[Math] Complex number – how to find the angle between the imaginary axis and real axis

Question

Assume I have complex number $z = a + ib$.

$z$ can be represented by a polar representation as $r(\cos \theta+i\sin \theta)$,

when $r$ is the absolute value of $z$, $\sqrt{a^2 + b^2}$.

But how can I find $\theta$?

Answer 1

Consider the following Argand-diagram

The y-axis is the imaginary axis and the x-axis is the real one. The complex number in question is

$$x + yi$$

To figure out $\theta$, consider the right-triangle formed by the two-coordinates on the plane (illustrated in red). Let $\theta$ be the angle formed with the real axis.

$$\tan\theta = \frac{y}{x}$$

$$\implies \boxed{\tan^{-1}\left(\frac{y}{x}\right)}$$

The hypotenuse of the triangle will be

$$\sqrt{x^2 + y^2}$$

Therefore,

$$\sin\theta = \frac{y}{\sqrt{x^2 + y^2}}$$