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 |
√8 sin θ = 2 |
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 = 1 cos θ = 1/√2 |
√2 sin θ = 1 sin θ = 1/√2 |
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
√8 cos θ = 2 cos θ = 1/√2 |
√8 sin θ = -2 sin 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 = 1 cos θ = 1/√2 |
√2 sin θ = -1 sin θ = -1/√2 |
θ lies in the fourth quadrant.
θ = 2π - α
α = 45°
θ = 2π - π/4
θ = 7π/4
Jun 05, 23 09:20 PM
Jun 05, 23 08:05 AM
Jun 05, 23 07:42 AM