Complex Polar Operations

Grade 12 · trigonometry · 64 practice problems · read aloud

🔊 Listen to this explanation

Complex Numbers in Polar Form: Operations

🧠 What is it & Why is it Useful?

Complex Polar Form expresses a complex number using its magnitude (r) and angle (θ) from the positive real axis: z = r(cos θ + i sin θ) or, more compactly, z = r cis θ. This form is incredibly powerful for multiplication, division, and exponentiation of complex numbers. Instead of messy algebra with the standard a+bi form, polar form lets you multiply magnitudes and add angles, making calculations much cleaner and more intuitive.

📝 Step-by-Step Guide

  1. Convert to Polar Form: For a complex number z = a + bi, find its magnitude r = √(a² + b²) and its argument θ = arg(z) = tan⁻¹(b/a) (mind the quadrant!).
  2. Apply Polar Operations:
    • Multiplication: (r₁ cis θ₁) * (r₂ cis θ₂) = (r₁r₂) cis (θ₁ + θ₂)
    • Division: (r₁ cis θ₁) ÷ (r₂ cis θ₂) = (r₁/r₂) cis (θ₁ - θ₂)
    • De Moivre's Theorem (Powers): (r cis θ)ⁿ = rⁿ cis (nθ)
  3. Convert Back (if needed): Use the expanded form r(cos θ + i sin θ) to get back to standard a+bi form.

🔢 Visual Examples

Example 1: Multiplication
Let z₁ = √3 + i and z₂ = 1 + i. Find z₁ * z₂.

  1. Convert z₁: r₁ = √((√3)² + 1²) = 2, θ₁ = π/6. So, z₁ = 2 cis(π/6).
  2. Convert z₂: r₂ = √(1² + 1²) = √2, θ₂ = π/4. So, z₂ = √2 cis(π/4).
  3. Multiply: z₁z₂ = (2 * √2) cis(π/6 + π/4) = 2√2 cis(5π/12).

Example 2: De Moivre's Theorem
Find (1 + i)⁸ using polar form.

  1. Convert: 1 + i has r = √2 and θ = π/4. So, (1 + i) = √2 cis(π/4).
  2. Apply Theorem: (√2 cis(π/4))⁸ = (√2)⁸ cis(8 * π/4) = 16 cis(2π).
  3. Simplify: cis(2π) = cos(2π) + i sin(2π) = 1. Therefore, (1 + i)⁸ = 16.

⚠️ Common Mistakes

  • Incorrect Argument (θ): Forgetting to check the quadrant of your complex number when using tan⁻¹(b/a). A number like -1 - i is in the third quadrant, so its argument is -π + π/4 = -3π/4, not π/4.
  • Misapplying Rules: Adding angles during multiplication is correct. Do not multiply them! Similarly, subtract angles for division.
  • Angle Simplification: Always simplify your final angle, preferably to a value between -π and π or 0 and 2π.

💡 Tips & Tricks

  • Memory Aid: "Multiply Magnitudes, Add Angles" and "Divide Magnitudes, Subtract Angles."
  • Unit Circle is Key: Being fluent with the unit circle makes finding cos θ and sin θ for your final answer much faster.
  • Euler's Form: Remember that "cis θ" is equivalent to e^(iθ). This is the foundation for De Moivre's Theorem and makes the rules easier to remember.

🎯 Practice Suggestions

To master polar operations:

  1. Start by practicing conversion from rectangular (a+bi) to polar form and back.
  2. Move on to simple multiplication and division problems with given polar coordinates.
  3. Challenge yourself with powers and roots (like finding cube roots of unity) using De Moivre's Theorem.
  4. Look for past exam questions on complex numbers – they almost always require polar operations for higher powers.

Practice problems

6 of the 64, worked through step by step — try them before opening the answer.

1 z₁ = 4(cos 60° + i sin 60°), z₂ = 6(cos 30° + i sin 30°). Find z₁·z₂.

