CONVERT THE GIVEN COMPLEX NUMBER FROM RECTANGULAR TO POLAR FORM

Example 1 :

Convert the given complex number from rectangular to polar form

- √2 + i√2

Solution :

- √2 + i√2  =  r(cosθ  + i sinθ) ----(1)

r = √[(-√2)² + (√2)²] = √(2 + 2) = √4 = 2

r = 2

Apply the value of r in the first equation

- √2 + i√2  =  2(cosθ  + isinθ)

- √2 + i√2  =  2cosθ + i2sinθ

Equating the real and imaginary parts separately

Since sinθ is positive and cosθ is negative the required angle θ lies in the second quadrant.

θ  =  Π - α

Here α is nothing but the angles of sin and cos for which we get the value 1/√2

θ  =  Π - (Π/4)

θ  =  (4Π-Π)/4  =  3Π/4

Modulus = 2 and argument = 3Π/4

Hence, the polar form of the given complex number :

2(cos 3Π/4  + i sin 3Π/4)

Example 2 :

Convert the given complex number from rectangular to polar form

1 + i√3

Solution :

1 + i√3 = r(cosθ + isinθ) ----(1)

r = √[(1)² + (√3)²] = √(1 + 3) = √4 = 2

r = 2

Apply the value of r in the first equation

1 + i√3  =  2(cosθ + isinθ)

1 + i√3  =  2cosθ + i2sinθ

Equating the real and imaginary parts separately

Since sinθ and cosθ are positive, the required angle θ lies in the first quadrant.

θ  =  α

Here α is nothing but the angles of sin and cos for which we get the values 1/2 and √3/2 respectively.

θ  =  Π/3

Modulus = 2 and argument = Π/3

Hence, the polar form of the given complex number :

2(cos Π/3 + i sin Π/3)

Example 3 :

Convert the given complex number from rectangular to polar form

-1 - i√3

Solution :

-1 - i√3  =  r(cosθ + isinθ) ----(1)

r  =  √[(-1)² + (-√3)²]  =  √(1 + 3)  =  √4  =  2

r  =  2

Apply the value of r in the first equation

-1 - i√3  =  2(cosθ + isinθ)

-1 - i√3  =  2cosθ + i2sinθ

Equating the real and imaginary parts separately

Since sin θ and cos θ are negative the required angle θ lies in the third quadrant.

θ  =  -Π + α

Here α is nothing but the angles of sin and cos for which we get the values √3/2 and 1/2 respectively.

θ  =  -Π + α

     =  -Π + Π/3  =  (-3Π+Π)/3  =  -2Π/3

Modulus = 2 and argument =-2Π/3

Hence, the polar form of the given complex number :

2(cos (-2Π/3)  + i sin (-2Π/3))

Example 4 :

Convert the given complex number from rectangular to polar form

√21 + i√7

Solution :

√21 + i√7 =  r(cosθ + isinθ) ----(1)

r  =  √[(√21)² + (√7)²]  =  √[21 + 7]  = √28

r = 2√7

Apply the value of r in the first equation

√21 + i√7= 2√7(cosθ + isinθ)

Equating the real and imaginary parts separately

Since sin θ and cos θ are negative the required angle θ lies in the first quadrant.

θ = α

Here α is nothing but the angles of sin and cos for which we get the values 1/2 and √3/2 respectively.

θ = α

     = Π/3

Modulus = 2√7 and argument = Π/3

Hence, the polar form of the given complex number :

2√7[cos(Π/3)  + i sin (Π/3)]

Example 5 :

Convert the given complex number from rectangular to polar form

-5√2/2 - i(5√2/2)

Solution :

-5√2/2 - i(5√2/2) = r(cosθ + isinθ) ----(1)

r  =  √[-5√2/2)² + (-5√2/2)²]  =  √(50 + 50)/4  = √25

r = 5

Apply the value of r in the first equation

-5√2/2 - i(5√2/2) = 5(cosθ + isinθ)

Equating the real and imaginary parts separately

Since sinθ is negative and cos θ is negative, the angle lies in the third quadrant.

θ  =  -Π + α

Here α is nothing but the angles of sin and cos for which we get the values √2/2 and √2/2 respectively.

θ = -Π + (Π/4)

= (-4Π + Π)/4

= -3Π/4

Modulus = 5 and argument = -3Π/4

Hence, the polar form of the given complex number :

2√7[cos (-3Π/4)  + i sin (-3Π/4)]

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.