Complex Numbers

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Answer Key & Explanations

Complex Numbers · Grade 12 · Worksheet 1

1. Convert the complex number 3 + 4i to polar form (r(cosθ + isinθ)) = ? Answer: 5(cos(0.9273) + isin(0.9273)) Solution: To convert the complex number 3 + 4i to polar form r(cosθ + i sinθ), follow these steps: Find the modulus r. The modulus r is the distance from the origin to the point (3, 4) in the complex plane.
   Full step-by-step solution

   To convert the complex number 3 + 4i to polar form r(cosθ + i sinθ), follow these steps: Step 1: Find the modulus r. The modulus r is the distance from the origin to the point (3, 4) in the complex plane. Formula: r = sqrt(a^2 + b^2), where a is the real part (3) and b is the imaginary part (4). Calculation: r = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. So, r = 5. Step 2: Find the argument θ. The argument θ is the angle the line from the origin to the point (3, 4) makes with the positive real axis. Formula: θ = arctan(b/a), where a=3 and b=4. Calculation: θ = arctan(4/3). Since the point (3, 4) is in the first quadrant (both coordinates positive), the angle we get from the arctan function is correct and no adjustment is needed. Numerical value: arctan(4/3) ≈ arctan(1.3333) ≈ 0.9273 radians. So, θ ≈ 0.9273 radians. Step 3: Write the polar form. Substitute the values of r and θ into the polar form r(cosθ + i sinθ). Result: 5(cos(0.9273) + i sin(0.9273)). Therefore, the polar form of 3 + 4i is 5(cos(0.9273) + i sin(0.9273)).

2. An electrical engineer is analyzing an alternating current circuit with two voltage sources. The first source has a voltage of 8√3 + 8i volts, and the second source has a voltage of 4√2 - 4√2i volts. To find the total voltage in the circuit, the engineer needs to convert both complex numbers to polar form and then add them. What is the total voltage expressed in polar form (r∠θ) with θ in degrees? Answer: 12√2∠15° Solution: In electrical engineering, complex numbers are often used to represent AC voltages and currents. The polar form r∠θ shows the magnitude (r) and phase angle (θ) of the signal.
   Full step-by-step solution

   In electrical engineering, complex numbers are often used to represent AC voltages and currents. The polar form r∠θ shows the magnitude (r) and phase angle (θ) of the signal. To add complex numbers in polar form, it's typically easier to convert them to rectangular form first, perform the addition, then convert the result back to polar form. This approach is particularly useful when analyzing circuits with multiple voltage sources that have different phase relationships.

3. Convert the complex number 3 - 4i to polar form (r(cosθ + isinθ)) = ? Answer: 5(cos(-0.9273) + isin(-0.9273)) Solution: Find the modulus r. The modulus r is the distance from the origin to the point (3, -4) in the complex plane. Formula: r = sqrt(a^2 + b^2) where a = 3 and b = -4.
   Full step-by-step solution

   Let's convert the complex number 3 - 4i to polar form r(cosθ + i sinθ). Step 1: Find the modulus r. The modulus r is the distance from the origin to the point (3, -4) in the complex plane. Formula: r = sqrt(a^2 + b^2) where a = 3 and b = -4. So r = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5. Step 2: Find the argument θ. The argument θ is the angle the line from the origin to (3, -4) makes with the positive real axis. We can find it using: θ = arctan(b/a) = arctan(-4/3). Since the point (3, -4) is in the 4th quadrant (positive real, negative imaginary), the angle will be negative or between 270° and 360° in degrees. θ = arctan(-4/3) ≈ -0.9273 radians. We can verify this is correct because: cos(-0.9273) ≈ 0.6 = 3/5 sin(-0.9273) ≈ -0.8 = -4/5 These match our complex number when multiplied by r=5. Step 3: Write in polar form. The polar form is r(cosθ + i sinθ). Substituting our values: 5(cos(-0.9273) + i sin(-0.9273)). Therefore, the polar form of 3 - 4i is 5(cos(-0.9273) + i sin(-0.9273)).

4. Liam is analyzing an AC circuit where the voltage across a capacitor is represented by the complex number V = -5 + 12i volts. To determine the phase relationship between voltage and current, he needs to convert this voltage to polar form. What is the polar representation of this voltage in the form r∠θ, where r is the magnitude in volts and θ is the phase angle in degrees between -180° and 180°? Answer: 13∠112.6° Solution: To convert a complex number from rectangular to polar form, we calculate the magnitude using the Pythagorean theorem and determine the angle using inverse trigonometric functions. When using arctangent, we need to adjust for the correct quadrant since the standard arctangent function only gives…
   Full step-by-step solution

   To convert a complex number from rectangular to polar form, we calculate the magnitude using the Pythagorean theorem and determine the angle using inverse trigonometric functions. The angle's quadrant depends on the signs of both components - negative real and positive imaginary places the angle in the second quadrant. When using arctangent, we need to adjust for the correct quadrant since the standard arctangent function only gives results between -90° and 90°.

