DeMoivre Theorem

Grade 12 · trigonometry · 91 practice problems · read aloud

🔊 Listen to this explanation

DeMoivre's Theorem: Powers of Complex Numbers

📌 What is it and Why is it Useful?

DeMoivre's Theorem is a powerful formula used to raise complex numbers in polar form to any integer power n. It states:

[r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ))

This theorem simplifies complex calculations that would be extremely tedious using standard (a+bi) form. It's essential for finding roots of complex numbers and has applications in engineering and physics.

🛠️ Step-by-Step Guide

  1. Convert to Polar Form: Express your complex number as z = r(cos θ + i sin θ). Find r (modulus) and θ (argument).
  2. Apply the Theorem: Raise r to the power n, and multiply θ by n.
  3. Simplify: Write the result in polar form, or convert back to a+bi form if required.

🔢 Worked Examples

Example 1: Find (1 + i)⁴

  1. Convert to polar: r = √(1²+1²) = √2, θ = π/4. So 1+i = √2(cos π/4 + i sin π/4)
  2. Apply theorem: [√2(cos π/4 + i sin π/4)]⁴ = (√2)⁴ [cos(4·π/4) + i sin(4·π/4)]
  3. Simplify: 4 [cos π + i sin π] = 4[-1 + i·0] = -4

Example 2: Find (√3 - i)³

  1. Convert to polar: r = √((√3)²+(-1)²) = 2, θ = -π/6 (or 11π/6). So z = 2[cos(-π/6) + i sin(-π/6)]
  2. Apply theorem: 2³ [cos(3·-π/6) + i sin(3·-π/6)] = 8 [cos(-π/2) + i sin(-π/2)]
  3. Simplify: 8 [0 + i(-1)] = -8i

⚠️ Common Mistakes to Avoid

  • Incorrect Argument: Always find the correct θ (argument) using the complex number's quadrant.
  • Forgetting to Multiply θ: Remember: [r cis θ]ⁿ = rⁿ cis (nθ), not rⁿ cis θ.
  • Angle Simplification: After multiplying, reduce nθ to its principal value (between 0 and 2π or -π and π).

💡 Tips & Tricks

  • Use the shorthand: (r cis θ)ⁿ = rⁿ cis (nθ)
  • For roots, remember: ⁿ√[r cis θ] = ⁿ√r cis[(θ + 2πk)/n] for k=0,1,2,...,n-1
  • When converting back to a+bi form, use the unit circle for common angles.

🎯 Practice Suggestions

  • Start with complex numbers with moduli of 1 to focus on the angle multiplication.
  • Practice converting between rectangular (a+bi) and polar forms quickly.
  • Try finding both powers AND roots of complex numbers.
  • Use problems that result in "nice" answers (like -8, 8i, etc.) to check your work.

Practice problems

6 of the 91, worked through step by step — try them before opening the answer.

1 (√3 + i)^4 = ?

Hint: Convert the complex number to polar form using r = √(a² + b²) and θ = arctan(b/a), then apply De Moivre's Theorem to raise it to the given power.

Show the answer

Answer: -8 + 8√3 i

  1. Convert √3 + i to polar form r = √((√3)² + 1²) = √(3 + 1) = √4 = 2 θ = arctan(1/√3) = π/6 So √3 + i = 2(cos(π/6) + i sin(π/6))
  2. Apply De Moivre's Theorem (√3 + i)^4 = [2(cos(π/6) + i sin(π/6))]^4 = 2^4 (cos(4×π/6) + i sin(4×π/6)) = 16 (cos(2π/3) + i sin(2π/3))
  3. Evaluate trigonometric functions cos(2π/3) = -1/2 sin(2π/3) = √3/2
  4. Multiply 16 × (-1/2 + i√3/2) = -8 + 8√3 i

The answer is -8 + 8√3 i.

2 (√3 + i)^6 = ?

