Coordinate Conversion

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Answer Key & Explanations

Coordinate Conversion · Grade 12 · Worksheet 3

1. Liam is analyzing the position of a drone flying over a city grid. The drone's coordinates are recorded as (4, -4√3) meters from the control center. To program the drone's return path using polar navigation, Liam needs to convert these rectangular coordinates to polar form (r, θ), where r ≥ 0 and θ is measured in radians between 0 and 2π. What are the polar coordinates of the drone's position? Answer: (8, 5π/3) Solution: The formula is: r = √(x² + y²) Here x = 4, y = -4√3 r = √(4² + (-4√3)²) r = √(16 + (16 × 3)) r = √(16 + 48) r = √64 r = 8 Find the reference angle (the angle in the first quadrant with the same x and y magnitudes) We use: tan(θ_ref) = |y| / |x| = (4√3) / 4 = √3 We know tan(θ_ref) = √3 when θ_ref…
   Full step-by-step solution

   Let's convert the rectangular coordinates (4, -4√3) to polar coordinates (r, θ). Step 1: Calculate r (the radial distance from the origin) The formula is: r = √(x² + y²) Here x = 4, y = -4√3 r = √(4² + (-4√3)²) r = √(16 + (16 × 3)) r = √(16 + 48) r = √64 r = 8 Step 2: Find the reference angle (the angle in the first quadrant with the same x and y magnitudes) We use: tan(θ_ref) = |y| / |x| = (4√3) / 4 = √3 We know tan(θ_ref) = √3 when θ_ref = π/3 radians (since tan(60°) = √3) So θ_ref = π/3 Step 3: Determine the correct quadrant for θ Our point is (4, -4√3), which means: x = 4 (positive) y = -4√3 (negative) This places the point in Quadrant IV (where x > 0, y < 0) Step 4: Find θ in Quadrant IV In Quadrant IV, θ = 2π - θ_ref θ = 2π - π/3 θ = 6π/3 - π/3 θ = 5π/3 Step 5: Verify our answer We have r = 8 and θ = 5π/3 Check: x = r cos θ = 8 cos(5π/3) = 8 × (1/2) = 4 ✓ y = r sin θ = 8 sin(5π/3) = 8 × (-√3/2) = -4√3 ✓ Therefore, the polar coordinates are (8, 5π/3).

2. Convert the rectangular coordinates (-2, 2) to polar form (r, θ) where r > 0 and 0 ≤ θ < 2π. Answer: (2√2, 3π/4) Solution: Step 1: Calculate r using the formula r = √(x² + y²) r = √((-2)² + (2)²) = √(4 + 4) = √8 = 2√2 Step 2: Calculate θ using the formula θ = arctan(y/x) θ = arctan(2/(-2)) = arctan(-1) = -π/4 Step 3: Since the point (-2, 2) is in Quadrant II, we need to add π to the angle θ = -π/4 + π = 3π/4 Step 4:…
   Full step-by-step solution

   Step 1: Calculate r using the formula r = √(x² + y²) r = √((-2)² + (2)²) = √(4 + 4) = √8 = 2√2 Step 2: Calculate θ using the formula θ = arctan(y/x) θ = arctan(2/(-2)) = arctan(-1) = -π/4 Step 3: Since the point (-2, 2) is in Quadrant II, we need to add π to the angle θ = -π/4 + π = 3π/4 Step 4: Verify that 3π/4 is between 0 and 2π 0 ≤ 3π/4 < 2π ✓ The polar coordinates are (2√2, 3π/4).

3. A point P is located in the complex plane at coordinates (3, 4). Convert this rectangular coordinate representation to polar form (r, θ), where r ≥ 0 and θ is measured in radians between 0 and 2π. Express your answer in exact form where possible. Answer: (5, arctan(4/3)) Solution: We are given the rectangular coordinates (3, 4) for point P in the complex plane. We want to convert this to polar form (r, θ), with r ≥ 0 and θ in radians between 0 and 2π.
   Full step-by-step solution

