Coordinate Conversion

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Answer Key & Explanations

Coordinate Conversion · Grade 12 · Worksheet 2

1. Liam is designing a navigation system for a drone that needs to fly from its current position at coordinates (4, -4√3) to a charging station located at the origin. To program the drone's flight path, he needs to convert the rectangular coordinates of its current position to polar coordinates (r, θ) where r ≥ 0 and 0 ≤ θ < 2π. What polar coordinates should Liam use for the drone's programming? Answer: (8, 5π/3) Solution: x = 4 y = -4√3 We need polar coordinates (r, θ) with r ≥ 0 and 0 ≤ θ < 2π. r = √(x² + y²) x² = 4² = 16 y² = (-4√3)² = 16 × 3 = 48 r = √(16 + 48) = √64 = 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 We need polar coordinates (r, θ) with r ≥ 0 and 0 ≤ θ < 2π. --- **Step 1: Calculate r** The formula is: r = √(x² + y²) x² = 4² = 16 y² = (-4√3)² = 16 × 3 = 48 So: r = √(16 + 48) = √64 = 8 So r = 8. --- **Step 2: Find the reference angle** The reference angle θ_ref is given by: θ_ref = arctan(|y/x|) |y/x| = | -4√3 / 4 | = | -√3 | = √3 So: θ_ref = arctan(√3) We know tan(π/3) = √3, so θ_ref = π/3. --- **Step 3: Determine the correct quadrant for θ** x = 4 (positive) y = -4√3 (negative) Positive x and negative y means the point is in Quadrant IV. In Quadrant IV: θ = 2π - θ_ref So: θ = 2π - π/3 = (6π/3 - π/3) = 5π/3. --- **Step 4: Verify** We have r = 8, 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 Matches the given coordinates. --- **Final answer:** (8, 5π/3)

2. Liam is designing a drone navigation system that uses polar coordinates. His drone is currently positioned at the point (4, -4√3) in the rectangular coordinate system. To program the drone's movement, he needs to convert this position to polar coordinates (r, θ) where r > 0 and 0 ≤ θ < 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. x² = 4² = 16 y² = (-4√3)² = (16 × 3) = 48 So r = √(16 + 48) = √64 = 8.
   Full step-by-step solution

   Let's convert the rectangular coordinates (4, -4√3) to polar coordinates (r, θ) with r > 0 and 0 ≤ θ < 2π. Step 1: Calculate r (the radial distance from the origin) The formula is r = √(x² + y²). Here x = 4, y = -4√3. x² = 4² = 16 y² = (-4√3)² = (16 × 3) = 48 So r = √(16 + 48) = √64 = 8. So r = 8. Step 2: Find the reference angle θ' We use tan(θ) = y/x = (-4√3)/4 = -√3. The reference angle θ' is the acute angle whose tangent is √3 (ignoring the negative sign for now). We know tan(π/3) = √3, so θ' = π/3. Step 3: Determine the correct quadrant for θ Since x = 4 (positive) and y = -4√3 (negative), the point is in Quadrant IV. In Quadrant IV, θ = 2π - θ' (because angles are measured counterclockwise from the positive x-axis). So θ = 2π - π/3 = (6π/3 - π/3) = 5π/3. Step 4: Verify the coordinates We can 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. This matches the given rectangular coordinates. Final answer: (8, 5π/3)

3. A navigation system tracks a ship's position on a radar screen. The ship is located at rectangular coordinates (12, -16) relative to the lighthouse at the origin, where coordinates are measured in kilometers. Convert these rectangular coordinates to polar coordinates (r, θ), where r ≥ 0 and θ is measured in radians between 0 and 2π. Express r as an exact value and θ as an exact radian measure. Answer: (20, 5.1760) or (20, 5.176) Solution: Calculate r using r = sqrt(x^2 + y^2). r = sqrt(12^2 + (-16)^2) = sqrt(144 + 256) = sqrt(400) = 20. Find the reference angle using tan(theta_ref) = |y/x| = |(-16)/12| = 16/12 = 4/3.
   Full step-by-step solution

