Complex Numbers

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Answer Key & Explanations

Complex Numbers · Grade 12 · Worksheet 2

1. Convert the complex number -4 + 4i to polar form r(cosθ + isinθ) = ? Answer: 4√2(cos(3π/4) + isin(3π/4)) Solution: Calculate the modulus r = √(a² + b²) = √((-4)² + 4²) = √(16 + 16) = √32 = 4√2 Determine the angle θ. Since a = -4 and b = 4, the point is in Quadrant II.
   Full step-by-step solution

   Step 1: Calculate the modulus r = √(a² + b²) = √((-4)² + 4²) = √(16 + 16) = √32 = 4√2 Step 2: Determine the angle θ. Since a = -4 and b = 4, the point is in Quadrant II. Step 3: Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(4/4) = tan⁻¹(1) = π/4 Step 4: In Quadrant II, θ = π - π/4 = 3π/4 Step 5: Write in polar form: r(cosθ + isinθ) = 4√2(cos(3π/4) + isin(3π/4)) The answer is 4√2(cos(3π/4) + isin(3π/4)).

2. Tane is designing a suspension bridge and models the force on a support cable using the complex number 9 + 12i kilonewtons. To analyze the magnitude and direction of this force, he must convert it to polar form r(cos θ + i sin θ) with θ in degrees between 0° and 360°. What is the polar form of this force? Answer: 15(cos 53.1° + i sin 53.1°) Solution: Identify the real part a = 9 and imaginary part b = 12. Calculate the magnitude r = sqrt(9² + 12²) = sqrt(81 + 144) = sqrt(225) = 15. Find the reference angle using tan θ = b/a = 12/9 = 4/3.
   Full step-by-step solution

   Step 1: Identify the real part a = 9 and imaginary part b = 12. Step 2: Calculate the magnitude r = sqrt(9² + 12²) = sqrt(81 + 144) = sqrt(225) = 15. Step 3: Find the reference angle using tan θ = b/a = 12/9 = 4/3. So θ_ref = arctan(4/3) ≈ 53.1°. Step 4: Since both a = 9 > 0 and b = 12 > 0, the point (9, 12) lies in the first quadrant, so θ = θ_ref = 53.1°. Step 5: Write in polar form: 15(cos 53.1° + i sin 53.1°).

3. Charlotte is a robotics engineer designing a control system for an industrial arm. The position of the arm's endpoint is represented by the complex number -8 + 8√3 i centimeters relative to the base. To program the arm's movement, she needs to express this position in polar form r(cos θ + i sin θ), where θ is measured in degrees between 0° and 360°. What is the polar representation of the arm's position? Answer: 16(cos 120° + i sin 120°) Solution: Identify the real part a = -8 and the imaginary part b = 8√3. Calculate the magnitude r = sqrt(a^2 + b^2) = sqrt((-8)^2 + (8√3)^2) = sqrt(64 + 64*3) = sqrt(64 + 192) = sqrt(256) = 16.
   Full step-by-step solution

   Step 1: Identify the real part a = -8 and the imaginary part b = 8√3. Step 2: Calculate the magnitude r = sqrt(a^2 + b^2) = sqrt((-8)^2 + (8√3)^2) = sqrt(64 + 64*3) = sqrt(64 + 192) = sqrt(256) = 16. Step 3: Find the reference angle using the arctangent of the absolute values: tan(θ_ref) = |b|/|a| = (8√3)/8 = √3. So θ_ref = arctan(√3) = 60°. Step 4: Since a is negative and b is positive, the point lies in the second quadrant. The angle measured from the positive real axis is θ = 180° - 60° = 120°. Step 5: Write the polar form: r(cos θ + i sin θ) = 16(cos 120° + i sin 120°). The answer is 16(cos 120° + i sin 120°).

