Finding polar coordinates angle for complex numbers given cartesian form

Question

I have the following formula for finding $\theta$ given cartesian form of complex numbers.

$$\theta = \begin{cases} \tan^{-1}(\frac{y}{x}) & x \leq 0 \\ \tan^{-1}(\frac{y}{x}) & x \geq 0 \\ \pm\frac{\pi}{2} & x = 0 \end{cases} $$

My confusion is when $x=0$. How do I tell if my angle is $\frac{\pi}{2}$ or $-\frac{\pi}{2}$

For example, let's say my complex number is $z = -i$

In that case $x=0$, so is the angle $\frac{\pi}{2}$ or $-\frac{\pi}{2}$?

Answer 1

You need to try to understand the reason why that formula is given in the first place. The angle $\theta$ is the angle the complex number in the plane makes with the positive real axis (positive $x$-axis). Counting the angle in the positive direction is done counter clockwise. So if $z = i$, then it is clearly $\displaystyle \frac{\pi}{2}$ away from the positive $x$-axis purely from looking at the diagram.

In sum, if $x = 0$ then we have that $z = iy$ for some real number $y$. If $y > 0$ then $\displaystyle \theta = \frac{\pi}{2}$ and if $y < 0$ we have $\displaystyle \theta = -\frac{\pi}{2}$, try to convince yourself of this!