Complex Polar Operations

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Answer Key & Explanations

Complex Polar Operations · Grade 12 · Worksheet 2

1. A complex number is represented on the complex plane with a magnitude of 8 units and an angle of 150° from the positive real axis. If this number is multiplied by another complex number with magnitude 2 and angle 60°, what is the resulting complex number in polar form (r, θ)? Answer: (16, 210°) Solution: Represent the first complex number in polar form. The first number has magnitude \( r_1 = 8 \) and angle \( \theta_1 = 150^\circ \). \( z_1 = (8, 150^\circ) \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Represent the first complex number in polar form.** The first number has magnitude \( r_1 = 8 \) and angle \( \theta_1 = 150^\circ \). So in polar form: \( z_1 = (8, 150^\circ) \). --- **Step 2: Represent the second complex number in polar form.** The second number has magnitude \( r_2 = 2 \) and angle \( \theta_2 = 60^\circ \). So in polar form: \( z_2 = (2, 60^\circ) \). --- **Step 3: Multiply the magnitudes.** When multiplying two complex numbers in polar form, multiply the magnitudes: \( r = r_1 \times r_2 = 8 \times 2 = 16 \). --- **Step 4: Add the angles.** When multiplying two complex numbers in polar form, add the angles: \( \theta = \theta_1 + \theta_2 = 150^\circ + 60^\circ = 210^\circ \). --- **Step 5: Write the result in polar form.** The product is \( (16, 210^\circ) \). --- **Step 6: Check if the angle needs adjustment.** Angles are usually given between \( 0^\circ \) and \( 360^\circ \). Here \( 210^\circ \) is already in that range, so no adjustment is needed. --- **Final Answer:** (16, 210°)

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 complex number is represented on the Argand diagram as a point with coordinates (3, 4). Convert this complex number to polar form (r, θ), where r is the magnitude and θ is the principal argument in radians. Answer: (5, 0.9273) Solution: We are given the complex number represented by the point (3, 4) on the Argand diagram. That means the real part is 3 and the imaginary part is 4.
   Full step-by-step solution

   We are given the complex number represented by the point (3, 4) on the Argand diagram. That means the real part is 3 and the imaginary part is 4. So the complex number is: z = 3 + 4i --- **Step 1: Find the magnitude r** The magnitude r is the distance from the origin to the point (3, 4). Formula: r = sqrt(real^2 + imag^2) So: r = sqrt(3^2 + 4^2) r = sqrt(9 + 16) r = sqrt(25) r = 5 --- **Step 2: Find the principal argument θ** The argument θ is the angle measured counterclockwise from the positive real axis to the point. Formula: θ = arctan(imag / real) = arctan(4 / 3) Since the point (3, 4) is in the first quadrant (both coordinates positive), the arctan result is already the principal argument. So: θ = arctan(4 / 3) Using a calculator: 4 / 3 = 1.333333... arctan(1.333333...) ≈ 0.9273 radians --- **Step 3: Write the polar form** Polar form is (r, θ) = (5, 0.9273) --- **Final Answer:** (5, 0.9273)

4. A complex number is represented on the complex plane with a magnitude of 8 units and an angle of 150° from the positive real axis. What is the rectangular form (a + bi) of this complex number? Answer: -4√3 + 4i Solution: Magnitude r = 8 Angle θ = 150° from the positive real axis. Recall the conversion from polar to rectangular form. a = r * cos(θ) b = r * sin(θ) So the complex number is a + bi.
   Full step-by-step solution

   We are given a complex number in polar form: Magnitude r = 8 Angle θ = 150° from the positive real axis. Step 1: Recall the conversion from polar to rectangular form. The rectangular form is: a = r * cos(θ) b = r * sin(θ) So the complex number is a + bi. Step 2: Find the cosine and sine of 150°. 150° is in the second quadrant, where cosine is negative and sine is positive. We can use the reference angle: 180° - 150° = 30°. cos(150°) = -cos(30°) = -√3 / 2 sin(150°) = sin(30°) = 1 / 2 Step 3: Compute a and b. a = r * cos(θ) = 8 * (-√3 / 2) = -4√3 b = r * sin(θ) = 8 * (1 / 2) = 4 Step 4: Write the rectangular form. a + bi = -4√3 + 4i This matches the correct answer: -4√3 + 4i.

