Complex Polar Operations

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Answer Key & Explanations

Complex Polar Operations · Grade 12 · Worksheet 1

1. A complex number is represented in the complex plane with magnitude 8 and an angle of 45° from the positive real axis. This number is then multiplied by another complex number with magnitude 2 and an angle of 15°. What is the magnitude and angle (in degrees) of the resulting product? Express your answer in the form: magnitude, angle. Answer: 16, 60 Solution: Represent the first complex number in polar form. A complex number with magnitude \( r_1 \) and angle \( \theta_1 \) can be written as: z_1 = r_1 \cdot e^{i \theta_1} Given: \( r_1 = 8 \), \( \theta_1 = 45^\circ \).
   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 magnitude \( r_1 \) and angle \( \theta_1 \) can be written as: \[ z_1 = r_1 \cdot e^{i \theta_1} \] Given: \( r_1 = 8 \), \( \theta_1 = 45^\circ \). So: \[ z_1 = 8 \cdot e^{i \cdot 45^\circ} \] --- **Step 2: Represent the second complex number in polar form.** Given: \( r_2 = 2 \), \( \theta_2 = 15^\circ \). So: \[ z_2 = 2 \cdot e^{i \cdot 15^\circ} \] --- **Step 3: Multiply the two complex numbers in polar form.** When multiplying in polar form: - Multiply the magnitudes: \( r = r_1 \times r_2 \) - Add the angles: \( \theta = \theta_1 + \theta_2 \) So: \[ z_1 \cdot z_2 = (8 \times 2) \cdot e^{i (45^\circ + 15^\circ)} \] \[ z_1 \cdot z_2 = 16 \cdot e^{i \cdot 60^\circ} \] --- **Step 4: Read the magnitude and angle from the result.** Magnitude = \( 16 \) Angle = \( 60^\circ \) --- **Final Answer:** 16, 60

2. On an Argand diagram, Noah plots a complex number z₁ with a modulus of 9 and an argument of 135°. He then plots another complex number z₂ with a modulus of 3 and an argument of 45°. If Noah multiplies z₁ by z₂, what is the modulus and argument (in degrees) of the resulting complex number? Express your answer as: modulus, argument. Answer: 27, 180 Solution: Recall the rule for multiplying complex numbers in polar form: For z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), the product is r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)].
   Full step-by-step solution

   Step 1: Recall the rule for multiplying complex numbers in polar form: For z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), the product is r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]. Step 2: Identify the given values: r₁ = 9, θ₁ = 135°, r₂ = 3, θ₂ = 45°. Step 3: Multiply the moduli: r = 9 × 3 = 27. Step 4: Add the arguments: θ = 135° + 45° = 180°. Step 5: The resulting complex number has modulus 27 and argument 180°. Since 180° is within the standard range of 0° to 360°, no adjustment is needed. The answer is 27, 180.

3. On an Argand diagram, Hana draws two complex numbers. The first number, z₁, has a modulus of 6 and an argument of 210°. The second number, z₂, has a modulus of 4 and an argument of 135°. Hana multiplies z₁ by z₂ and then divides the result by a third complex number, z₃, which has a modulus of 3 and an argument of 60°. What is the modulus and argument (in degrees) of the final complex number? Express your answer in the form: modulus, argument. Answer: 8, 285 Solution: Represent z₁ and z₂ in polar form. z₁ = 6(cos 210° + i sin 210°) z₂ = 4(cos 135° + i sin 135°) Multiply z₁ by z₂. When multiplying in polar form, multiply the moduli and add the arguments.
   Full step-by-step solution

   Step 1: Represent z₁ and z₂ in polar form. z₁ = 6(cos 210° + i sin 210°) z₂ = 4(cos 135° + i sin 135°) Step 2: Multiply z₁ by z₂. When multiplying in polar form, multiply the moduli and add the arguments. Modulus: 6 × 4 = 24 Argument: 210° + 135° = 345° So z₁ × z₂ = 24(cos 345° + i sin 345°) Step 3: Divide the product by z₃. z₃ = 3(cos 60° + i sin 60°) When dividing in polar form, divide the moduli and subtract the arguments. Modulus: 24 ÷ 3 = 8 Argument: 345° - 60° = 285° Step 4: The final complex number has modulus 8 and argument 285°. The answer is 8, 285.

4. Sophia is analyzing the interference pattern of two electromagnetic waves in a physics experiment. The first wave is represented by the complex number 6(cos(2π/3) + i sin(2π/3)) and the second wave by 4(cos(π/4) + i sin(π/4)). 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(cos(11π/12) + i sin(11π/12)) Solution: Identify the magnitudes and angles from the polar forms. First complex number: magnitude = 6, angle = 2π/3 Second complex number: magnitude = 4, angle = π/4 Multiply the magnitudes: 6 × 4 = 24 Add the angles: 2π/3 + π/4 = 8π/12 + 3π/12 = 11π/12 Write the product in polar form: 24(cos(11π/12) + i…
   Full step-by-step solution

   Step 1: Identify the magnitudes and angles from the polar forms. First complex number: magnitude = 6, angle = 2π/3 Second complex number: magnitude = 4, angle = π/4 Step 2: Multiply the magnitudes: 6 × 4 = 24 Step 3: Add the angles: 2π/3 + π/4 = 8π/12 + 3π/12 = 11π/12 Step 4: Write the product in polar form: 24(cos(11π/12) + i sin(11π/12)) The answer is 24(cos(11π/12) + i sin(11π/12)).

5. A physicist is analyzing quantum states represented by complex numbers in polar form. The first quantum state has amplitude 6 and phase angle 45°, while the second state has amplitude 4 and phase angle 60°. When these states interact in a quantum system, their combined effect is calculated by multiplying their complex representations. What is the resulting quantum state in polar form? Express your answer as 'r∠θ' where r is the magnitude and θ is the angle in degrees. Answer: 24∠105° Solution: Identify the given complex numbers in polar form: 6∠45° and 4∠60° To multiply complex numbers in polar form, multiply the magnitudes: 6 × 4 = 24 Add the phase angles: 45° + 60° = 105° Combine the results to get the product in polar form: 24∠105° The resulting quantum state is 24∠105°.
   Full step-by-step solution

   Step 1: Identify the given complex numbers in polar form: 6∠45° and 4∠60° Step 2: To multiply complex numbers in polar form, multiply the magnitudes: 6 × 4 = 24 Step 3: Add the phase angles: 45° + 60° = 105° Step 4: Combine the results to get the product in polar form: 24∠105° The resulting quantum state is 24∠105°.

6. Matiu is a telecommunications engineer designing a phased-array antenna system. Two signal vectors from different antenna elements are represented in polar form as z₁ = 12∠80° and z₂ = 7∠-50°. To calculate the combined signal strength for a beamforming algorithm, Matiu needs to multiply these complex numbers. What is the product in polar form (r∠θ) with the angle in degrees? Answer: 84∠30° Solution: Identify the magnitudes and angles from the polar forms. z₁ has magnitude r₁ = 12 and angle θ₁ = 80°. z₂ has magnitude r₂ = 7 and angle θ₂ = -50°.
   Full step-by-step solution

   Step 1: Identify the magnitudes and angles from the polar forms. z₁ has magnitude r₁ = 12 and angle θ₁ = 80°. z₂ has magnitude r₂ = 7 and angle θ₂ = -50°. Step 2: Multiply the magnitudes. r₁ × r₂ = 12 × 7 = 84. Step 3: Add the angles. θ₁ + θ₂ = 80° + (-50°) = 30°. Step 4: Write the product in polar form. The product is 84∠30°. The answer is 84∠30°.