Complex Numbers

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Answer Key & Explanations

Complex Numbers · Grade 12 · Worksheet 1

1. Liam is designing a drone navigation system that uses complex numbers to represent positions. His drone starts at the origin and flies to a point represented by the complex number -5 + 5√3 i. He needs to convert this position to polar form (r∠θ) where r is the distance from the origin and θ is the angle measured in degrees from the positive real axis, with -180° < θ ≤ 180°. What is the polar form of the drone's position? Answer: 10∠120° Solution: When converting complex numbers to polar form, the magnitude represents the distance from the origin, while the angle indicates the direction. The angle depends on which quadrant the point is located in - points with negative real parts and positive imaginary parts are in the second quadrant,…
   Full step-by-step solution

   When converting complex numbers to polar form, the magnitude represents the distance from the origin, while the angle indicates the direction. The angle depends on which quadrant the point is located in - points with negative real parts and positive imaginary parts are in the second quadrant, where angles range from 90° to 180°. The reference angle is always measured from the nearest x-axis.

2. Convert the complex number -4 - 4i to polar form r(cosθ + isinθ) = ? Answer: 4√2(cos(5π/4) + isin(5π/4)) Solution: Calculate the modulus r = √(a² + b²) = √((-4)² + (-4)²) = √(16 + 16) = √32 = 4√2 Determine the angle θ using tanθ = b/a = (-4)/(-4) = 1 Since both real and imaginary parts are negative, the complex number lies in the third quadrant The reference angle is π/4, so the actual angle is π + π/4 =…
   Full step-by-step solution

   Step 1: Calculate the modulus r = √(a² + b²) = √((-4)² + (-4)²) = √(16 + 16) = √32 = 4√2 Step 2: Determine the angle θ using tanθ = b/a = (-4)/(-4) = 1 Step 3: Since both real and imaginary parts are negative, the complex number lies in the third quadrant Step 4: The reference angle is π/4, so the actual angle is π + π/4 = 5π/4 Step 5: Write in polar form: r(cosθ + isinθ) = 4√2(cos(5π/4) + isin(5π/4))

3. Olivia is a sound engineer analyzing a complex audio signal. The signal's amplitude and phase at a certain frequency are represented by the complex number -5√2 + 5√2 i. To calibrate the sound system, she needs to express this signal in polar form r(cos θ + i sin θ), where r is the magnitude and θ is the phase angle in degrees between 0° and 360°. What is the polar form of this complex signal? Answer: 10(cos 135° + i sin 135°) Solution: Identify the real and imaginary parts. The complex number is -5√2 + 5√2 i, so a = -5√2 and b = 5√2. Calculate the magnitude r.
   Full step-by-step solution

   Step 1: Identify the real and imaginary parts. The complex number is -5√2 + 5√2 i, so a = -5√2 and b = 5√2. Step 2: Calculate the magnitude r. r = sqrt(a^2 + b^2) = sqrt((-5√2)^2 + (5√2)^2) = sqrt(25*2 + 25*2) = sqrt(50 + 50) = sqrt(100) = 10. Step 3: Determine the angle θ. Since a is negative and b is positive, the point lies in the second quadrant. The reference angle is arctan(|b/a|) = arctan((5√2)/(5√2)) = arctan(1) = 45°. In the second quadrant, θ = 180° - 45° = 135°. Step 4: Write the polar form. r(cos θ + i sin θ) = 10(cos 135° + i sin 135°). The answer is 10(cos 135° + i sin 135°).

