Complex Polar Operations

lessonbunny.com

Answer Key & Explanations

Complex Polar Operations · Grade 12 · Worksheet 3

1. Sophia is analyzing the interference pattern of two electromagnetic waves in a physics experiment. The first wave is represented by the complex number 6∠45° and the second wave by 4∠15°. To determine the combined wave amplitude at a specific point, she needs to multiply these complex numbers in polar form. What is the product in polar form? Answer: 24∠60° Solution: Identify the magnitudes and angles from the polar forms First complex number: magnitude = 6, angle = 45° Second complex number: magnitude = 4, angle = 15° 6 × 4 = 24 45° + 15° = 60° 24∠60° The answer is 24∠60°.
   Full step-by-step solution

   Step 1: Identify the magnitudes and angles from the polar forms First complex number: magnitude = 6, angle = 45° Second complex number: magnitude = 4, angle = 15° Step 2: Multiply the magnitudes 6 × 4 = 24 Step 3: Add the angles 45° + 15° = 60° Step 4: Write the result in polar form 24∠60° The answer is 24∠60°.

2. A physicist is studying wave interference patterns in a quantum system. The wave function from the first source is represented by the complex number 6∠45° in polar form, while the wave function from the second source is 4∠-30°. To analyze their constructive interference at a specific point, she needs to multiply these complex wave functions. What is the product in polar form? Answer: 24∠15° Solution: In quantum mechanics, wave functions are often represented as complex numbers in polar form. When analyzing interference patterns, multiplying wave functions helps determine their combined effect.
   Full step-by-step solution

   In quantum mechanics, wave functions are often represented as complex numbers in polar form. When analyzing interference patterns, multiplying wave functions helps determine their combined effect. The multiplication follows the rule where magnitudes multiply and angles add, which is different from how real numbers behave.

3. An electrical engineer is designing a filter circuit that processes two signal components. The first signal is represented by the complex number 5∠45° volts and the second signal is 2∠-30° volts. To analyze how these signals interact in the filter, she needs to divide the first signal by the second signal using polar form operations. What is the result of this division in polar form? Answer: 2.5∠75° Solution: In electrical engineering, complex numbers in polar form are often used to represent signals with both magnitude and phase.
   Full step-by-step solution

   In electrical engineering, complex numbers in polar form are often used to represent signals with both magnitude and phase. When dividing two such numbers, the resulting magnitude is the quotient of the individual magnitudes, and the resulting angle is the difference between the angles. This operation is fundamental in analyzing how signals combine in circuits like filters and impedance networks.

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. 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 amplitude and phase shift, she needs to multiply these complex numbers. What is the product in polar form? Answer: 24∠75° 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° = 75° The product is 24∠75° The answer is 24∠75°.
   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° = 75° Step 4: Write the result in polar form The product is 24∠75° The answer is 24∠75°.

6. 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

7. z₁ = 4(cos 60° + i sin 60°), z₂ = 6(cos 30° + i sin 30°). Find z₁·z₂. Answer: 24(cos 90° + i sin 90°) Solution: Identify the moduli and angles. r₁ = 4, θ₁ = 60°; r₂ = 6, θ₂ = 30°. Multiply the moduli: r₁ × r₂ = 4 × 6 = 24.
   Full step-by-step solution

   Step 1: Identify the moduli and angles. r₁ = 4, θ₁ = 60°; r₂ = 6, θ₂ = 30°. Step 2: Multiply the moduli: r₁ × r₂ = 4 × 6 = 24. Step 3: Add the angles: θ₁ + θ₂ = 60° + 30° = 90°. Step 4: Write the product in polar form: z₁·z₂ = 24(cos 90° + i sin 90°). The answer is 24(cos 90° + i sin 90°).