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Coordinate Conversion

Grade 12 · Geometry · Worksheet 1

  1. 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: ______________
  2. Convert the rectangular coordinates (-7, 7√3) to polar form (r, θ) where r > 0 and 0 ≤ θ < 2π. Answer: ______________
  3. A radar station tracks an aircraft at point P on its screen, which is a Cartesian grid overlaid on the display. The aircraft is located at rectangular coordinates (-7, -7√3). 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. Answer: ______________
  4. Emma is programming an autonomous lawn mower that navigates using polar coordinates. The mower's current position relative to its charging station is at rectangular coordinates (-3√3, -3) meters. To calculate the mower's next navigation command, Emma 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 Emma use for the mower's current position? Answer: ______________
  5. Liam is designing a solar panel array and needs to position a support beam from the origin (0,0) to a critical connection point at coordinates (-4, 4√3) meters. To program the robotic arm that will install the beam, he needs to convert these rectangular coordinates to polar coordinates (r, θ), where r is the distance in meters and θ is the angle in radians measured from the positive x-axis. What are the polar coordinates for this connection point? Answer: ______________
  6. Liam is designing a solar panel array and needs to position a support pole at coordinates (4, -4√3) meters from the control center. To program the automated installation system, he 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 polar coordinates should Liam use for the support pole? Answer: ______________
  7. Convert (8, 6) to polar coordinates (r, θ) where r > 0 and 0 ≤ θ < 2π. Answer: ______________
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Answer Key & Explanations

Coordinate Conversion · Grade 12 · Worksheet 1

  1. 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, 5.3559) Solution: r = sqrt(x^2 + y^2) = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5. θ_ref = arctan(|y/x|) = arctan(|-4/3|) = arctan(4/3). This is not a standard exact angle, so we keep it as arctan(4/3) numerically for now.
    Full step-by-step solution

    Let's convert the rectangular coordinates (3, -4) to polar form (r, θ). Step 1: Calculate r (the magnitude or modulus) r = sqrt(x^2 + y^2) = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5. Step 2: Find the reference angle θ_ref θ_ref = arctan(|y/x|) = arctan(|-4/3|) = arctan(4/3). This is not a standard exact angle, so we keep it as arctan(4/3) numerically for now. Step 3: Determine the correct quadrant for θ The point (3, -4) has x > 0 and y < 0, so it is in Quadrant IV. In Quadrant IV, θ = 2π - θ_ref (since θ_ref is positive and measured from the positive x-axis). So θ = 2π - arctan(4/3). Step 4: Compute numerical value for θ First, arctan(4/3) ≈ 0.9273 radians. Then θ = 2π - 0.9273 ≈ 6.2832 - 0.9273 = 5.3559 radians. Step 5: Final polar coordinates (r, θ) = (5, 5.3559) Answer: (5, 5.3559)

  2. Convert the rectangular coordinates (-7, 7√3) to polar form (r, θ) where r > 0 and 0 ≤ θ < 2π. Answer: (14, 2π/3) Solution: Calculate r using r = √(x² + y²). r = √((-7)² + (7√3)²) = √(49 + 49·3) = √(49 + 147) = √196 = 14. Calculate the reference angle using tan θ = y/x.
    Full step-by-step solution

    Step 1: Calculate r using r = √(x² + y²). r = √((-7)² + (7√3)²) = √(49 + 49·3) = √(49 + 147) = √196 = 14. Step 2: Calculate the reference angle using tan θ = y/x. tan θ = (7√3)/(-7) = -√3. The reference angle where tan = √3 is π/3. Step 3: Determine the correct quadrant. x = -7 (negative), y = 7√3 (positive), so the point is in Quadrant II. In Quadrant II, θ = π - reference angle = π - π/3 = 2π/3. Step 4: Write the polar coordinates. (r, θ) = (14, 2π/3). The answer is (14, 2π/3).

  3. A radar station tracks an aircraft at point P on its screen, which is a Cartesian grid overlaid on the display. The aircraft is located at rectangular coordinates (-7, -7√3). 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. Answer: (14, 4π/3) Solution: Calculate the magnitude r using the formula r = sqrt(x^2 + y^2). Here x = -7 and y = -7√3. So r = sqrt((-7)^2 + (-7√3)^2) = sqrt(49 + 49 * 3) = sqrt(49 + 147) = sqrt(196) = 14.
    Full step-by-step solution

    Step 1: Calculate the magnitude r using the formula r = sqrt(x^2 + y^2). Here x = -7 and y = -7√3. So r = sqrt((-7)^2 + (-7√3)^2) = sqrt(49 + 49 * 3) = sqrt(49 + 147) = sqrt(196) = 14. Step 2: Find the reference angle using tan(θ_ref) = |y/x| = |(-7√3)/(-7)| = |√3| = √3. The reference angle with tan = √3 is π/3 (since tan(π/3) = √3). Step 3: Determine the actual angle θ based on the quadrant. Since x = -7 (negative) and y = -7√3 (negative), the point lies in Quadrant III. In Quadrant III, θ = π + θ_ref = π + π/3 = 4π/3. Step 4: Write the final polar coordinates: (r, θ) = (14, 4π/3).

