Coordinate Conversion

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Answer Key & Explanations

Coordinate Conversion · Grade 12 · Worksheet 2

1. A point P is located at coordinates (-5, 5√3) on the Cartesian plane. Convert this rectangular coordinate 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: (10, 2π/3) Solution: Calculate r using the formula r = sqrt(x^2 + y^2) r = sqrt((-5)^2 + (5√3)^2) = sqrt(25 + 75) = sqrt(100) = 10 Find the reference angle using tan(θ_ref) = |y/x| θ_ref = arctan(|5√3/(-5)|) = arctan(√3) = π/3 The point (-5, 5√3) is in Quadrant II (negative x, positive y) In Quadrant II, θ = π -…
   Full step-by-step solution

   Step 1: Calculate r using the formula r = sqrt(x^2 + y^2) r = sqrt((-5)^2 + (5√3)^2) = sqrt(25 + 75) = sqrt(100) = 10 Step 2: Find the reference angle using tan(θ_ref) = |y/x| θ_ref = arctan(|5√3/(-5)|) = arctan(√3) = π/3 Step 3: Determine the correct quadrant and angle θ The point (-5, 5√3) is in Quadrant II (negative x, positive y) In Quadrant II, θ = π - θ_ref = π - π/3 = 2π/3 Step 4: Write the polar coordinates The polar coordinates are (r, θ) = (10, 2π/3)

2. Liam is designing a drone navigation system that uses polar coordinates. His drone is currently at position (4, 4√3) in rectangular coordinates. To program the drone's flight path, he needs to convert this position to polar coordinates (r, θ) where r ≥ 0 and 0 ≤ θ < 2π. What polar coordinates should Liam use for his drone's current position? Answer: (8, π/3) Solution: Recall the formulas for polar coordinates. r = √(x² + y²) θ = arctan(y/x) (but we must choose the correct quadrant) Calculate r.
   Full step-by-step solution

   Let's convert the rectangular coordinates (4, 4√3) to polar coordinates (r, θ). Step 1: Recall the formulas for polar coordinates. For a point (x, y): r = √(x² + y²) θ = arctan(y/x) (but we must choose the correct quadrant) Step 2: Calculate r. x = 4, y = 4√3 r = √(4² + (4√3)²) r = √(16 + 16 * 3) r = √(16 + 48) r = √64 r = 8 Step 3: Determine the angle θ. First, compute tan(θ) = y/x = (4√3)/4 = √3. So θ = arctan(√3). Step 4: Identify the reference angle. We know tan(π/3) = √3, so the reference angle is π/3. Step 5: Determine the correct quadrant. The point is (4, 4√3). Since x = 4 > 0 and y = 4√3 > 0, the point is in Quadrant I. In Quadrant I, the angle is simply the reference angle. So θ = π/3. Step 6: Final polar coordinates. (r, θ) = (8, π/3) This matches the given correct answer.

3. A drone is conducting a search pattern over a rectangular field. The drone's current position relative to its control station is at Cartesian coordinates (-4, 4√3) meters. The search coordinator needs to convert this position to polar coordinates to program the drone's next movement. What are the polar coordinates (r, θ) of the drone's position, with r measured in meters and θ measured in radians between 0 and 2π? Answer: (8, 2π/3) Solution: x = -4 y = 4√3 r = √(x² + y²) r = √( (-4)² + (4√3)² ) r = √( 16 + (16 × 3) ) r = √( 16 + 48 ) r = √64 r = 8 So r = 8 meters.
   Full step-by-step solution

   Let's solve step by step. We are given Cartesian coordinates: x = -4 y = 4√3 --- **Step 1: Calculate r (the radial distance)** The formula is: r = √(x² + y²) Substitute: r = √( (-4)² + (4√3)² ) r = √( 16 + (16 × 3) ) r = √( 16 + 48 ) r = √64 r = 8 So r = 8 meters. --- **Step 2: Calculate θ using the tangent formula** The formula is: θ = arctan(y/x) Substitute: θ = arctan( (4√3) / (-4) ) θ = arctan( -√3 ) --- **Step 3: Determine the correct quadrant** x = -4 (negative) y = 4√3 (positive) This means the point is in **Quadrant II**. In Quadrant II, the arctan function (which returns values between -π/2 and π/2) will give a negative angle because y/x is negative. arctan(-√3) = -π/3 radians. To get the Quadrant II angle, add π: θ = -π/3 + π θ = (-π/3 + 3π/3) θ = 2π/3 radians. --- **Step 4: Check if θ is between 0 and 2π** 2π/3 ≈ 2.094, which is between 0 and 2π. --- **Final Answer:** (r, θ) = (8, 2π/3)

4. 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.

