The polar form of a complex number z = a + ib is given by:
A
z = r(cos θ + i sin θ)
B
z = a + ib
C
z = r + θi
D
z = a cos b + i sin b
Analysis & Theory
In polar form, z = r(cos θ + i sin θ), where r = |z| and θ = arg(z).
In polar form z = r(cos θ + i sin θ), r represents:
A
Modulus of z
B
Argument of z
C
Real part of z
D
Imaginary part of z
Analysis & Theory
r = |z| is the modulus (distance from origin) of the complex number.
In polar form z = r(cos θ + i sin θ), θ represents:
A
Argument of z
B
Modulus of z
C
Real part of z
D
Imaginary part of z
Analysis & Theory
θ = arg(z) is the angle made by the line joining origin to z with the positive x-axis.
If z = 1 + i, then the modulus r is:
A
√2
B
1
C
2
D
1/√2
Analysis & Theory
|z| = √(1² + 1²) = √2.
If z = 1 + i, then the argument θ is:
A
π/4
B
π/2
C
π/3
D
π/6
Analysis & Theory
θ = arctan(b/a) = arctan(1/1) = π/4.
The polar form of z = 1 + i is:
A
√2(cos π/4 + i sin π/4)
B
√2(cos π/2 + i sin π/2)
C
1(cos π/4 + i sin π/4)
D
2(cos π/4 + i sin π/4)
Analysis & Theory
z = 1 + i ⇒ r = √2, θ = π/4, so polar form: √2(cos π/4 + i sin π/4).
If z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), then z₁ * z₂ in polar form is:
A
r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]
B
r₁r₂[cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)]
C
r₁ + r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]
D
r₁r₂[cos θ₁ cos θ₂ + i sin θ₁ sin θ₂]
Analysis & Theory
Multiplying in polar form: multiply moduli and add arguments.
If z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), then z₁ / z₂ in polar form is:
A
r₁/r₂ [cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)]
B
r₁/r₂ [cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]
C
r₁r₂ [cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)]
D
r₁ + r₂ [cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)]
Analysis & Theory
Dividing in polar form: divide moduli and subtract arguments.
Euler's form of a complex number z in polar form is:
A
z = r e^(iθ)
B
z = r cos θ + i sin θ
C
z = a + ib
D
z = r(cos θ - i sin θ)
Analysis & Theory
Euler’s formula: e^(iθ) = cos θ + i sin θ, so z = r e^(iθ).
If z = 2(cos π/3 + i sin π/3), then z in a + ib form is:
A
1 + √3 i
B
2 + √3 i
C
1 - √3 i
D
√3 + i
Analysis & Theory
z = 2(cos π/3 + i sin π/3) = 2(1/2 + i√3/2) = 1 + √3 i.