Complex Numbers

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Answer Key & Explanations

Complex Numbers · Grade 12 · Worksheet 2

1. On the complex plane, Matiu observes a point representing a complex number located at coordinates (-7, 7√3). Express this complex number in polar form r(cos θ + i sin θ), where r > 0 and θ is the principal argument in radians between 0 and 2π. Answer: 14(cos(2π/3) + i sin(2π/3)) Solution: Identify the rectangular coordinates a = -7 and b = 7√3. Calculate the modulus r = sqrt(a^2 + b^2) = sqrt((-7)^2 + (7√3)^2) = sqrt(49 + 49*3) = sqrt(49 + 147) = sqrt(196) = 14.
   Full step-by-step solution

   Step 1: Identify the rectangular coordinates a = -7 and b = 7√3. Step 2: Calculate the modulus r = sqrt(a^2 + b^2) = sqrt((-7)^2 + (7√3)^2) = sqrt(49 + 49*3) = sqrt(49 + 147) = sqrt(196) = 14. Step 3: Find the reference angle using tan^(-1)(|b/a|) = tan^(-1)((7√3)/7) = tan^(-1)(√3) = π/3. Step 4: The point (-7, 7√3) lies in Quadrant II (negative real, positive imaginary), so the principal argument θ = π - π/3 = 2π/3. Step 5: Write the polar form: z = r(cos θ + i sin θ) = 14(cos(2π/3) + i sin(2π/3)). The answer is 14(cos(2π/3) + i sin(2π/3)).

2. Convert the complex number 3 + 4i to polar form (r(cosθ + i sinθ)) = ? Answer: 5(cos(0.9273) + i sin(0.9273)) Solution: To convert the complex number 3 + 4i to polar form r(cosθ + i sinθ), we follow these steps: The modulus r is the distance from the origin to the point (3,4) in the complex plane.
   Full step-by-step solution

   To convert the complex number 3 + 4i to polar form r(cosθ + i sinθ), we follow these steps: Step 1: Calculate the modulus r The modulus r is the distance from the origin to the point (3,4) in the complex plane. Formula: r = sqrt(a^2 + b^2) where a = 3 and b = 4. Calculation: r = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. So, r = 5. Step 2: Calculate the argument θ The argument θ is the angle the line from the origin to (3,4) makes with the positive real axis. Since the point (3,4) is in the first quadrant (both coordinates positive), we use: θ = arctan(b/a) = arctan(4/3). Calculation: θ = arctan(4/3) ≈ arctan(1.3333) ≈ 0.9273 radians. We use radians because polar form typically uses radians. Step 3: Write in polar form Substitute r = 5 and θ ≈ 0.9273 into the polar form formula: Polar form = r(cosθ + i sinθ) = 5(cos(0.9273) + i sin(0.9273)). Therefore, the polar form of 3 + 4i is 5(cos(0.9273) + i sin(0.9273)).

3. Noah is analyzing a mechanical vibration system where the displacement of a spring-mass system is modeled by the complex number -7 + 7√3 i millimeters. To determine the amplitude and phase angle of the oscillation, he must convert this displacement to polar form r(cos θ + i sin θ) with θ in radians between 0 and 2π. What is the polar representation of the displacement? Answer: 14(cos(2π/3) + i sin(2π/3)) Solution: Identify the real part a = -7 and imaginary part b = 7√3. Step 2: Calculate the magnitude r = sqrt(a^2 + b^2) = sqrt((-7)^2 + (7√3)^2) = sqrt(49 + 147) = sqrt(196) = 14.
   Full step-by-step solution

   Step 1: Identify the real part a = -7 and imaginary part b = 7√3. Step 2: Calculate the magnitude r = sqrt(a^2 + b^2) = sqrt((-7)^2 + (7√3)^2) = sqrt(49 + 147) = sqrt(196) = 14. Step 3: The point (-7, 7√3) lies in the second quadrant because a < 0 and b > 0. Step 4: Find the reference angle: tan φ = |b|/|a| = (7√3)/7 = √3, so φ = π/3. Step 5: In the second quadrant, θ = π - φ = π - π/3 = 2π/3. Step 6: Therefore, the polar form is r(cos θ + i sin θ) = 14(cos(2π/3) + i sin(2π/3)). The answer is 14(cos(2π/3) + i sin(2π/3)).

