Polar Form of Complex Numbers

[MathDefinition title="Polar Form"] The polar form of a complex number z = a + ib is z = r(cos θ + i sin θ), where r = |z| = √(a² + b²) and θ = arg(z) = tan⁻¹(b/a). [/MathDefinition]

[MathDefinition title="Exponential Form"] Using Euler's formula e^(iθ) = cos θ + i sin θ, the exponential form of a complex number is z = re^(iθ). [/MathDefinition]

[MathKeyFormula title="Key Formulas for Polar Form"]

1. r = |z| = √(a² + b²)
2. θ = arg(z) = tan⁻¹(b/a) (adjusted for quadrant)
3. z = r(cos θ + i sin θ) = r cis θ
4. z = re^(iθ) (exponential form)
5. For z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂): z₁ × z₂ = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)] z₁/z₂ = (r₁/r₂)[cos(θ₁ − θ₂) + i sin(θ₁ − θ₂)]
6. zⁿ = rⁿ(cos nθ + i sin nθ) [De Moivre's Theorem] [/MathKeyFormula]

[MathNote title="Converting to Polar Form"]

1. Find r = √(a² + b²)
2. Find θ using tan θ = b/a
3. Determine the correct quadrant based on signs of a and b
4. Write z = r(cos θ + i sin θ) [/MathNote]

[MathExample title="Example 1: Converting to Polar Form"] Problem: Convert z = 1 + i to polar form.

Solution: Given z = 1 + i

r = √(1² + 1²) = √2

tan θ = 1/1 = 1 θ = π/4 (since both a and b are positive, first quadrant)

Polar form: z = √2(cos(π/4) + i sin(π/4)) = √2 cis(π/4) = √2 e^(iπ/4) [/MathExample]

[MathExample title="Example 2: Multiplication in Polar Form"] Problem: Find the product of z₁ = 2(cos 30° + i sin 30°) and z₂ = 3(cos 45° + i sin 45°).

Solution: Using the formula: z₁ × z₂ = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]

r₁ = 2, θ₁ = 30° r₂ = 3, θ₂ = 45°

z₁ × z₂ = 2 × 3[cos(30° + 45°) + i sin(30° + 45°)] = 6[cos 75° + i sin 75°]

cos 75° = cos(45° + 30°) = (√6 − √2)/4 sin 75° = sin(45° + 30°) = (√6 + √2)/4

Therefore, z₁ × z₂ = 6[(√6 − √2)/4 + i(√6 + √2)/4] = (3/2)(√6 − √2) + i(3/2)(√6 + √2) [/MathExample]

[MathExample title="Example 3: Division in Polar Form"] Problem: Find (4 cis 60°)/(2 cis 30°).

Solution: Using the formula: z₁/z₂ = (r₁/r₂)[cos(θ₁ − θ₂) + i sin(θ₁ − θ₂)]

r₁/r₂ = 4/2 = 2 θ₁ − θ₂ = 60° − 30° = 30°

Therefore, (4 cis 60°)/(2 cis 30°) = 2 cis 30° = 2(cos 30° + i sin 30°) = 2(√3/2 + i/2) = √3 + i [/MathExample]

[MathExample title="Example 4: Finding Power using Polar Form"] Problem: Find (1 + i)⁸ using polar form.

Solution: First, convert 1 + i to polar form: r = √(1² + 1²) = √2 θ = π/4

So, 1 + i = √2 cis(π/4)

Using De Moivre's Theorem: zⁿ = rⁿ cis(nθ)

(1 + i)⁸ = (√2)⁸ cis(8 × π/4) = 2⁴ cis(2π) = 16(cos 2π + i sin 2π) = 16(1 + 0i) = 16

Therefore, (1 + i)⁸ = 16 [/MathExample]