# CONVERT COMPLEX NUMBERS FROM POLAR TO RECTANGULAR FORM

## About "Convert complex numbers from polar to rectangular form"

Convert complex numbers from polar to rectangular form :

A complex number in the form of r(cos θ + i sin θ) can be written in the rectangular form z = x + i y using the following formulas.

Here x = r cos θ, y = r sinθ and r √(x²+ y²)

Let us see some example problems to understand how to convert complex numbers from polar to rectangular form.

Example 1 :

Convert the given polar form to rectangular form

2 (cos 3Π/4  + i sin 3Π/4)

Solution :

By comparing the given polar form to the general equation of polar form r(cos θ + i sin θ), we get r = 2 and θ = 3Π/4.

Rectangular form of a complex number is x + iy

x = r cos θ and y = r sin θ

Finding real part :

x = 2 cos 3Π/4

3Π/4 = 135°

x = 2 cos (90°+45°)

Since 135 lies in second quadratic, we have to put positive sign only for sin θ and its reciprocal cosec θ only. Here we have to put negative sign

x = -2 sin 45° ==> -2 x 1/√2 ==> -√2

Hence the real part of the complex number is -√2. Now we have to find the imaginary part.

Finding imaginary part :

y = 2 sin 3Π/4

y = 2 cos (90°+45°)

y = 2 sin 45° ==> 2 x 1/√2 ==> √2

So the rectangular form of the complex number is

-√2 + i√2

Example 2 :

Convert the given polar form to rectangular form

2 (cos Π/3  + i sin Π/3)

Solution :

Rectangular form of a complex number is x + iy

x + iy =  2 cos Π/3  + i 2 sin Π/3

x = r cos θ and y = r sin θ

r = 2 and  θ = Π/3

Finding real part :

x = 2 cos Π/3

Π/3 = 60°

Since 60° lies in the first quadrant, we have to put positive sign for all trigonometric ratios.

x = 2 cos 60° ==> 2 x 1/2 ==> 1

Hence the real part of the complex number is 1. Now we have to find the imaginary part.

Finding imaginary part :

y = 2 sin Π/3

y = 2 sin 60° ==> 2 x (√3/2) ==> √3

So the rectangular form of the complex number is

1 + i√3

Example 2 :

Convert the given polar form to rectangular form

2 (cos (-2Π/3)  + i sin (-2Π/3))

Solution :

Rectangular form of a complex number is x + iy

x + iy =  2 cos (-2Π/3)  + i 2 sin (-2Π/3)

x = r cos θ and y = r sin θ

r = 2 and  θ = -2Π/3

Finding real part :

x = 2 cos (-2Π/3)

2Π/3 = 120°

2 cos (-2Π/3) = 2 cos (-120°) ==> 2 cos 120°

2 cos (90 + 30) ==> 2 sin 30° ==> 2 x (1/2) ==> 1

Hence the real part of the complex number is 1. Now we have to find the imaginary part.

Finding imaginary part :

y = 2 sin (-2Π/3)

2 sin (-2Π/3) = 2 sin (-120°) ==> -2 sin 120°

-2 sin (90 + 30) ==> -2 cos 30° ==> -2 x (√3/2) ==> -√3

So the rectangular form of the complex number is

1 - i√3

After having gone through the stuff given above, we hope that the students would have understood "Convert complex numbers from polar to rectangular form".

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