Question

How do you convert complex numbers to polar form?

Answer 1

Step 1. Conversion of complex number to polar form :

A complex number is a number of the form $x+iy$ , where $x$ and $y$ are real numbers and $i$ is the imaginary number.

$z=x+yi$ (complex number ) $\dots ..\left(1\right)$

is called the rectangular coordinate form of a complex number.

Shown in graph , where

$x=$ The real part of $z$

$y=$the imaginary part of $z$

$r=$ length of the vector

$\theta =$angle made with the real axis.

Step 2. Compute $r$ value by Pythagoras theorem :

From, Pythagoras' theorem $r^2=x^2+y^2$

According to the graph

$\cos \theta =\left(\frac{x}{r}\right)$

$\Rightarrow$ $x=r\cos \theta \dots ..\left(2\right)$

$\sin \theta =\left(\frac{y}{r}\right)$

$\Rightarrow$ $y=r\sin \theta \dots ...\left(3\right)$

Step 3. Putting these values in equation 1 :

we have $z=x+yi$

$\Rightarrow$ $z=r\cos \theta +\left(r\sin \theta \right)i$

$\Rightarrow$ $z=r(\cos \theta +i\sin \theta )$

Hence, $\mathit{z}\mathbf{=}\mathit{r}\mathbf{}\mathbf{(}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathbf{}\mathbf{+}\mathit{i}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathbf{)}$ is the standard form of polar form of complex number .