Complex Numbers

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Answer Key & Explanations

Complex Numbers · Grade 12 · Worksheet 3

1. Hana is a sound engineer calibrating a phase shifter for a concert's audio system. The device processes a signal represented by the complex number -7√3 + 7i. To analyze the phase shift and amplitude, she must convert this signal into polar form r(cos θ + i sin θ), where θ is in radians between 0 and 2π. What is the polar representation? Answer: 14(cos(5π/6) + i sin(5π/6)) Solution: Identify a = -7√3 and b = 7. Calculate the magnitude r = sqrt((-7√3)^2 + 7^2) = sqrt(147 + 49) = sqrt(196) = 14. Find the reference angle using tan α = |b/a| = 7/(7√3) = 1/√3 = √3/3, so α = π/6.
   Full step-by-step solution

   Step 1: Identify a = -7√3 and b = 7. Step 2: Calculate the magnitude r = sqrt((-7√3)^2 + 7^2) = sqrt(147 + 49) = sqrt(196) = 14. Step 3: Find the reference angle using tan α = |b/a| = 7/(7√3) = 1/√3 = √3/3, so α = π/6. Step 4: Since a < 0 and b > 0, the point is in Quadrant II. The angle θ = π - α = π - π/6 = 5π/6. Step 5: Write polar form: 14(cos(5π/6) + i sin(5π/6)). The answer is 14(cos(5π/6) + i sin(5π/6)).

2. Convert the complex number 5 - 5√3i to polar form r(cosθ + isinθ) = ? Answer: 10(cos(5π/3) + isin(5π/3)) Solution: Identify the real and imaginary parts: a = 5, b = -5√3 Calculate the modulus r = √(a² + b²) = √(5² + (-5√3)²) = √(25 + 75) = √100 = 10 Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(5√3/5) = tan⁻¹(√3) = π/3 Determine the actual angle θ: Since the point (5, -5√3) is in Quadrant IV, θ = 2π…
   Full step-by-step solution

   Step 1: Identify the real and imaginary parts: a = 5, b = -5√3 Step 2: Calculate the modulus r = √(a² + b²) = √(5² + (-5√3)²) = √(25 + 75) = √100 = 10 Step 3: Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(5√3/5) = tan⁻¹(√3) = π/3 Step 4: Determine the actual angle θ: Since the point (5, -5√3) is in Quadrant IV, θ = 2π - π/3 = 5π/3 Step 5: Write in polar form: r(cosθ + isinθ) = 10(cos(5π/3) + isin(5π/3)) The answer is 10(cos(5π/3) + isin(5π/3)).

3. A complex number is represented on the complex plane as a vector from the origin to the point (-5, 5). Express this complex number in polar form r(cosθ + i sinθ), where r > 0 and θ is the principal argument in radians between 0 and 2π. Answer: 5√2(cos(3π/4) + i sin(3π/4)) Solution: Find the modulus r using the formula r = √(a² + b²) where a = -5 and b = 5 r = √((-5)² + 5²) = √(25 + 25) = √50 = 5√2 Reference angle = tan⁻¹(5/5) = tan⁻¹(1) = π/4 Determine the principal argument θ based on the quadrant The point (-5, 5) is in Quadrant II, so θ = π - π/4 = 3π/4 z = r(cosθ + i…
   Full step-by-step solution

   Step 1: Find the modulus r using the formula r = √(a² + b²) where a = -5 and b = 5 r = √((-5)² + 5²) = √(25 + 25) = √50 = 5√2 Step 2: Find the reference angle using tan⁻¹(|b/a|) Reference angle = tan⁻¹(5/5) = tan⁻¹(1) = π/4 Step 3: Determine the principal argument θ based on the quadrant The point (-5, 5) is in Quadrant II, so θ = π - π/4 = 3π/4 Step 4: Write the polar form z = r(cosθ + i sinθ) = 5√2(cos(3π/4) + i sin(3π/4)) The answer is 5√2(cos(3π/4) + i sin(3π/4))

4. Convert the complex number 12 - 16i to polar form r(cosθ + isinθ) = ? Answer: 20(cos(5.3559) + isin(5.3559)) Solution: Identify the real and imaginary parts: a = 12, b = -16 Calculate the modulus r = √(a² + b²) = √(12² + (-16)²) = √(144 + 256) = √400 = 20 Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(16/12) = tan⁻¹(4/3) ≈ 0.9273 radians Determine the actual angle θ: Since the point (12, -16) is in…
   Full step-by-step solution