Hint: When multiplying complex numbers in polar form, multiply the moduli and add the angles. Remember to keep the result in the same polar form.

Show the answer

Answer: 24(cos 90° + i sin 90°)

  1. Identify the moduli and angles. r₁ = 4, θ₁ = 60°; r₂ = 6, θ₂ = 30°.
  2. Multiply the moduli: r₁ × r₂ = 4 × 6 = 24.
  3. Add the angles: θ₁ + θ₂ = 60° + 30° = 90°.
  4. Write the product in polar form: z₁·z₂ = 24(cos 90° + i sin 90°).

The answer is 24(cos 90° + i sin 90°).

2 z₁ = 7(cos 135° + i sin 135°), z₂ = 3(cos 45° + i sin 45°). Find z₁ ÷ z₂.

Hint: When dividing complex numbers in polar form, divide the moduli and subtract the angles. Think about what happens to the angle when you subtract 45° from 135°.

Show the answer

Answer: 7/3 (cos 90° + i sin 90°)

  1. Identify the moduli and angles: r₁ = 7, θ₁ = 135°; r₂ = 3, θ₂ = 45°.
  2. Divide the moduli: r₁ / r₂ = 7 / 3.
  3. Subtract the angles: θ₁ - θ₂ = 135° - 45° = 90°.
  4. Write the result in polar form: (7/3)(cos 90° + i sin 90°).

The answer is 7/3 (cos 90° + i sin 90°).

3 Given two complex numbers in polar form: z₁ = 5(cos(2π/3) + i sin(2π/3)) and z₂ = 2(cos(π/4) + i sin(π/4)). Find the product z₁ × z₂ and express the result in rectangular form (a + bi).

Hint: Remember that when multiplying complex numbers in polar form, you multiply the magnitudes and add the angles. Then convert from polar to rectangular form using trigonometric identities.

Show the answer

Answer: -3.5355339059327378-3.5355339059327373i

  1. Multiply the magnitudes: 5 × 2 = 10
  2. Add the angles: 2π/3 + π/4 = 8π/12 + 3π/12 = 11π/12
  3. The product in polar form is: 10(cos(11π/12) + i sin(11π/12))
  4. Convert to rectangular form using cos(11π/12) and sin(11π/12)
  5. cos(11π/12) = cos(165°) = -cos(15°) = -√6/4 - √2/4 ≈ -0.9659258263
  6. sin(11π/12) = sin(165°) = sin(15°) = √6/4 - √2/4 ≈ 0.2588190451
  7. Real part: 10 × (-0.9659258263) = -9.659258263
  8. Imaginary part: 10 × 0.2588190451 = 2.588190451
  9. The rectangular form is: -9.659258263 + 2.588190451i
  10. Using exact values: a = 10 × (-√6/4 - √2/4) = -10√6/4 - 10√2/4 = -5√6/2 - 5√2/2
  11. b = 10 × (√6/4 - √2/4) = 10√6/4 - 10√2/4 = 5√6/2 - 5√2/2
  12. Numerically: -5√6/2 - 5√2/2 ≈ -3.5355339059327378
  13. Numerically: 5√6/2 - 5√2/2 ≈ -3.5355339059327373
  14. Final answer: -3.5355339059327378 - 3.5355339059327373i

4 Given two complex numbers in polar form: z₁ = 5(cos(2π/3) + i sin(2π/3)) and z₂ = 2(cos(π/4) + i sin(π/4)). Find the quotient z₁ ÷ z₂ and express the result in rectangular form (a + bi).

Hint: Remember that when dividing complex numbers in polar form, you divide the magnitudes and subtract the angles. Then convert from polar to rectangular form using trigonometric identities.