Hint: Convert the complex number to polar form using r = √(a² + b²) and θ = arctan(b/a), then apply De Moivre's Theorem: (r(cosθ + i sinθ))^n = r^n(cos(nθ) + i sin(nθ))

Show the answer

Answer: -64

  1. Convert √3 + i to polar form r = √((√3)² + 1²) = √(3 + 1) = √4 = 2 θ = arctan(1/√3) = π/6 So √3 + i = 2(cos(π/6) + i sin(π/6))
  2. Apply De Moivre's Theorem (2(cos(π/6) + i sin(π/6)))^6 = 2^6(cos(6 × π/6) + i sin(6 × π/6)) = 64(cos(π) + i sin(π))
  3. Evaluate the trigonometric functions cos(π) = -1 sin(π) = 0 So 64(-1 + i × 0) = 64 × -1 = -64

The answer is -64.

3 [2(cos 27° + i sin 27°)]⁵ = ?

Hint: Apply De Moivre's Theorem: raise the modulus to the power and multiply the angle by the exponent. Then simplify the angle if it exceeds 360°.

Show the answer

Answer: 32(cos 135° + i sin 135°)

  1. Identify r = 2, θ = 27°, n = 5.
  2. Apply De Moivre's Theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ).
  3. Compute rⁿ = 2⁵ = 32.
  4. Compute nθ = 5 × 27° = 135°.
  5. Write the result: 32(cos 135° + i sin 135°).

The answer is 32(cos 135° + i sin 135°).

4 [2(cos 36° + i sin 36°)]¹⁰ = ?

Hint: Apply De Moivre's Theorem: raise the modulus to the power and multiply the angle by the exponent. Then simplify the resulting angle and trigonometric values.

Show the answer

Answer: 1024(cos 360° + i sin 360°) = 1024

  1. Identify r = 2, θ = 36°, n = 10.
  2. Apply De Moivre's Theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ).
  3. Compute rⁿ = 2¹⁰ = 1024.
  4. Compute nθ = 10 × 36° = 360°.
  5. Write the result: 1024(cos 360° + i sin 360°).
  6. Simplify: cos 360° = 1, sin 360° = 0.
  7. Final answer: 1024(1 + 0i) = 1024.

The answer is 1024.

5 [4(cos(π/8) + i sin(π/8))]⁴ = ?

Hint: Apply De Moivre's Theorem: raise the modulus to the power and multiply the angle by the power. Then simplify the resulting angle to find exact trigonometric values.

Show the answer

Answer: 256(cos(π/2) + i sin(π/2)) = 256i

  1. Identify r = 4 and θ = π/8.
  2. Apply De Moivre's Theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ). Here n = 4.
  3. Compute rⁿ = 4⁴ = 256.
  4. Compute nθ = 4 × (π/8) = 4π/8 = π/2.
  5. The result in polar form is 256(cos(π/2) + i sin(π/2)).
  6. Evaluate cos(π/2) = 0 and sin(π/2) = 1.
  7. The final answer is 256(0 + i·1) = 256i.

6 (2(cos(π/4) + i sin(π/4)))^8 = ?

Hint: When raising a complex number in polar form to a power, apply the exponent to the modulus and multiply the argument by the exponent. For example, (r(cosθ + i sinθ))^n = r^n(cos(nθ) + i sin(nθ)).

Show the answer

Answer: 256

  1. Recognize the complex number form** The expression inside the parentheses is: 2(cos(π/4) + i sin(π/4)) This is in polar form: r(cos θ + i sin θ) where r = 2 and θ = π/4. --- **
  2. Apply De Moivre’s Theorem** De Moivre’s Theorem says: [r(cos θ + i sin θ)]^n = r^n (cos(nθ) + i sin(nθ)) Here, n = 8, r = 2, θ = π/4. So: (2(cos(π/4) + i sin(π/4)))^8 = 2^8 (cos(8 × π/4) + i sin(8 × π/4)) --- **
  3. Compute the power of the modulus** 2^8 = 256. So we have: 256 (cos(8 × π/4) + i sin(8 × π/4)) --- **
  4. Compute the new angle** 8 × π/4 = 8π/4 = 2π. So: cos(2π) = 1 sin(2π) = 0 --- **
  5. Substitute back** 256 (1 + i × 0) = 256 × 1 = 256. --- **Final Answer:** 256

Let's solve step-by-step. We are given: (2(cos(π/4) + i sin(π/4)))^8 --- **

Practise this topic — 10 free problems, no signup →