5. Convert the complex number -2 + 2i to polar form (r(cosθ + isinθ)) = ? Answer: 2√2(cos(3π/4) + isin(3π/4)) Solution: Calculate the modulus r = sqrt(a² + b²) = sqrt((-2)² + 2²) = sqrt(4 + 4) = sqrt(8) = 2√2 Find the angle θ using tanθ = b/a = 2/(-2) = -1 Since the complex number is in quadrant II (negative real, positive imaginary), θ = π - π/4 = 3π/4 Write in polar form: r(cosθ + isinθ) = 2√2(cos(3π/4) +…
   Full step-by-step solution

   Step 1: Calculate the modulus r = sqrt(a² + b²) = sqrt((-2)² + 2²) = sqrt(4 + 4) = sqrt(8) = 2√2 Step 2: Find the angle θ using tanθ = b/a = 2/(-2) = -1 Step 3: Since the complex number is in quadrant II (negative real, positive imaginary), θ = π - π/4 = 3π/4 Step 4: Write in polar form: r(cosθ + isinθ) = 2√2(cos(3π/4) + isin(3π/4)) The answer is 2√2(cos(3π/4) + isin(3π/4)).

6. Convert the complex number 5 - 7i to polar form r(cosθ + isinθ) = ? Answer: √74(cos(5.352) + isin(5.352)) Solution: Identify the real and imaginary parts: a = 5, b = -7 Calculate the modulus r = √(a² + b²) = √(5² + (-7)²) = √(25 + 49) = √74 Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(7/5) ≈ 0.9505 radians Determine the actual angle θ: Since the point (5, -7) is in Quadrant IV, θ = 2π - 0.9505 ≈…
   Full step-by-step solution

   Step 1: Identify the real and imaginary parts: a = 5, b = -7 Step 2: Calculate the modulus r = √(a² + b²) = √(5² + (-7)²) = √(25 + 49) = √74 Step 3: Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(7/5) ≈ 0.9505 radians Step 4: Determine the actual angle θ: Since the point (5, -7) is in Quadrant IV, θ = 2π - 0.9505 ≈ 6.2832 - 0.9505 = 5.3327 radians Step 5: Write in polar form: r(cosθ + isinθ) = √74(cos(5.3327) + isin(5.3327)) The answer is √74(cos(5.3327) + isin(5.3327)).

7. Convert the complex number 7 - 7√3i to polar form r(cosθ + isinθ) = ? Answer: 14(cos(5π/3) + isin(5π/3)) Solution: Identify the real and imaginary parts: a = 7, b = -7√3 Calculate the modulus r = √(a² + b²) = √(7² + (-7√3)²) = √(49 + 147) = √196 = 14 Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(7√3/7) = tan⁻¹(√3) = π/3 Determine the actual angle θ: Since the point (7, -7√3) is in Quadrant IV…
   Full step-by-step solution

   Step 1: Identify the real and imaginary parts: a = 7, b = -7√3 Step 2: Calculate the modulus r = √(a² + b²) = √(7² + (-7√3)²) = √(49 + 147) = √196 = 14 Step 3: Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(7√3/7) = tan⁻¹(√3) = π/3 Step 4: Determine the actual angle θ: Since the point (7, -7√3) is in Quadrant IV (positive real, negative imaginary), θ = 2π - π/3 = 5π/3 Step 5: Write in polar form: r(cosθ + isinθ) = 14(cos(5π/3) + isin(5π/3)) The answer is 14(cos(5π/3) + isin(5π/3)).

8. An electrical engineer is analyzing an alternating current circuit with a capacitor and inductor in series. The voltage across the capacitor is represented by the complex number -3 + 4i volts, while the voltage across the inductor is 2 - 6i volts. If these two components are in series, what is the total voltage across both components expressed in polar form (r∠θ) with the angle in degrees? Answer: √65∠-135° Solution: In electrical engineering, AC circuit analysis often uses complex numbers to represent voltages and currents.
   Full step-by-step solution

   In electrical engineering, AC circuit analysis often uses complex numbers to represent voltages and currents. The real part typically represents the resistive component, while the imaginary part represents the reactive component. When converting from rectangular form (a + bi) to polar form (r∠θ), the magnitude r is found using the Pythagorean theorem, and the angle θ is determined using inverse trigonometric functions, considering which quadrant the complex number lies in. This representation is particularly useful for understanding phase relationships in AC circuits.