Complex Polar Operations

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Complex Polar Operations · Grade 12 · Worksheet 3

1. Mason is an electrical engineer designing a power distribution network for a large industrial facility. He needs to analyze the combined effect of two alternating current sources connected in series. The first source has a voltage represented by the complex number 8∠75° volts, and the second source has a voltage represented by the complex number 9∠-20° volts. To determine the total voltage output of the series combination, Mason must multiply these complex numbers in polar form. What is the product in polar form (r∠θ) with the angle in degrees? Answer: 72∠55° Solution: Identify the magnitudes and angles from the polar forms. First source: magnitude = 8, angle = 75° Second source: magnitude = 9, angle = -20° Multiply the magnitudes. 8 × 9 = 72 Add the angles.
   Full step-by-step solution

   Step 1: Identify the magnitudes and angles from the polar forms. First source: magnitude = 8, angle = 75° Second source: magnitude = 9, angle = -20° Step 2: Multiply the magnitudes. 8 × 9 = 72 Step 3: Add the angles. 75° + (-20°) = 55° Step 4: Write the result in polar form. 72∠55° The answer is 72∠55°.

2. On an Argand diagram, Noah represents a complex number z₁ with a vector of magnitude 12 units making an angle of 210° with the positive real axis. He then multiplies z₁ by another complex number z₂, represented by a vector of magnitude 4 units making an angle of 135° with the positive real axis. What is the magnitude and angle (in degrees) of the resulting product vector? Express your answer in the form: magnitude, angle. Answer: 48, 345 Solution: Write the complex numbers in polar form. z₁ = 12(cos 210° + i sin 210°) z₂ = 4(cos 135° + i sin 135°) Recall the multiplication rule for polar form. Multiply the magnitudes.
   Full step-by-step solution

   Step 1: Write the complex numbers in polar form. z₁ = 12(cos 210° + i sin 210°) z₂ = 4(cos 135° + i sin 135°) Step 2: Recall the multiplication rule for polar form. When multiplying z₁ and z₂: - Magnitude: r₁ × r₂ - Angle: θ₁ + θ₂ Step 3: Multiply the magnitudes. r = 12 × 4 = 48 Step 4: Add the angles. θ = 210° + 135° = 345° Step 5: Check if the angle is within the principal range (0° to 360°). 345° is between 0° and 360°, so no adjustment is needed. Step 6: The product vector has magnitude 48 and angle 345°. The answer is 48, 345.

3. A complex number is represented in the complex plane with modulus 8 and argument 2π/3 radians. If this number is multiplied by another complex number with modulus 2 and argument π/6 radians, what is the modulus and argument of the resulting complex number? Express your answer in the form 'modulus: [value], argument: [value] radians'. Answer: modulus: 16, argument: 5π/6 radians Solution: A complex number with modulus \( r_1 \) and argument \( \theta_1 \) can be written as: z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) Given: \( r_1 = 8 \), \( \theta_1 = 2\pi/3 \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Represent the first complex number in polar form** A complex number with modulus \( r_1 \) and argument \( \theta_1 \) can be written as: \[ z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) \] Given: \( r_1 = 8 \), \( \theta_1 = 2\pi/3 \). So: \[ z_1 = 8 \left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right) \] --- **Step 2: Represent the second complex number in polar form** Given: \( r_2 = 2 \), \( \theta_2 = \pi/6 \). So: \[ z_2 = 2 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) \] --- **Step 3: Multiply the two complex numbers in polar form** When multiplying two complex numbers in polar form: - Multiply the moduli: \( r = r_1 \times r_2 \) - Add the arguments: \( \theta = \theta_1 + \theta_2 \) So: \[ r = 8 \times 2 = 16 \] \[ \theta = \frac{2\pi}{3} + \frac{\pi}{6} \] --- **Step 4: Add the arguments** \[ \frac{2\pi}{3} + \frac{\pi}{6} = \frac{4\pi}{6} + \frac{\pi}{6} = \frac{5\pi}{6} \] --- **Step 5: Check if argument needs adjustment** The argument \( \frac{5\pi}{6} \) is between \( 0 \) and \( \pi \), so it's already in the standard range for the principal value. No adjustment needed. --- **Step 6: Final answer** Modulus: 16 Argument: \( \frac{5\pi}{6} \) radians --- **Final:** modulus: 16, argument: 5π/6 radians

4. An aerospace engineer is designing a navigation system that uses complex numbers to represent signal vectors. The system receives two signals: the first signal is represented by 5∠45° and the second by 3∠15°. To analyze the combined signal strength in a particular component, the engineer needs to multiply these complex numbers in polar form. What is the product in polar form (r∠θ)? Answer: 15∠60° Solution: Identify the magnitudes and angles from the polar forms. First signal: magnitude = 5, angle = 45° Second signal: magnitude = 3, angle = 15° Multiply the magnitudes. 5 × 3 = 15 Add the angles.
   Full step-by-step solution

   Step 1: Identify the magnitudes and angles from the polar forms. First signal: magnitude = 5, angle = 45° Second signal: magnitude = 3, angle = 15° Step 2: Multiply the magnitudes. 5 × 3 = 15 Step 3: Add the angles. 45° + 15° = 60° Step 4: Combine the results to form the product in polar form. 15∠60° The answer is 15∠60°.

5. Isabella is an electrical engineer analyzing the combined effect of two alternating current signals in a complex circuit. The first signal is represented by the complex number 12(cos 70° + i sin 70°) volts, and the second signal is represented by 7(cos 40° + i sin 40°) volts. To determine the total voltage when these signals are combined through a specific multiplier component, she needs to multiply these two complex numbers in polar form. What is the product in polar form, expressed as r(cos θ + i sin θ), with the angle in degrees? Answer: 84(cos 110° + i sin 110°) Solution: Identify the magnitudes and angles from the polar forms. First signal: magnitude = 12, angle = 70° Second signal: magnitude = 7, angle = 40° Multiply the magnitudes. 12 × 7 = 84 Add the angles.
   Full step-by-step solution

   Step 1: Identify the magnitudes and angles from the polar forms. First signal: magnitude = 12, angle = 70° Second signal: magnitude = 7, angle = 40° Step 2: Multiply the magnitudes. 12 × 7 = 84 Step 3: Add the angles. 70° + 40° = 110° Step 4: Write the product in polar form. 84(cos 110° + i sin 110°) The answer is 84(cos 110° + i sin 110°).

6. 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). Answer: -1.7678-4.0315i Solution: Write the quotient in polar form: z₁ ÷ z₂ = (5/2)[cos(2π/3 - π/4) + i sin(2π/3 - π/4)] Calculate the magnitude: 5 ÷ 2 = 2.5 Calculate the angle difference: 2π/3 - π/4 = 8π/12 - 3π/12 = 5π/12 The quotient in polar form is: 2.5[cos(5π/12) + i sin(5π/12)] Convert to rectangular form: a = 2.5 ×…
   Full step-by-step solution

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