4. An electrical engineer is analyzing an alternating current circuit with a voltage source that produces a complex voltage of 24 + 10i volts and a load impedance of 3 - 4i ohms. To calculate the current using Ohm's Law (I = V/Z), she needs to represent both the voltage and impedance in polar form. What is the polar form representation of the voltage V = 24 + 10i volts? Express your answer in the form r(cosθ + isinθ) with θ in degrees rounded to one decimal place. Answer: 26(cos22.6° + isin22.6°) Solution: Identify the rectangular form components. The voltage is V = 24 + 10i. Calculate the magnitude r.
   Full step-by-step solution

   Let's find the polar form of V = 24 + 10i volts. Step 1: Identify the rectangular form components. The voltage is V = 24 + 10i. So the real part a = 24, and the imaginary part b = 10. Step 2: Calculate the magnitude r. The magnitude r is given by r = sqrt(a^2 + b^2). r = sqrt(24^2 + 10^2) = sqrt(576 + 100) = sqrt(676) = 26. So r = 26 volts. Step 3: Calculate the angle θ in degrees. The angle θ is given by θ = arctan(b/a) = arctan(10/24) = arctan(5/12). Using a calculator: arctan(5/12) ≈ arctan(0.4166667) ≈ 22.61986 degrees. Rounded to one decimal place: θ ≈ 22.6 degrees. Step 4: Write the polar form. The polar form is r(cosθ + i sinθ). Substituting r = 26 and θ = 22.6°, we get: 26(cos22.6° + i sin22.6°) Final answer: 26(cos22.6° + i sin22.6°)

5. Liam is analyzing an AC circuit where the voltage across a component is given by V = 5√3 - 5i volts. To determine the amplitude and phase shift of the voltage signal, he needs to convert this complex number to polar form. What is the polar representation of this voltage in the form r∠θ, where θ is measured in degrees between -180° and 180°? Answer: 10∠-30° Solution: When converting complex numbers to polar form, we find the distance from the origin (magnitude) and the angle from the positive real axis. The angle depends on which quadrant the point is located in.
   Full step-by-step solution

   When converting complex numbers to polar form, we find the distance from the origin (magnitude) and the angle from the positive real axis. The angle depends on which quadrant the point is located in. For example, a complex number like 3 - 4i would be in the fourth quadrant, so its angle would be negative.

6. Convert the complex number -6 + 6i to polar form r(cosθ + isinθ) = ? Answer: 6√2(cos(3π/4) + isin(3π/4)) Solution: Identify the real and imaginary parts: a = -6, b = 6 Calculate the modulus r = √(a² + b²) = √((-6)² + 6²) = √(36 + 36) = √72 = 6√2 Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(6/6) = tan⁻¹(1) = π/4 Determine the actual angle θ: Since the point (-6, 6) is in Quadrant II, θ = π - π/4 =…
   Full step-by-step solution

   Step 1: Identify the real and imaginary parts: a = -6, b = 6 Step 2: Calculate the modulus r = √(a² + b²) = √((-6)² + 6²) = √(36 + 36) = √72 = 6√2 Step 3: Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(6/6) = tan⁻¹(1) = π/4 Step 4: Determine the actual angle θ: Since the point (-6, 6) is in Quadrant II, θ = π - π/4 = 3π/4 Step 5: Write in polar form: r(cosθ + isinθ) = 6√2(cos(3π/4) + isin(3π/4)) The answer is 6√2(cos(3π/4) + isin(3π/4)).

7. 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).