4. Emma is an electrical engineer designing a filter for a radio receiver. The impedance of a circuit component is given by the complex number Z = -5√3 + 5i ohms. To analyze the phase shift introduced by this component, she must convert this impedance to polar form. What is the polar representation of Z in the form r(cosθ + i sinθ), where θ is in degrees between 0° and 360°? Answer: 10(cos150° + i sin150°) Solution: Identify the real part a = -5√3 and the imaginary part b = 5. Calculate the magnitude r = sqrt(a² + b²) = sqrt((-5√3)² + 5²) = sqrt(75 + 25) = sqrt(100) = 10.
   Full step-by-step solution

   Step 1: Identify the real part a = -5√3 and the imaginary part b = 5. Step 2: Calculate the magnitude r = sqrt(a² + b²) = sqrt((-5√3)² + 5²) = sqrt(75 + 25) = sqrt(100) = 10. Step 3: Find the reference angle α = arctan(|b|/|a|) = arctan(5 / (5√3)) = arctan(1/√3) = arctan(√3/3) = 30°. Step 4: Since a < 0 and b > 0, the point lies in the second quadrant. The angle θ = 180° - α = 180° - 30° = 150°. Step 5: Write the polar form: Z = 10(cos150° + i sin150°). Answer: 10(cos150° + i sin150°)

5. Convert the complex number 1 - √3i to polar form r(cosθ + isinθ) = ? Answer: 2(cos(5π/3) + isin(5π/3)) Solution: Identify the real and imaginary parts: a = 1, b = -√3 Calculate the modulus r = √(a² + b²) = √(1² + (-√3)²) = √(1 + 3) = √4 = 2 Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(√3/1) = π/3 Determine the actual angle θ: Since the point (1, -√3) is in Quadrant IV, θ = 2π - π/3 = 5π/3 Write in…
   Full step-by-step solution

   Step 1: Identify the real and imaginary parts: a = 1, b = -√3 Step 2: Calculate the modulus r = √(a² + b²) = √(1² + (-√3)²) = √(1 + 3) = √4 = 2 Step 3: Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(√3/1) = π/3 Step 4: Determine the actual angle θ: Since the point (1, -√3) is in Quadrant IV, θ = 2π - π/3 = 5π/3 Step 5: Write in polar form: r(cosθ + isinθ) = 2(cos(5π/3) + isin(5π/3)) The answer is 2(cos(5π/3) + isin(5π/3)).

6. 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)).

7. A complex number z is represented as a point in the complex plane with coordinates (3, 4). Express z in polar form, r(cos θ + i sin θ), where r is the modulus and θ is the argument in radians between 0 and 2π. Answer: 5(cos 0.9273 + i sin 0.9273) Solution: We are given the complex number z with coordinates (3, 4) in the complex plane. z = 3 + 4i The modulus r is the distance from the origin to the point (3, 4).
   Full step-by-step solution

   We are given the complex number z with coordinates (3, 4) in the complex plane. That means: z = 3 + 4i --- **Step 1: Find the modulus r** The modulus r is the distance from the origin to the point (3, 4). Formula: r = sqrt(x^2 + y^2) So: r = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 Thus: r = 5 --- **Step 2: Find the argument θ** The argument θ is the angle measured counterclockwise from the positive real axis to the point (3, 4). Since x = 3 > 0 and y = 4 > 0, the point is in the first quadrant. We use: θ = arctan(y/x) So: θ = arctan(4/3) We compute: arctan(4/3) ≈ 0.9273 radians (since 4/3 ≈ 1.3333, and arctan(1.3333) is about 0.9273). Because the point is in the first quadrant, θ is exactly this value, between 0 and π/2. --- **Step 3: Write in polar form** Polar form: z = r(cos θ + i sin θ) Substitute r = 5 and θ ≈ 0.9273: z = 5(cos 0.9273 + i sin 0.9273) --- **Final Answer:** 5(cos 0.9273 + i sin 0.9273)

8. Convert the complex number -2 + 2i to polar form r(cosθ + isinθ) = ? Answer: 2√2(cos(3π/4) + isin(3π/4)) Solution: Calculate the magnitude r = √(a² + b²) = √((-2)² + 2²) = √(4 + 4) = √8 = 2√2 Calculate 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) + isin(3π/4))…
   Full step-by-step solution

   Step 1: Calculate the magnitude r = √(a² + b²) = √((-2)² + 2²) = √(4 + 4) = √8 = 2√2 Step 2: Calculate 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)).