4. Mason is an audio engineer analyzing a sound wave captured by a microphone. The wave is represented by the complex number -8 - 8√3 i in the complex plane, where the real axis represents the in-phase component and the imaginary axis represents the quadrature component. To determine the amplitude and phase shift of the wave, Mason needs to convert this complex number to polar form r(cos θ + i sin θ), where r is the amplitude and θ is the phase angle in degrees between -180° and 180°. What is the polar representation of the sound wave? Answer: 16(cos(-120°) + i sin(-120°)) or 16∠-120° Solution: Identify the real part a = -8 and imaginary part b = -8√3. Find the magnitude r = sqrt(a^2 + b^2) = sqrt((-8)^2 + (-8√3)^2) = sqrt(64 + 64*3) = sqrt(64 + 192) = sqrt(256) = 16.
   Full step-by-step solution

   Step 1: Identify the real part a = -8 and imaginary part b = -8√3. Step 2: Find the magnitude r = sqrt(a^2 + b^2) = sqrt((-8)^2 + (-8√3)^2) = sqrt(64 + 64*3) = sqrt(64 + 192) = sqrt(256) = 16. Step 3: Find the reference angle φ: tan φ = |b/a| = |(-8√3)/(-8)| = √3, so φ = 60°. Step 4: Determine the quadrant: a < 0 and b < 0, so the point is in the third quadrant. The principal argument θ (between -180° and 180°) is θ = -180° + φ = -180° + 60° = -120°. Step 5: Write the polar form: 16(cos(-120°) + i sin(-120°)). Final answer: 16(cos(-120°) + i sin(-120°)) or 16∠-120°.

5. Liam is designing a drone navigation system that uses complex numbers to represent positions. His drone starts at the origin and flies to a point represented by the complex number -5 + 12i. To program the return flight path, he needs to convert this position to polar form (r∠θ), where r is the distance from the origin and θ is the angle measured in degrees from the positive real axis, with -180° < θ ≤ 180°. What is the polar form of the drone's position? Answer: 13∠112.6° Solution: The polar form of a complex number represents it as a magnitude and an angle.
   Full step-by-step solution

   The polar form of a complex number represents it as a magnitude and an angle. The magnitude is found using the distance formula, and the angle is determined using inverse trigonometric functions, adjusted for the correct quadrant based on the signs of the real and imaginary parts. This form is useful in applications like navigation and electrical engineering for analyzing magnitude and direction.

6. Convert the complex number 1 - √3i to polar form r(cosθ + isinθ) = ? Answer: 2(cos(5π/3) + isin(5π/3)) Solution: Identify the real and imaginary parts: a = 1, b = -√3 Calculate the modulus r = √(a² + b²) = √(1² + (-√3)²) = √(1 + 3) = √4 = 2 Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(√3/1) = π/3 Determine the actual angle θ: Since the point (1, -√3) is in Quadrant IV, θ = 2π - π/3 = 5π/3 Write in…
   Full step-by-step solution

   Step 1: Identify the real and imaginary parts: a = 1, b = -√3 Step 2: Calculate the modulus r = √(a² + b²) = √(1² + (-√3)²) = √(1 + 3) = √4 = 2 Step 3: Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(√3/1) = π/3 Step 4: Determine the actual angle θ: Since the point (1, -√3) is in Quadrant IV, θ = 2π - π/3 = 5π/3 Step 5: Write in polar form: r(cosθ + isinθ) = 2(cos(5π/3) + isin(5π/3)) The answer is 2(cos(5π/3) + isin(5π/3)).

7. Convert the complex number -6 + i to polar form r(cos θ + i sin θ) = ? Answer: √37(cos(2.9845130209103035) + i sin(2.9845130209103035)) Solution: Identify the real and imaginary parts: a = -6, b = 1 Calculate the modulus r = √(a² + b²) = √((-6)² + 1²) = √(36 + 1) = √37 Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(1/6) ≈ 0.165148677 Determine the actual angle θ: Since the point (-6, 1) is in Quadrant II, θ = π - 0.165148677 ≈…
   Full step-by-step solution

   Step 1: Identify the real and imaginary parts: a = -6, b = 1 Step 2: Calculate the modulus r = √(a² + b²) = √((-6)² + 1²) = √(36 + 1) = √37 Step 3: Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(1/6) ≈ 0.165148677 Step 4: Determine the actual angle θ: Since the point (-6, 1) is in Quadrant II, θ = π - 0.165148677 ≈ 2.976443977 Step 5: Write in polar form: r(cosθ + isinθ) = √37(cos(2.976443977) + isin(2.976443977)) The answer is √37(cos(2.976443977) + isin(2.976443977)).