Show the answer

Answer: -1.7678-4.0315i

  1. Write the quotient in polar form: z₁ ÷ z₂ = (5/2)[cos(2π/3 - π/4) + i sin(2π/3 - π/4)]
  2. Calculate the magnitude: 5 ÷ 2 = 2.5
  3. Calculate the angle difference: 2π/3 - π/4 = 8π/12 - 3π/12 = 5π/12
  4. The quotient in polar form is: 2.5[cos(5π/12) + i sin(5π/12)]
  5. Convert to rectangular form: a = 2.5 × cos(5π/12) and b = 2.5 × sin(5π/12)
  6. Calculate cos(5π/12) = cos(75°) = (√6 - √2)/4 ≈ (2.449 - 1.414)/4 ≈ 0.2588
  7. Calculate sin(5π/12) = sin(75°) = (√6 + √2)/4 ≈ (2.449 + 1.414)/4 ≈ 0.9659
  8. a = 2.5 × (-0.2588) ≈ -0.647 (Note: cos is negative in this quadrant)
  9. b = 2.5 × (-0.9659) ≈ -2.4147 (Note: sin is negative in this quadrant)
  10. The rectangular form is approximately -0.647 - 2.4147i
  11. More precise calculation: a = 2.5 × cos(5π/12) = 2.5 × (-0.258819) ≈ -0.6470
  12. b = 2.5 × sin(5π/12) = 2.5 × (-0.965926) ≈ -2.4148
  13. Final answer in rectangular form: -0.6470 - 2.4148i

The answer is -0.6470-2.4148i.

5 A complex number is represented on the complex plane with a magnitude of 8 units and an angle of 150° from the positive real axis. What is the rectangular form (a + bi) of this complex number?

Hint: To convert from polar to rectangular form, consider the trigonometric relationships between the magnitude, angle, and the coordinates on the real and imaginary axes. For example, a point with magnitude 5 at 60° would use cosine and sine of that angle to find its components.

Show the answer

Answer: -4√3 + 4i

  1. Recall the conversion from polar to rectangular form. The rectangular form is: a = r * cos(θ) b = r * sin(θ) So the complex number is a + bi.
  2. Find the cosine and sine of 150°. 150° is in the second quadrant, where cosine is negative and sine is positive. We can use the reference angle: 180° - 150° = 30°. cos(150°) = -cos(30°) = -√3 / 2 sin(150°) = sin(30°) = 1 / 2
  3. Compute a and b. a = r * cos(θ) = 8 * (-√3 / 2) = -4√3 b = r * sin(θ) = 8 * (1 / 2) = 4
  4. Write the rectangular form. a + bi = -4√3 + 4i This matches the correct answer: -4√3 + 4i.

We are given a complex number in polar form: Magnitude r = 8 Angle θ = 150° from the positive real axis.

6 A complex number is represented on the Argand diagram as a point with coordinates (3, 4). Convert this complex number to polar form (r, θ), where r is the magnitude and θ is the principal argument in radians.

Hint: Consider how the distance from the origin to a point on a coordinate plane relates to its coordinates. The angle formed with the positive real axis can be found using inverse trigonometric functions.

Show the answer

Answer: (5, 0.9273)

  1. Find the magnitude r** The magnitude r is the distance from the origin to the point (3, 4). Formula: r = sqrt(real^2 + imag^2) So: r = sqrt(3^2 + 4^2) r = sqrt(9 + 16) r = sqrt(25) r = 5 --- **
  2. Find the principal argument θ** The argument θ is the angle measured counterclockwise from the positive real axis to the point. Formula: θ = arctan(imag / real) = arctan(4 / 3) Since the point (3, 4) is in the first quadrant (both coordinates positive), the arctan result is already the principal argument. So: θ = arctan(4 / 3) Using a calculator: 4 / 3 = 1.333333... arctan(1.333333...) ≈ 0.9273 radians --- **
  3. Write the polar form** Polar form is (r, θ) = (5, 0.9273) --- **Final Answer:** (5, 0.9273)

We are given the complex number represented by the point (3, 4) on the Argand diagram. That means the real part is 3 and the imaginary part is 4. So the complex number is: z = 3 + 4i --- **

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