   Step 1: Calculate r using r = sqrt(x^2 + y^2). r = sqrt(12^2 + (-16)^2) = sqrt(144 + 256) = sqrt(400) = 20. Step 2: Find the reference angle using tan(theta_ref) = |y/x| = |(-16)/12| = 16/12 = 4/3. So theta_ref = arctan(4/3). Step 3: Determine the quadrant. Since x = 12 > 0 and y = -16 < 0, the point is in Quadrant IV. In Quadrant IV, theta = 2π - theta_ref. Step 4: Compute theta. theta = 2π - arctan(4/3). Using a calculator, arctan(4/3) ≈ 0.9273 radians. So theta ≈ 2π - 0.9273 = 6.2832 - 0.9273 = 5.3559 radians. Step 5: Write the polar coordinates as (r, θ) = (20, 2π - arctan(4/3)) or approximately (20, 5.3559). The exact answer is (20, 2π - arctan(4/3)).

4. Convert (4√3, -4) to polar coordinates (r, θ) where r > 0 and 0 ≤ θ < 2π. Answer: (8, 11π/6) Solution: Calculate r using r = √(x² + y²) r = √((4√3)² + (-4)²) = √(48 + 16) = √64 = 8 Calculate θ using tanθ = y/x tanθ = (-4)/(4√3) = -1/√3 The reference angle is π/6 since tan(π/6) = 1/√3 Since x = 4√3 > 0 and y = -4 < 0, the point is in Quadrant IV θ = 2π - π/6 = 12π/6 - π/6 = 11π/6 (r, θ) = (8, 11π/6)
   Full step-by-step solution

   Step 1: Calculate r using r = √(x² + y²) r = √((4√3)² + (-4)²) = √(48 + 16) = √64 = 8 Step 2: Calculate θ using tanθ = y/x tanθ = (-4)/(4√3) = -1/√3 Step 3: Determine the reference angle The reference angle is π/6 since tan(π/6) = 1/√3 Step 4: Determine the correct quadrant Since x = 4√3 > 0 and y = -4 < 0, the point is in Quadrant IV Step 5: Find θ in Quadrant IV θ = 2π - π/6 = 12π/6 - π/6 = 11π/6 Step 6: Final polar coordinates (r, θ) = (8, 11π/6)

5. A point in the coordinate plane has rectangular coordinates (-4, 4√3). Convert this point to polar coordinates (r, θ) where r > 0 and 0 ≤ θ < 2π. What is the value of θ in radians? Answer: 2.094 Solution: We are given rectangular coordinates (x, y) = (-4, 4√3) and need to convert to polar coordinates (r, θ) with r > 0 and 0 ≤ θ < 2π. The formula is r = √(x² + y²).
   Full step-by-step solution

   We are given rectangular coordinates (x, y) = (-4, 4√3) and need to convert to polar coordinates (r, θ) with r > 0 and 0 ≤ θ < 2π. Step 1: Compute r The formula is r = √(x² + y²). x² = (-4)² = 16 y² = (4√3)² = 16 × 3 = 48 x² + y² = 16 + 48 = 64 r = √64 = 8 So r = 8. Step 2: Compute θ The formula is tan θ = y/x. Here, y/x = (4√3)/(-4) = -√3. So tan θ = -√3. Step 3: Determine the quadrant x = -4 (negative), y = 4√3 (positive). This means the point is in Quadrant II. Step 4: Find the reference angle We know tan θ = √3 when θ = π/3 (60°) in Quadrant I. So the reference angle is π/3. Step 5: Adjust for Quadrant II In Quadrant II, θ = π - π/3 = 2π/3. Let's check: 2π/3 radians is 120°, which is in Quadrant II, and tan(120°) = -√3, which matches. Step 6: Verify θ is between 0 and 2π 2π/3 ≈ 2.094, which is between 0 and 2π. Final answer: θ = 2π/3 ≈ 2.094. Thus, the polar coordinates are (8, 2π/3) and θ = 2.094.