5. Liam is designing a satellite dish that needs to be positioned using polar coordinates. The dish's receiver is located at the rectangular coordinates (4, -4√3) meters. To properly align the dish with the satellite, Liam needs to convert these coordinates to polar form (r, θ) where r > 0 and 0 ≤ θ < 2π. What are the polar coordinates of the receiver? Answer: (8, 5π/3) Solution: Recall the conversion formulas from rectangular coordinates (x, y) to polar coordinates (r, θ). r = sqrt(x^2 + y^2) θ = arctan(y/x) (but we must choose the correct quadrant) Identify x and y from the given rectangular coordinates.
   Full step-by-step solution

   Step 1: Recall the conversion formulas from rectangular coordinates (x, y) to polar coordinates (r, θ). We have: r = sqrt(x^2 + y^2) θ = arctan(y/x) (but we must choose the correct quadrant) Step 2: Identify x and y from the given rectangular coordinates. x = 4 y = -4√3 Step 3: Calculate r. r = sqrt( (4)^2 + (-4√3)^2 ) = sqrt( 16 + (16 * 3) ) = sqrt( 16 + 48 ) = sqrt(64) = 8 Step 4: Find the reference angle. First compute tan(θ) = y/x = (-4√3) / 4 = -√3. The reference angle is the angle whose tangent is √3, which is π/3. Step 5: Determine the correct quadrant for θ. Since x = 4 (positive) and y = -4√3 (negative), the point is in Quadrant IV. Step 6: Find θ in Quadrant IV. In Quadrant IV, θ = 2π - reference angle. So θ = 2π - π/3 = (6π/3 - π/3) = 5π/3. Step 7: Write the polar coordinates. We have r = 8 and θ = 5π/3, with 0 ≤ θ < 2π. Thus the polar coordinates are (8, 5π/3). Final answer: (8, 5π/3)

6. Convert the rectangular coordinates (3, 4) to polar coordinates (r, θ) where r > 0 and 0 ≤ θ < 2π. Answer: (5, 0.9273) Solution: To convert rectangular coordinates (3, 4) to polar coordinates (r, θ), we use the formulas: r = sqrt(x^2 + y^2) θ = arctan(y/x) r = sqrt(3^2 + 4^2) r = sqrt(9 + 16) r = sqrt(25) r = 5 θ = arctan(4/3) θ = arctan(1.3333...) θ ≈ 0.9273 radians Since x = 3 (positive) and y = 4 (positive), the point…
   Full step-by-step solution

   To convert rectangular coordinates (3, 4) to polar coordinates (r, θ), we use the formulas: r = sqrt(x^2 + y^2) θ = arctan(y/x) Step 1: Calculate r r = sqrt(3^2 + 4^2) r = sqrt(9 + 16) r = sqrt(25) r = 5 Step 2: Calculate θ θ = arctan(4/3) θ = arctan(1.3333...) Using a calculator, we find: θ ≈ 0.9273 radians Step 3: Verify the quadrant Since x = 3 (positive) and y = 4 (positive), the point is in the first quadrant. The angle we calculated (0.9273 radians) is already in the first quadrant, so no adjustment is needed. Step 4: Final answer The polar coordinates are (5, 0.9273), where r = 5 and θ ≈ 0.9273 radians. Note: 0.9273 radians is approximately 53.13 degrees, which makes sense for a 3-4-5 right triangle.

7. Convert the rectangular coordinates (3, 4) to polar coordinates (r, θ) where θ is in radians and 0 ≤ θ < 2π. Answer: (5, 0.9273) Solution: To convert rectangular coordinates (3, 4) to polar coordinates (r, θ), we use the formulas: r = sqrt(x^2 + y^2) θ = arctan(y/x) r = sqrt(3^2 + 4^2) r = sqrt(9 + 16) r = sqrt(25) r = 5 θ_ref = arctan(4/3) θ_ref = arctan(1.3333) θ_ref ≈ 0.9273 radians Since x = 3 (positive) and y = 4 (positive),…
   Full step-by-step solution

   To convert rectangular coordinates (3, 4) to polar coordinates (r, θ), we use the formulas: r = sqrt(x^2 + y^2) θ = arctan(y/x) Step 1: Calculate r r = sqrt(3^2 + 4^2) r = sqrt(9 + 16) r = sqrt(25) r = 5 Step 2: Calculate θ First, find the reference angle using arctan: θ_ref = arctan(4/3) θ_ref = arctan(1.3333) θ_ref ≈ 0.9273 radians Step 3: Determine the correct quadrant for θ Since x = 3 (positive) and y = 4 (positive), the point is in the first quadrant. In the first quadrant, θ = θ_ref, so: θ ≈ 0.9273 radians Step 4: Verify the angle is in the correct range 0.9273 radians is between 0 and 2π, so no adjustment is needed. Therefore, the polar coordinates are (5, 0.9273).

8. Convert the rectangular coordinates (-5, 5√3) to polar form (r, θ) where θ is in radians and 0 ≤ θ < 2π. Answer: (10, 2π/3) 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 Since x = -5 (negative) and y = 5√3 (positive), the…
   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 Since x = -5 (negative) and y = 5√3 (positive), the point is in Quadrant II Step 4: Find the reference angle arctan(√3) = π/3 Step 5: Adjust for Quadrant II In Quadrant II, θ = π - π/3 = 2π/3 Step 6: Verify the angle is in the range 0 ≤ θ < 2π 2π/3 is approximately 2.094, which satisfies 0 ≤ θ < 2π The polar coordinates are (10, 2π/3).