5. Sophia is an electrical engineer analyzing the voltage and current in an AC circuit. The voltage across a component is represented by the complex number 11∠36° volts, and the current through it is 6∠-24° amperes. To find the impedance (Z = V / I), she needs to divide these complex numbers in polar form. What is the impedance in polar form (r∠θ) with the angle in degrees? Answer: 1.833∠60° Solution: Identify the magnitudes and angles. Voltage V = 11∠36°, so magnitude r₁ = 11 and angle θ₁ = 36°. Current I = 6∠-24°, so magnitude r₂ = 6 and angle θ₂ = -24°.
   Full step-by-step solution

   Step 1: Identify the magnitudes and angles. Voltage V = 11∠36°, so magnitude r₁ = 11 and angle θ₁ = 36°. Current I = 6∠-24°, so magnitude r₂ = 6 and angle θ₂ = -24°. Step 2: Divide the magnitudes: r₁ / r₂ = 11 / 6 = 1.8333... which rounds to 1.833. Step 3: Subtract the angles: θ₁ - θ₂ = 36° - (-24°) = 36° + 24° = 60°. Step 4: Write the result in polar form: 1.833∠60°. The answer is 1.833∠60°.

6. An electrical engineer is analyzing two alternating currents in a circuit. The first current has a magnitude of 8 amperes and a phase angle of 30°, while the second current has a magnitude of 6 amperes and a phase angle of 60°. When these currents interact, their combined effect is calculated by multiplying their complex representations in polar form. What is the resulting magnitude and phase angle of the product? Express your answer in the form 'magnitude ∠ angle°'. Answer: 48 ∠ 90° Solution: Magnitude = 8 A, Phase angle = 30° I₁ = 8 ∠ 30° Magnitude = 6 A, Phase angle = 60° I₂ = 6 ∠ 60° Recall the rule for multiplying complex numbers in polar form When multiplying two complex numbers in polar form: (r₁ ∠ θ₁) × (r₂ ∠ θ₂) = (r₁ × r₂) ∠ (θ₁ + θ₂) Magnitude of product = 8 × 6 = 48 Phase…
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Represent the currents in polar form** The first current: Magnitude = 8 A, Phase angle = 30° So in polar form: I₁ = 8 ∠ 30° The second current: Magnitude = 6 A, Phase angle = 60° So in polar form: I₂ = 6 ∠ 60° --- **Step 2: Recall the rule for multiplying complex numbers in polar form** When multiplying two complex numbers in polar form: (r₁ ∠ θ₁) × (r₂ ∠ θ₂) = (r₁ × r₂) ∠ (θ₁ + θ₂) --- **Step 3: Multiply the magnitudes** Magnitude of product = 8 × 6 = 48 --- **Step 4: Add the phase angles** Phase angle of product = 30° + 60° = 90° --- **Step 5: Write the final answer in polar form** Product = 48 ∠ 90° --- **Final Answer:** 48 ∠ 90°

7. Isabella is an electrical engineer designing a power distribution system. She measures the voltage across a component as V = 8∠50° volts and the current through it as I = 2∠-20° amperes, both in polar form. To determine the impedance Z of the component, she uses Ohm's law in complex form: Z = V / I. What is the impedance in polar form (r∠θ) with the angle in degrees? Answer: 4∠70° Solution: Identify the magnitudes and angles. Voltage: magnitude = 8, angle = 50° Current: magnitude = 2, angle = -20° Divide the magnitudes. 8 ÷ 2 = 4 Subtract the angles.
   Full step-by-step solution

   Step 1: Identify the magnitudes and angles. Voltage: magnitude = 8, angle = 50° Current: magnitude = 2, angle = -20° Step 2: Divide the magnitudes. 8 ÷ 2 = 4 Step 3: Subtract the angles. 50° - (-20°) = 50° + 20° = 70° Step 4: Write the result in polar form. 4∠70° The answer is 4∠70°.