Polar Form of Complex Numbers

Introduction

In addition to the rectangular form, the polar form of complex numbers is an additional way to represent complex numbers. The usual way to show complex numbers is in z = x+iy. The ‘i’ in the given equation is the imaginary number. In the polar form, the complex numbers are defined as the aggregate of modulus and argument. 

The modulus or mod of a complex number is also known as the absolute value. This polar form is shown using the polar coordinates of imaginary and real numbers in the coordinate system.

The horizontal axis is also known as the real axis, while the vertical axis means the imaginary axis. r and θ are the representation of real and complex components, where

r equals the vector length and

θ stands for the angle which the vector makes with the real axis.

With the help of the Pythagorean Theorem, we can also write:

r^2 = a^2 + b^2     (where complex number was a + ib)

With the knowledge of trigonometric ratios, we are aware that;

Cos θ = Adjoining side of the angle θ/Hypotenuse

Cos θ = a/r

Also, sin θ = Opposite side of the angle θ/Hypotenuse

Sin θ = b/r

Each side is being multiplied by r :

rcosθ = a and rsinθ = b

A complex number’s rectangular form is represented by:

z = a+ib

Putting in the values of a and b:

z = a+ib

z  =  r (cosθ + i sinθ)

For a complex number, r stands for the absolute value, also referred to as the modulus, and the angle θ means the argument of the complex number.

The Polar Form of Complex Numbers

The polar form of complex number z = a + ib has the equation:

z = r (cosθ+isinθ)

,where

r = |z| = √(a^2+b^2)

a = r cosθ

b = r sinθ

θ = tan^-1(b/a) for a,b>0

θ = tan^-1(b/a) + π for  a<0, b>0

θ = tan^-1(b/a) – π for a<0,b<0

θ = tan^-1(b/a) for a>0,b<0

Here we take the value of tan^-1(b/a) from  – 2 to 2

Rectangular form to Polar Form of Complex Numbers Calculator

With the help of an example, let us help you convert rectangular form to polar form of Complex Numbers.

Example: Find the polar form of the complex number 5+2i.

 5+2i is the rectangular form of a complex number.

To convert into polar form modulus and argument, i.e., r and θ.

The modulus or absolute value of the complex number is given by:

r = |z| = √a^2+b^2

r = √(5^2+2^2)

r = √25+4

r = √29

r ≈ 5.39

To find the argument of a complex number, we need to check the condition first, such as:

Here a>0, hence, we make use of the formula,

θ = tan^-1(b/a) = θ = tan^-1(2/5) = 0.38°

Hence, we can conclude that the polar form of 5+2i is:

5+2i = 5.39(cos(0.38)+isin(0.38)) 

Adding Polar Form of Complex Numbers example

Take the example of 2 complex numbers, one in a polar form and another in the rectangular form. You need to combine, i.e., add the above two numbers and eventually show them in the opposite form again. Here is an example of adding complex numbers in polar form

Let 3+4i and 7(cos300 + i sin300 )be the two complex numbers.

The first step is to change 7(cos300 + i sin300 ) into a rectangular form.

7(cos300 + i sin300 ) = a+ib

Hence,

a= 7 cos 30° = 6.06

b = 7 sin 30° =3.5

So,

7(cos300 + i sin300 ) = 6.06 + 3.5i

Therefore, on adding the two complex numbers given above in the question, we get:

(3+4i)+ (6.06 + 3.5i) = 9.06 + 7.5i

Moving forward to change the found complex number in a polar form of complex numbers, derive the modulus and argument of the above resultant. Hence,

Modulus equals to;

r = |z| = √(a^2+b^2)

r = √(7.5^2+9.06^2)

r =11.76

Also, the argument equals to;

θ = tan^-1(b/a)

θ = tan^-1(7.5/9.06)

θ = 39.8°

Therefore, we have the solution where the resultant complex number is 11.76(cos39.80 + 

i sin39.80) .

Polar Form of Complex Numbers examples

Example 1: 

The distance of point B from the origin is eight units, and the angle made with the positive x-axis is π/6. Next, find the polar coordinates of point B using the formula for the opposite form of complex numbers.

Solution:

 Distance of point B from the origin, r = 8 units

Angle made with the positive x-axis, θ = π/6

The polar coordinates of complex number at point B are (8, π/6)

Answer: Polar coordinates of complex number at point B are (8, π/6)

Example 2:

Calculate the modulus and argument of z = 2 + 4i using the formula for the polar form of complex numbers.

Solution:

The modulus of z = 2 + 4i is |z| = √(2^2 + (4)^2) = √(4+16) = √20

Now, since both  real and imaginary parts are positive, z lies in the fourth first.

The angle θ is given by

θ = tan^-1(4 / 2) =  tan^-1 (2) = 63.43°

The significance of the minus sign is in the direction in which the angle needs to be measured.

Answer: The modulus and argument of z = 2 + 4i are √20 and 63.43°, respectively.

Conclusion

Here we have learned how to find the modulus and argument of a complex number, and also how to work with complex numbers in polar form. Solved some examples including normal arithmetic operations on complex numbers. This shall help in learning to work on complex numbers in euler form as well.