Coordinate Conversion

lessonbunny.com

Answer Key & Explanations

Coordinate Conversion · Grade 12 · Worksheet 3

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. 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.

3. 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).

4. A point in the rectangular coordinate system has 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 the rectangular coordinates (x, y) = (-4, 4√3) and need to convert to polar coordinates (r, θ) with r > 0 and 0 ≤ θ < 2π. Find r. The formula is r = √(x² + y²).
   Full step-by-step solution

   We are given the 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: Find r. The formula is r = √(x² + y²). x = -4, y = 4√3. x² = (-4)² = 16 y² = (4√3)² = 16 * 3 = 48 x² + y² = 16 + 48 = 64 r = √64 = 8 So r = 8. Step 2: Find θ. The formula is tan θ = y/x. y/x = (4√3)/(-4) = -√3. So tan θ = -√3. Step 3: Determine the correct quadrant. x = -4 (negative), y = 4√3 (positive). Negative x and positive y means the point is in Quadrant II. Step 4: Find the reference angle. 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. But let's check: tan(2π/3) = tan(120°) = -√3, yes. So θ = 2π/3 ≈ 2.094. Step 6: Verify the range. 2π/3 is between 0 and 2π, so it's valid. Final answer: θ = 2π/3 ≈ 2.094.

5. Convert the rectangular coordinates (-6, 6√3) to polar form (r, θ) where r > 0 and 0 ≤ θ < 2π. Answer: (12, 2π/3) Solution: Calculate r using r = sqrt(x^2 + y^2). Here x = -6 and y = 6√3. So r = sqrt((-6)^2 + (6√3)^2) = sqrt(36 + 108) = sqrt(144) = 12.
   Full step-by-step solution

   Step 1: Calculate r using r = sqrt(x^2 + y^2). Here x = -6 and y = 6√3. So r = sqrt((-6)^2 + (6√3)^2) = sqrt(36 + 108) = sqrt(144) = 12. Step 2: Calculate the reference angle using tan θ = y/x. tan θ = (6√3)/(-6) = -√3. The reference angle where tan = √3 is π/3. Step 3: Determine the quadrant. Since x = -6 (negative) and y = 6√3 (positive), the point lies in Quadrant II. In Quadrant II, the angle θ = π - reference angle = π - π/3 = 2π/3. Step 4: Write the polar coordinates as (r, θ) = (12, 2π/3). The answer is (12, 2π/3).

6. 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).

7. Convert the rectangular coordinates (4, 4√3) to polar form (r, θ) where θ is in radians and 0 ≤ θ < 2π. Answer: (8, π/3) Solution: Recall the formulas for conversion. r = sqrt(x^2 + y^2) θ = arctan(y/x) (but we must consider the quadrant) Calculate r. Here, x = 4 and y = 4√3.
   Full step-by-step solution

   Let's convert the rectangular coordinates (4, 4√3) to polar form (r, θ). Step 1: Recall the formulas for conversion. In polar coordinates: r = sqrt(x^2 + y^2) θ = arctan(y/x) (but we must consider the quadrant) Step 2: Calculate r. Here, x = 4 and y = 4√3. r = sqrt( (4)^2 + (4√3)^2 ) r = sqrt( 16 + (16 * 3) ) r = sqrt( 16 + 48 ) r = sqrt(64) r = 8 Step 3: Find the angle θ. First, compute the reference angle using the tangent ratio: tan(θ) = y/x = (4√3) / 4 = √3 We know that tan(θ) = √3 when θ = π/3 (or 60 degrees) in the first quadrant. Step 4: Determine the correct quadrant for θ. Our point is (4, 4√3). Since x = 4 (positive) and y = 4√3 (positive), the point is in the first quadrant. Therefore, the reference angle π/3 is already the correct angle. There is no need to adjust. Step 5: Write the polar coordinates. We have r = 8 and θ = π/3, with 0 ≤ θ < 2π. Final answer: (8, π/3)

8. Convert the rectangular coordinates (3, -4) to polar form (r, θ) where r > 0 and 0 ≤ θ < 2π. Answer: (5, 5.3559) Solution: Calculate r using the formula r = √(x² + y²) r = √(3² + (-4)²) = √(9 + 16) = √25 = 5 Calculate the reference angle θ' using θ' = arctan(|y/x|) θ' = arctan(|-4/3|) = arctan(4/3) ≈ 0.9273 radians Since x = 3 (positive) and y = -4 (negative), the point is in Quadrant IV In Quadrant IV: θ = 2π - θ'…
   Full step-by-step solution

   Step 1: Calculate r using the formula r = √(x² + y²) r = √(3² + (-4)²) = √(9 + 16) = √25 = 5 Step 2: Calculate the reference angle θ' using θ' = arctan(|y/x|) θ' = arctan(|-4/3|) = arctan(4/3) ≈ 0.9273 radians Step 3: Determine the correct quadrant for θ Since x = 3 (positive) and y = -4 (negative), the point is in Quadrant IV In Quadrant IV: θ = 2π - θ' = 2π - 0.9273 ≈ 6.2832 - 0.9273 = 5.3559 radians Step 4: Write the polar coordinates (r, θ) = (5, 5.3559) The answer is (5, 5.3559).