Complex Polar Operations

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Complex Polar Operations · Grade 12 · Worksheet 2

1. Liam is designing a complex electrical circuit and needs to multiply two alternating current phasors. The first phasor has magnitude 4 and angle 30°, while the second has magnitude 3 and angle 60°. Express the product in polar form (r∠θ) where r is the magnitude and θ is the angle in degrees. Answer: 12∠90° Solution: When multiplying two phasors in polar form, we multiply their magnitudes and add their angles. Identify the given phasors. First phasor: magnitude r1 = 4, angle θ1 = 30° Second phasor: magnitude r2 = 3, angle θ2 = 60° Multiply the magnitudes.
   Full step-by-step solution

   When multiplying two phasors in polar form, we multiply their magnitudes and add their angles. Step 1: Identify the given phasors. First phasor: magnitude r1 = 4, angle θ1 = 30° Second phasor: magnitude r2 = 3, angle θ2 = 60° Step 2: Multiply the magnitudes. The magnitude of the product, r, is r1 * r2. r = 4 * 3 r = 12 Step 3: Add the angles. The angle of the product, θ, is θ1 + θ2. θ = 30° + 60° θ = 90° Step 4: Write the final answer in polar form. The product is r ∠ θ = 12 ∠ 90° Therefore, the product of the two phasors is 12∠90°.

2. Dr. Chen is analyzing quantum states in a physics experiment. The first quantum state is represented by the complex number 5(cos(π/4) + i sin(π/4)) and the second state is 3(cos(π/12) + i sin(π/12)). To determine their entanglement probability, she needs to multiply these complex numbers in polar form. What is the product in polar form? Answer: 15(cos(π/3) + i sin(π/3)) Solution: Identify the magnitudes and angles from the polar forms First complex number: magnitude = 5, angle = π/4 Second complex number: magnitude = 3, angle = π/12 5 × 3 = 15 π/4 + π/12 = 3π/12 + π/12 = 4π/12 = π/3 15(cos(π/3) + i sin(π/3)) The answer is 15(cos(π/3) + i sin(π/3)).
   Full step-by-step solution

   Step 1: Identify the magnitudes and angles from the polar forms First complex number: magnitude = 5, angle = π/4 Second complex number: magnitude = 3, angle = π/12 Step 2: Multiply the magnitudes 5 × 3 = 15 Step 3: Add the angles π/4 + π/12 = 3π/12 + π/12 = 4π/12 = π/3 Step 4: Write the product in polar form 15(cos(π/3) + i sin(π/3)) The answer is 15(cos(π/3) + i sin(π/3)).

3. A physicist is studying wave interference patterns using complex numbers. She has two waves represented in polar form: Wave A = 6∠45° and Wave B = 4∠-30°. To analyze their combined effect at a specific point, she needs to multiply these complex numbers. What is the product in polar form (r∠θ) with the angle in degrees? Answer: 24∠15° Solution: Identify the magnitudes and angles from the polar forms Wave A: magnitude = 6, angle = 45° Wave B: magnitude = 4, angle = -30° 6 × 4 = 24 45° + (-30°) = 15° 24∠15° The answer is 24∠15°.
   Full step-by-step solution

   Step 1: Identify the magnitudes and angles from the polar forms Wave A: magnitude = 6, angle = 45° Wave B: magnitude = 4, angle = -30° Step 2: Multiply the magnitudes 6 × 4 = 24 Step 3: Add the angles 45° + (-30°) = 15° Step 4: Write the result in polar form 24∠15° The answer is 24∠15°.

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). 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.