6. Convert the rectangular coordinates (-5, 5) to polar form (r, θ) where θ is in radians and 0 ≤ θ < 2π. Answer: (5√2, 3π/4) Solution: Step 1: Calculate r using the formula r = √(x² + y²) r = √((-5)² + 5²) = √(25 + 25) = √50 = 5√2 Step 2: Calculate θ using the formula θ = arctan(y/x) θ = arctan(5/(-5)) = arctan(-1) = -π/4 Step 3: Adjust the angle based on the quadrant Since x = -5 and y = 5, the point is in Quadrant II In…
   Full step-by-step solution

   Step 1: Calculate r using the formula r = √(x² + y²) r = √((-5)² + 5²) = √(25 + 25) = √50 = 5√2 Step 2: Calculate θ using the formula θ = arctan(y/x) θ = arctan(5/(-5)) = arctan(-1) = -π/4 Step 3: Adjust the angle based on the quadrant Since x = -5 and y = 5, the point is in Quadrant II In Quadrant II, we add π to the angle: θ = -π/4 + π = 3π/4 Step 4: Verify the angle is in the range 0 ≤ θ < 2π 3π/4 is between 0 and 2π, so no further adjustment is needed Final answer: (5√2, 3π/4)

7. Convert the rectangular coordinates (-5, 5√3) to polar form (r, θ) where r > 0 and 0 ≤ θ < 2π. Answer: (10, 2π/3) Solution: Calculate r using the formula r = √(x² + y²) r = √((-5)² + (5√3)²) = √(25 + 25×3) = √(25 + 75) = √100 = 10 Calculate θ using the formula θ = arctan(y/x) θ = arctan((5√3)/(-5)) = arctan(-√3) Since x = -5 (negative) and y = 5√3 (positive), the point is in Quadrant II The reference angle is π/3…
   Full step-by-step solution

   Step 1: Calculate r using the formula r = √(x² + y²) r = √((-5)² + (5√3)²) = √(25 + 25×3) = √(25 + 75) = √100 = 10 Step 2: Calculate θ using the formula θ = arctan(y/x) θ = arctan((5√3)/(-5)) = arctan(-√3) Step 3: Determine the correct quadrant for θ Since x = -5 (negative) and y = 5√3 (positive), the point is in Quadrant II The reference angle is π/3 since tan(π/3) = √3 In Quadrant II, θ = π - π/3 = 2π/3 Step 4: Write the final polar coordinates (r, θ) = (10, 2π/3)

8. Convert the rectangular coordinates (6, -8) to polar form (r, θ) where r > 0 and 0 ≤ θ < 2π. Answer: (10, 5.1760) Solution: Step 1: Calculate r using the formula r = √(x² + y²) r = √(6² + (-8)²) = √(36 + 64) = √100 = 10 Step 2: Calculate the reference angle using θ = arctan(y/x) θ_ref = arctan(-8/6) = arctan(-4/3) ≈ -0.9273 radians Step 3: Determine the correct quadrant for θ Since x = 6 (positive) and y = -8…
   Full step-by-step solution

   Step 1: Calculate r using the formula r = √(x² + y²) r = √(6² + (-8)²) = √(36 + 64) = √100 = 10 Step 2: Calculate the reference angle using θ = arctan(y/x) θ_ref = arctan(-8/6) = arctan(-4/3) ≈ -0.9273 radians Step 3: Determine the correct quadrant for θ Since x = 6 (positive) and y = -8 (negative), the point is in Quadrant IV In Quadrant IV, θ = 2π + θ_ref = 2π - 0.9273 ≈ 5.3559 radians Step 4: Verify the angle is in the range 0 ≤ θ < 2π 5.3559 radians satisfies 0 ≤ θ < 2π Step 5: Write the final polar coordinates (r, θ) = (10, 5.3559) The answer is (10, 5.3559).