   Step 1: Identify the real and imaginary parts: a = 12, b = -16 Step 2: Calculate the modulus r = √(a² + b²) = √(12² + (-16)²) = √(144 + 256) = √400 = 20 Step 3: Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(16/12) = tan⁻¹(4/3) ≈ 0.9273 radians Step 4: Determine the actual angle θ: Since the point (12, -16) is in Quadrant IV, θ = 2π - 0.9273 ≈ 6.2832 - 0.9273 = 5.3559 radians Step 5: Write in polar form: r(cosθ + isinθ) = 20(cos(5.3559) + isin(5.3559)) The answer is 20(cos(5.3559) + isin(5.3559)).

5. Emma is calibrating a sonar system for an underwater research drone. The system detects an object at a position represented by the complex number -5√2 + 5√2 i meters relative to the drone. To calculate the object's distance and bearing, Emma must convert this position to polar form r(cos θ + i sin θ), where θ is in degrees between 0° and 360°. What is the polar representation? Answer: 10(cos 135° + i sin 135°) Solution: Identify the real part a = -5√2 and the imaginary part b = 5√2. Calculate the magnitude r = sqrt(a^2 + b^2) = sqrt((-5√2)^2 + (5√2)^2) = sqrt(25*2 + 25*2) = sqrt(50 + 50) = sqrt(100) = 10. Determine the angle θ.
   Full step-by-step solution

   Step 1: Identify the real part a = -5√2 and the imaginary part b = 5√2. Step 2: Calculate the magnitude r = sqrt(a^2 + b^2) = sqrt((-5√2)^2 + (5√2)^2) = sqrt(25*2 + 25*2) = sqrt(50 + 50) = sqrt(100) = 10. Step 3: Determine the angle θ. Since a is negative and b is positive, the point lies in the second quadrant. The reference angle is arctan(|b/a|) = arctan(|5√2 / (-5√2)|) = arctan(1) = 45°. Step 4: For the second quadrant, θ = 180° - 45° = 135°. Step 5: Write the polar form: 10(cos 135° + i sin 135°). Final answer: 10(cos 135° + i sin 135°)

6. 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 4 + 4i. 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 radians from the positive real axis, with -π < θ ≤ π. What is the polar form of the drone's position? Answer: (4√2, π/4) Solution: Identify the real and imaginary parts. The complex number is 4 + 4i. Calculate the modulus r.
   Full step-by-step solution

   Let's find the polar form of the complex number 4 + 4i. Step 1: Identify the real and imaginary parts. The complex number is 4 + 4i. So the real part a = 4, and the imaginary part b = 4. Step 2: Calculate the modulus r. The modulus r is the distance from the origin, given by the formula: r = sqrt(a^2 + b^2) Substitute a = 4 and b = 4: r = sqrt(4^2 + 4^2) = sqrt(16 + 16) = sqrt(32) Simplify sqrt(32): sqrt(32) = sqrt(16 * 2) = sqrt(16) * sqrt(2) = 4 * sqrt(2) So r = 4√2. Step 3: Calculate the argument θ. The argument θ is the angle measured from the positive real axis. We use the formula: θ = arctan(b/a), but we must consider which quadrant the point is in. Since a = 4 (positive) and b = 4 (positive), the point is in the first quadrant. θ = arctan(4/4) = arctan(1) We know that tan(π/4) = 1, so θ = π/4. Step 4: Verify the angle range. The problem specifies that θ must be between -π and π. Our calculated angle π/4 is within this range. Step 5: Write the final polar form. The polar form is (r, θ) = (4√2, π/4). Therefore, the drone's position in polar form is (4√2, π/4).

7. Noah is a marine biologist tracking the migration of a tagged shark using a sonar system. The shark's position relative to the research vessel is given by the complex number -8 + 8√3 i meters, where the real axis points east and the imaginary axis points north. To input this data into his navigation software, Noah must convert the position into polar form r(cos θ + i sin θ), with the angle θ measured in degrees between 0° and 360°. What is the polar representation of the shark'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 tan φ = |b/a| = (8√3)/8 = √3. Thus φ = arctan(√3) = 60°. Step 4: Determine the quadrant. Since a is negative and b is positive, the point lies in Quadrant II. In Quadrant II, the angle θ = 180° - φ = 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°).