5. An electrical engineer is analyzing alternating current in a circuit with two parallel components. The current in the first branch is represented by the complex number 3(cos(π/3) + i sin(π/3)) amperes, and the current in the second branch is 2(cos(π/6) + i sin(π/6)) amperes. To find the total current, she needs to multiply these complex numbers in polar form. What is the product in polar form? Answer: 6(cos(π/2) + i sin(π/2)) Solution: Write the given complex numbers in polar form. \( 3 \left( \cos(\pi/3) + i \sin(\pi/3) \right) \) \( 2 \left( \cos(\pi/6) + i \sin(\pi/6) \right) \) In polar form, a complex number is written as \( r (\cos \theta + i \sin \theta) \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Write the given complex numbers in polar form.** The first current is: \( 3 \left( \cos(\pi/3) + i \sin(\pi/3) \right) \) The second current is: \( 2 \left( \cos(\pi/6) + i \sin(\pi/6) \right) \) In polar form, a complex number is written as \( r (\cos \theta + i \sin \theta) \). So: - First number: \( r_1 = 3 \), \( \theta_1 = \pi/3 \) - Second number: \( r_2 = 2 \), \( \theta_2 = \pi/6 \) --- **Step 2: Recall the multiplication rule for complex numbers in polar form.** When multiplying two complex numbers in polar form: \[ [r_1 (\cos \theta_1 + i \sin \theta_1)] \cdot [r_2 (\cos \theta_2 + i \sin \theta_2)] = (r_1 r_2) \left[ \cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2) \right] \] --- **Step 3: Multiply the magnitudes.** \( r_1 r_2 = 3 \times 2 = 6 \) --- **Step 4: Add the angles.** \( \theta_1 + \theta_2 = \pi/3 + \pi/6 \) Write with common denominator: \( \pi/3 = 2\pi/6 \) \( 2\pi/6 + \pi/6 = 3\pi/6 = \pi/2 \) So the new angle is \( \pi/2 \). --- **Step 5: Write the product in polar form.** Magnitude \( 6 \), angle \( \pi/2 \): \[ 6 \left( \cos(\pi/2) + i \sin(\pi/2) \right) \] --- **Final Answer:** 6(cos(pi/2) + i sin(pi/2))

6. Liam is designing a complex electrical circuit where two alternating currents are represented as complex numbers in polar form. The first current is 8∠45° amperes and the second is 2∠15° amperes. When these currents combine in a particular circuit component, their combined effect is found by multiplying these complex numbers. What is the resulting current in polar form? Answer: 16∠60° Solution: First current: magnitude 8, angle 45° → 8∠45° Second current: magnitude 2, angle 15° → 2∠15° When combining them in this circuit component, we multiply the complex numbers in polar form.
   Full step-by-step solution

   Step 1: Understand the problem We have two alternating currents in polar form: First current: magnitude 8, angle 45° → 8∠45° Second current: magnitude 2, angle 15° → 2∠15° When combining them in this circuit component, we multiply the complex numbers in polar form. Step 2: Recall the multiplication rule for complex numbers in polar form For two complex numbers: r1∠θ1 and r2∠θ2 Their product is: (r1 × r2) ∠ (θ1 + θ2) That is: multiply the magnitudes, add the angles. Step 3: Apply the rule Magnitude: 8 × 2 = 16 Angle: 45° + 15° = 60° Step 4: Write the result in polar form Resulting current = 16∠60° Step 5: Conclusion The combined effect of the two currents is a current with magnitude 16 amperes and phase angle 60 degrees.

7. A physicist is modeling wave interference patterns using complex numbers. Two waves are represented by the complex numbers 5∠45° and 3∠-30° in polar form. To analyze their combined amplitude and phase shift when they interfere constructively, she needs to multiply these complex numbers. What is the product in polar form? Answer: 15∠15° Solution: In complex number multiplication using polar form, the magnitude of the product equals the product of the individual magnitudes, while the angle equals the sum of the individual angles.
   Full step-by-step solution

   In complex number multiplication using polar form, the magnitude of the product equals the product of the individual magnitudes, while the angle equals the sum of the individual angles. This principle is fundamental in analyzing wave interference, signal processing, and electrical engineering applications where phase relationships are critical.