $2\pi$ radians (a complete circle) are the same, as reflected by the figure below. Similarly, from the periodicity of $\sin (x)$ and $\cos (x)$, the definition $$ z=r\left( \cos \varphi +i \sin \varphi \right)=re^{i \varphi}$$

also has this property. The argument of zero is usually left undefined.

Answer 2

It will be a circle, since all points with argument in the interval $[0,2 \pi)$ satisfy the inequality.