   We are given the rectangular coordinates (3, 4) for point P in the complex plane. We want to convert this to polar form (r, θ), with r ≥ 0 and θ in radians between 0 and 2π. --- **Step 1: Find r (the magnitude)** In polar coordinates, r is the distance from the origin to the point. For a point (x, y), r = sqrt(x^2 + y^2) Here, x = 3, y = 4. So: r = sqrt(3^2 + 4^2) r = sqrt(9 + 16) r = sqrt(25) r = 5 --- **Step 2: Find θ (the angle)** The angle θ satisfies: tan(θ) = y / x = 4 / 3 So θ = arctan(4/3) is one possible value, but we must check the quadrant to ensure θ is between 0 and 2π. The point (3, 4) is in the first quadrant because both x and y are positive. In the first quadrant, arctan(4/3) gives an angle between 0 and π/2, which is between 0 and 2π. So θ = arctan(4/3) is correct. --- **Step 3: Final polar coordinates** Thus, the polar form is: (r, θ) = (5, arctan(4/3)) --- **Final answer:** (5, arctan(4/3))

4. A marine biologist is tracking a dolphin's position using an underwater coordinate system. The dolphin is currently located at rectangular coordinates (-3, -3√3) meters from the research station. To program an autonomous underwater vehicle to intercept the dolphin, the biologist needs to convert these coordinates to polar form (r, θ) where r > 0 and θ is measured in radians between 0 and 2π. What polar coordinates should the biologist use? Answer: (6, 4π/3) Solution: Converting rectangular coordinates to polar coordinates involves finding the distance from the origin using the Pythagorean theorem and determining the angle using trigonometric functions. For points with negative x and y values, the angle falls between π and 3π/2 radians.
   Full step-by-step solution

   Converting rectangular coordinates to polar coordinates involves finding the distance from the origin using the Pythagorean theorem and determining the angle using trigonometric functions. The angle must be adjusted based on which quadrant the point lies in, since the arctangent function alone doesn't account for the full coordinate plane. For points with negative x and y values, the angle falls between π and 3π/2 radians.

5. Noah is an aerospace engineer designing a radar system for a new satellite. The satellite's current position relative to the ground station is given in rectangular coordinates as (-1, √3) kilometers. To calibrate the radar's polar tracking system, Noah must convert this position to polar coordinates (r, θ), where r > 0 and θ is measured in radians between 0 and 2π. What polar coordinates should Noah use? Answer: (2, 2π/3) Solution: Calculate the radius r using r = sqrt(x^2 + y^2). Here x = -1 and y = sqrt(3), so r = sqrt((-1)^2 + (sqrt(3))^2) = sqrt(1 + 3) = sqrt(4) = 2.
   Full step-by-step solution

   Step 1: Calculate the radius r using r = sqrt(x^2 + y^2). Here x = -1 and y = sqrt(3), so r = sqrt((-1)^2 + (sqrt(3))^2) = sqrt(1 + 3) = sqrt(4) = 2. Step 2: Find the reference angle using tan(theta_ref) = |y/x| = |sqrt(3)/(-1)| = sqrt(3). The reference angle is arctan(sqrt(3)) = pi/3. Step 3: Determine the quadrant. Since x = -1 (negative) and y = sqrt(3) (positive), the point is in Quadrant II. In Quadrant II, the polar angle is theta = pi - reference angle = pi - pi/3 = 2pi/3. Step 4: Write the final polar coordinates: (r, theta) = (2, 2pi/3). The polar coordinates Noah should use are (2, 2π/3).

6. A marine biologist is tracking a dolphin that surfaces at coordinates (-5, 5√3) meters from her research boat. To plot the dolphin's position on her polar coordinate tracking system, she needs to convert these rectangular coordinates to polar form (r, θ), where r is the distance in meters and θ is the angle in radians measured counterclockwise from the positive x-axis. What polar coordinates should she record for the dolphin's position? Answer: (10, 2π/3) Solution: Calculate the distance r using the formula r = sqrt(x² + y²) Substitute the coordinates: r = sqrt((-5)² + (5√3)²) = sqrt(25 + 75) = sqrt(100) = 10 Calculate the reference angle using tan⁻¹(|y/x|) = tan⁻¹(|5√3/5|) = tan⁻¹(√3) = π/3 Determine the actual angle θ based on the quadrant.
   Full step-by-step solution

   Step 1: Calculate the distance r using the formula r = sqrt(x² + y²) Step 2: Substitute the coordinates: r = sqrt((-5)² + (5√3)²) = sqrt(25 + 75) = sqrt(100) = 10 Step 3: Calculate the reference angle using tan⁻¹(|y/x|) = tan⁻¹(|5√3/5|) = tan⁻¹(√3) = π/3 Step 4: Determine the actual angle θ based on the quadrant. Since x is negative and y is positive, the point is in Quadrant II. Step 5: In Quadrant II, θ = π - reference angle = π - π/3 = 2π/3 Step 6: The polar coordinates are (10, 2π/3) The answer is (10, 2π/3).