DIVISION OF COMPLEX NUMBERS IN POLAR FORM

Find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form. Express the argument as an angle between 0° and 360°.

Problem 1 :

z₁ = 20(cos 75° + i sin 75°)

z₂ = 4(cos 25° + i sin 25°)

Solution:

By using the z₁/z₂ formula,

Problem 2 :

z₁ = 50(cos 80° + i sin 80°)

z₂ = 10(cos 20° + i sin 20°)

Solution:

By using the z₁/z₂ formula,

Problem 3 :

Solution:

By using the z₁/z₂ formula,

Problem 4 :

Solution:

By using the z1/z2 formula,

Problem 5 :

z₁ = cos 80° + i sin 80°

z₂ = cos 200° + i sin 200°

Solution:

By using the z1/z2 formula,

Problem 6 :

z₁ = cos 70° + i sin 70°

z₂ = cos 230° + i sin 230°

Solution:

By using the z1/z2 formula,

Problem 7 :

z₁ = 2 + 2i

z₂ = 1 + i

Solution:

z₁ = 2 + 2i

x + iy = r(cos θ + i sin θ)

r = √(2² + 2²)

r = √8

2 + 2i = √8 cos θ  + √8 i sin θ 

The angle lies in the first quadrant.

θ = 45° (or) π/4

z₂ = 1 + i

r = √(1² + 1²)

r = √2

1 + i = √2 cos θ  + √2 i sin θ 

θ lies in first quadrant.

θ = 45° (or) π/4

Problem 8 :

z₁ = 2 - 2i

z₂ = 1 - i

Solution:

z₁ = 2 - 2i

x + iy = r(cos θ + i sin θ)

r = √(2² + (-2)²)

r = √8

2 - 2i = √8 cos θ  + √8 i sin θ 

The angle lies in the fourth quadrant.

θ = 2π - α

α = 45°

θ = 2π - π/4

θ = 7π/4

z₂ = 1 - i

r = √1² + (-1)²)

r = √2

1 - i = √2 cos θ  + √2 i sin θ 

θ lies in the fourth quadrant.

θ = 2π - α

α = 45°

θ = 2π - π/4

θ = 7π/4