In mathematics, complex numbers can be presented in two primary forms rectangular form ( also known as cartesian form). The polar form of a complex number expresses it in terms of its magnitude ( distance from the origin) and argument ( angle with positive real axis).

What is the polar form of a complex number? The polar form of a complex number is a way of representing a complex number in terms of its magnitude (or modulus) and its angle (or argument) with respect to the positive real axis.

Polar Form of Complex Numbers

Let 𝑧 = 𝑥 + 𝑦𝑖 = (𝑥, 𝑦) be a complex number in Cartesian (rectangular) form. (𝑥, 𝑦) are called Cartesian coordinates.
(𝑟, 𝜃) are called Polar coordinates. Now from figure: 
`\frac{x}{r}` = 𝑐𝑜𝑠𝜃 and `\frac{y}{r}` = 𝑠𝑖𝑛𝜃
Polar form of complex numbers
Where 𝑟 = |𝑧| is called modulus of ‘z’ and 𝜃 = arg 𝑧 is called argument of  'z'.
As the solutions of trigonometric equations are not unique, so 𝜃 ± 2𝜋, 𝜃 ± 4𝜋, 𝜃 ± 6𝜋, ⋯ are also arguments of ‘z’.
Note: All angles are measured in radians and positive in the counter-clockwise sense.


Principle Argument: 

The argument of a complex number ‘z’ which lies in the interval −𝜋 < 𝜃 ≤ 𝜋, is called the principle argument. It is denoted by 𝐴𝑟𝑔 𝑧 (Capital A).

Since the argument of a complex is not uniquely determined (because of the periodicity of the arctan function), we choose the principal argument as the argument with the smallest absolute value that lies in the interval (`-\pi, \pi`].
Note: arg 𝑧 = 𝐴𝑟𝑔 𝑧 ± 2𝑛𝜋 ( 𝑛 = ± 1, ±2, ±3, ⋯ ) 
argument in all quadrants
Now,

Examples of polar form of Complex Numbers

Example-1: 

Write 𝑧 = 1 + 𝑖 polar form. 
Solution: 

Here                                 𝑥 = 1, 𝑦 = 1 

                     𝑟 = `\sqrt{x^2 + y^2} = \sqrt{1+1} =  \sqrt{2}`

                    `\theta = tan^(-1) \frac{y}{x} = tan^(-1) (1) = \frac{\pi}{4}`

The required polar form is given by: 

                                   𝑧 = 𝑟(𝑐𝑜𝑠𝜃 + 𝑖 𝑠𝑖𝑛𝜃)

                                   z = `\sqrt{2}(\cos(\frac{\pi}{4}) + i \sin(\frac{\pi}{4}))`

Example-2: 

Write 𝑧 = 1 − 𝑖 polar form. 
Solution: 

Here                         𝑥 = 1, 𝑦 = −1 

                    𝑟 = `\sqrt{x^2 + y^2} = \sqrt{1+1} =  \sqrt{2}`

                  `\theta = - tan^(-1) \frac{y}{x} = - tan^(-1) (1) = - \frac{\pi}{4}`

The required polar form is given by: 

                                   𝑧 = 𝑟(𝑐𝑜𝑠𝜃 + 𝑖 𝑠𝑖𝑛𝜃)
                                   
                                   z = `\sqrt{2}(\cos(\frac{\pi}{4}) - i \sin(\frac{\pi}{4}))`

Example-3:

Determine the principle arguments of following complex numbers and show it graphically. 

                       1. `z_1` = −1 + 0.1𝑖                           2. `z_2` = −1 − 0.1𝑖 

Solution:

1. `z_1` = −1 + 0.1𝑖  

Here                              𝑥 = −1, 𝑦 = 0.1 

                 `\theta_1 = \pi - tan^(-1) \frac{y}{x} = \pi - tan^(-1) (\frac{0.1}{1}) = \pi - 0.099`

                 `Arg z_1 =  \pi - 0.099`

2. `z_2` = −1 − 0.1𝑖

Here                        𝑥 = −1, 𝑦 = −0.1 

`\theta_2 = -\pi + tan^(-1) \frac{y}{x} = -\pi + tan^(-1) (\frac{-0.1}{-1}) = -\pi + 0.099`

                 `Arg z_2 =  -\pi + 0.099`

Example-3:

Write z = `10(\cos(\frac{\pi}{5}) + i \sin(\frac{\pi}{5}))` in Cartesian form.
Solution:

Here                                 𝑟 = 10,               `\theta = \frac{\pi}{5}`

                                        𝑥 = 𝑟𝑐𝑜𝑠𝜃,             𝑦 = 𝑟𝑠𝑖𝑛𝜃

                                x = 10 \cos(\frac{\pi}{5})        y = 10 \sin(\frac{\pi}{5})

                                       𝑥 = 8.09                   𝑦 = 5.87 

The required Cartesian form is given by: 
                                                            
                                                            𝑧 = 𝑥 + 𝑦𝑖 

                                                        𝑧 = 8.09 + 5.87𝑖