  4. Emma is programming an autonomous lawn mower that navigates using polar coordinates. The mower's current position relative to its charging station is at rectangular coordinates (-3√3, -3) meters. To calculate the mower's next navigation command, Emma 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 Emma use for the mower's current position? Answer: (6, 7π/6) Solution: Calculate the radius r using the formula r = sqrt(x^2 + y^2) x = -3√3, y = -3 r = sqrt((-3√3)^2 + (-3)^2) r = sqrt(9*3 + 9) r = sqrt(27 + 9) r = sqrt(36) r = 6 Find the reference angle using tan(theta_ref) = |y/x| tan(theta_ref) = |-3 / (-3√3)| = |1/√3| = 1/√3 The reference angle is pi/6 (since…
    Full step-by-step solution

    Step 1: Calculate the radius r using the formula r = sqrt(x^2 + y^2) x = -3√3, y = -3 r = sqrt((-3√3)^2 + (-3)^2) r = sqrt(9*3 + 9) r = sqrt(27 + 9) r = sqrt(36) r = 6 Step 2: Find the reference angle using tan(theta_ref) = |y/x| tan(theta_ref) = |-3 / (-3√3)| = |1/√3| = 1/√3 The reference angle is pi/6 (since tan(pi/6) = 1/√3) Step 3: Determine the quadrant and find theta x is negative, y is negative, so the point lies in Quadrant III In Quadrant III: theta = pi + reference angle theta = pi + pi/6 = 6π/6 + π/6 = 7π/6 Step 4: Write the polar coordinates (r, θ) = (6, 7π/6) The answer is (6, 7π/6).

  5. Liam is designing a solar panel array and needs to position a support beam from the origin (0,0) to a critical connection point at coordinates (-4, 4√3) meters. To program the robotic arm that will install the beam, he needs to convert these rectangular coordinates to polar coordinates (r, θ), where r is the distance in meters and θ is the angle in radians measured from the positive x-axis. What are the polar coordinates for this connection point? Answer: (8, 2π/3) Solution: Calculate r (the distance from the origin to the point) The formula is: r = √(x² + y²) Here, x = -4 and y = 4√3. x² = (-4)² = 16 y² = (4√3)² = 16 * 3 = 48 So r = √(16 + 48) = √64 = 8 So r = 8 meters.
    Full step-by-step solution

    Let's find the polar coordinates (r, θ) for the rectangular coordinates (-4, 4√3). Step 1: Calculate r (the distance from the origin to the point) The formula is: r = √(x² + y²) Here, x = -4 and y = 4√3. x² = (-4)² = 16 y² = (4√3)² = 16 * 3 = 48 So r = √(16 + 48) = √64 = 8 So r = 8 meters. Step 2: Calculate θ (the angle from the positive x-axis) The formula is: θ = arctan(y/x) Here, y/x = (4√3)/(-4) = -√3 So θ = arctan(-√3) Step 3: Determine the correct quadrant for θ The point is at x = -4, y = 4√3. Since x is negative and y is positive, the point is in Quadrant II. Step 4: Find the reference angle arctan(√3) = π/3 (since tan(π/3) = √3) So the reference angle is π/3. Step 5: Adjust for the correct quadrant In Quadrant II, θ = π - (reference angle) So θ = π - π/3 = 2π/3 Step 6: Final polar coordinates (r, θ) = (8, 2π/3) Therefore, the polar coordinates for the connection point are (8, 2π/3).

  6. Liam is designing a solar panel array and needs to position a support pole at coordinates (4, -4√3) meters from the control center. To program the automated installation system, he 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 polar coordinates should Liam use for the support pole? Answer: (8, 5π/3) Solution: x = 4 y = -4√3 r = √(x² + y²) r = √(4² + (-4√3)²) r = √(16 + (16 × 3)) r = √(16 + 48) r = √64 r = 8 So r = 8.
    Full step-by-step solution

    Let's solve step by step. We are given rectangular coordinates: x = 4 y = -4√3 --- **Step 1: Find r (the radial distance)** The formula is: r = √(x² + y²) Substitute: r = √(4² + (-4√3)²) r = √(16 + (16 × 3)) r = √(16 + 48) r = √64 r = 8 So r = 8. --- **Step 2: Find θ using the tangent ratio** The formula is: tan θ = y / x Substitute: tan θ = (-4√3) / 4 tan θ = -√3 --- **Step 3: Determine the quadrant** x = 4 (positive) y = -4√3 (negative) So the point (4, -4√3) is in Quadrant IV. --- **Step 4: Find the reference angle** tan θ = √3 means the reference angle is π/3 (since tan(π/3) = √3). In Quadrant IV, θ = 2π - π/3 = 6π/3 - π/3 = 5π/3. --- **Step 5: Verify θ is between 0 and 2π** 5π/3 ≈ 5.236, which is between 0 and 2π (≈ 6.283). --- **Step 6: Final polar coordinates** (r, θ) = (8, 5π/3) --- ANSWER: (8, 5π/3)

  7. Convert (8, 6) to polar coordinates (r, θ) where r > 0 and 0 ≤ θ < 2π. Answer: (10, 0.6435) Solution: Calculate r using r = √(x² + y²) r = √(8² + 6²) = √(64 + 36) = √100 = 10 Calculate θ using θ = arctan(y/x) θ = arctan(6/8) = arctan(0.75) ≈ 0.6435 radians Since x = 8 > 0 and y = 6 > 0, the point is in Quadrant I, so θ = 0.6435 is correct.
    Full step-by-step solution

    Step 1: Calculate r using r = √(x² + y²) r = √(8² + 6²) = √(64 + 36) = √100 = 10 Step 2: Calculate θ using θ = arctan(y/x) θ = arctan(6/8) = arctan(0.75) ≈ 0.6435 radians Step 3: Verify the quadrant Since x = 8 > 0 and y = 6 > 0, the point is in Quadrant I, so θ = 0.6435 is correct. Step 4: Final answer (r, θ) = (10, 0.6435)