Complex Numbers

lessonbunny.com

Answer Key & Explanations

Complex Numbers · Grade 12 · Worksheet 3

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 + 12i. To program the return flight path, 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: 13∠112.6° Solution: The polar form of a complex number represents it as a magnitude and an angle.
   Full step-by-step solution

   The polar form of a complex number represents it as a magnitude and an angle. The magnitude is found using the distance formula, and the angle is determined using inverse trigonometric functions, adjusted for the correct quadrant based on the signs of the real and imaginary parts. This form is useful in applications like navigation and electrical engineering for analyzing magnitude and direction.

2. Liam is designing a drone navigation system that uses complex numbers to represent positions. His drone needs to fly from its current position at 2 + 2i to a target position at -1 + 3i. To calculate the flight vector, he needs to find the difference between these complex numbers and express it in polar form (r∠θ) where r is the distance and θ is the angle measured in degrees from the positive real axis. What is the polar form of the flight vector? Answer: -3 + i Solution: Complex numbers can represent positions in a 2D plane, where the real part is the x-coordinate and the imaginary part is the y-coordinate.
   Full step-by-step solution

   Complex numbers can represent positions in a 2D plane, where the real part is the x-coordinate and the imaginary part is the y-coordinate. The vector between two points is found by subtracting the starting position from the ending position. To convert from rectangular to polar form, we calculate the magnitude (distance from origin) and angle (direction). The angle calculation requires careful consideration of the quadrant where the point lies, as the standard inverse tangent function only gives correct results for points in the first and fourth quadrants.

3. Mason is an audio engineer analyzing a sound wave captured by a microphone. The wave is represented by the complex number -8 - 8√3 i in the complex plane, where the real axis represents the in-phase component and the imaginary axis represents the quadrature component. To determine the amplitude and phase shift of the wave, Mason needs to convert this complex number to polar form r(cos θ + i sin θ), where r is the amplitude and θ is the phase angle in degrees between -180° and 180°. What is the polar representation of the sound wave? Answer: 16(cos(-120°) + i sin(-120°)) or 16∠-120° Solution: Identify the real part a = -8 and imaginary part b = -8√3. Find the magnitude r = sqrt(a^2 + b^2) = sqrt((-8)^2 + (-8√3)^2) = sqrt(64 + 64*3) = sqrt(64 + 192) = sqrt(256) = 16.
   Full step-by-step solution

   Step 1: Identify the real part a = -8 and imaginary part b = -8√3. Step 2: Find the magnitude r = sqrt(a^2 + b^2) = sqrt((-8)^2 + (-8√3)^2) = sqrt(64 + 64*3) = sqrt(64 + 192) = sqrt(256) = 16. Step 3: Find the reference angle φ: tan φ = |b/a| = |(-8√3)/(-8)| = √3, so φ = 60°. Step 4: Determine the quadrant: a < 0 and b < 0, so the point is in the third quadrant. The principal argument θ (between -180° and 180°) is θ = -180° + φ = -180° + 60° = -120°. Step 5: Write the polar form: 16(cos(-120°) + i sin(-120°)). Final answer: 16(cos(-120°) + i sin(-120°)) or 16∠-120°.

4. 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)).

5. On the complex plane, Mason draws a vector from the origin to a point with coordinates (-3√3, -3). Express this complex number in polar form r(cos θ + i sin θ), where r > 0 and θ is the principal argument in radians between 0 and 2π. Answer: 6(cos(7π/6) + i sin(7π/6)) Solution: Identify a = -3√3 and b = -3. Calculate the modulus r = sqrt(a² + b²) = sqrt((-3√3)² + (-3)²) = sqrt(9*3 + 9) = sqrt(27 + 9) = sqrt(36) = 6. Find the reference angle.
   Full step-by-step solution

   Step 1: Identify a = -3√3 and b = -3. Step 2: Calculate the modulus r = sqrt(a² + b²) = sqrt((-3√3)² + (-3)²) = sqrt(9*3 + 9) = sqrt(27 + 9) = sqrt(36) = 6. Step 3: Find the reference angle. tan(φ) = |b|/|a| = 3/(3√3) = 1/√3. So φ = π/6. Step 4: Since both coordinates are negative, the point lies in Quadrant III. The principal argument θ = π + φ = π + π/6 = 7π/6. Step 5: Write the polar form: z = 6(cos(7π/6) + i sin(7π/6)). The answer is 6(cos(7π/6) + i sin(7π/6)).

6. Convert the complex number -2 + 2√3i to polar form r(cosθ + isinθ) = ? Answer: 4(cos(2π/3) + isin(2π/3)) Solution: Find the modulus r = sqrt(a² + b²) where a = -2 and b = 2√3 r = sqrt((-2)² + (2√3)²) = sqrt(4 + 12) = sqrt(16) = 4 Find the angle θ using tanθ = b/a tanθ = (2√3)/(-2) = -√3 Since a = -2 (negative) and b = 2√3 (positive), the complex number is in Quadrant II Reference angle = arctan(√3) = π/3 θ =…
   Full step-by-step solution

   Step 1: Find the modulus r = sqrt(a² + b²) where a = -2 and b = 2√3 r = sqrt((-2)² + (2√3)²) = sqrt(4 + 12) = sqrt(16) = 4 Step 2: Find the angle θ using tanθ = b/a tanθ = (2√3)/(-2) = -√3 Step 3: Determine the correct quadrant Since a = -2 (negative) and b = 2√3 (positive), the complex number is in Quadrant II Step 4: Find the reference angle Reference angle = arctan(√3) = π/3 Step 5: Find the actual angle in Quadrant II θ = π - π/3 = 2π/3 Step 6: Write in polar form r(cosθ + isinθ) = 4(cos(2π/3) + isin(2π/3)) The answer is 4(cos(2π/3) + isin(2π/3))

7. Convert the complex number 10 - 10√3i to polar form r(cosθ + isinθ) = ? Answer: 20(cos(5π/3) + isin(5π/3)) Solution: Identify the real and imaginary parts: a = 10, b = -10√3 Calculate the modulus r = √(a² + b²) = √(10² + (-10√3)²) = √(100 + 300) = √400 = 20 Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(10√3/10) = tan⁻¹(√3) = π/3 Determine the actual angle θ: Since the point (10, -10√3) is in Quadrant…
   Full step-by-step solution

   Step 1: Identify the real and imaginary parts: a = 10, b = -10√3 Step 2: Calculate the modulus r = √(a² + b²) = √(10² + (-10√3)²) = √(100 + 300) = √400 = 20 Step 3: Calculate the reference angle: tan⁻¹(|b|/|a|) = tan⁻¹(10√3/10) = tan⁻¹(√3) = π/3 Step 4: Determine the actual angle θ: Since the point (10, -10√3) is in Quadrant IV (positive real, negative imaginary), θ = 2π - π/3 = 5π/3 Step 5: Write in polar form: r(cosθ + isinθ) = 20(cos(5π/3) + isin(5π/3)) The answer is 20(cos(5π/3) + isin(5π/3)).