# 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

 √8 cos θ = 2 cos 𝜃=28cos 𝜃=222cos 𝜃=12 √8 sin θ = 2 sin 𝜃=28sin 𝜃=222sin 𝜃=12

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

 √2 cos θ = 1cos θ = 1/√2 √2 sin θ = 1sin θ = 1/√2

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

 √8 cos θ = 2cos θ = 1/√2 √8 sin θ = -2sin θ = -1/√2

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

 √2 cos θ = 1cos θ = 1/√2 √2 sin θ = -1sin θ = -1/√2

θ lies in the fourth quadrant.

θ = 2π - α

α = 45°

θ = 2π - π/4

θ = 7π/4

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