Coordinate Conversion

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Answer Key & Explanations

Coordinate Conversion · Grade 12 · Worksheet 1

1. A marine biologist is tracking a dolphin that surfaces at position (5, -5√3) meters relative to her research boat. To plot the dolphin's location on her polar coordinate tracking system, she 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 she record for the dolphin's position? Answer: (10, 5π/3) Solution: Calculate the radius r using the formula r = √(x² + y²) Given x = 5 and y = -5√3 r = √(5² + (-5√3)²) = √(25 + 25×3) = √(25 + 75) = √100 = 10 Find the reference angle using tan(θ) = y/x tan(θ) = (-5√3)/5 = -√3 Since x = 5 (positive) and y = -5√3 (negative), the point is in Quadrant IV The…
   Full step-by-step solution

   Step 1: Calculate the radius r using the formula r = √(x² + y²) Given x = 5 and y = -5√3 r = √(5² + (-5√3)²) = √(25 + 25×3) = √(25 + 75) = √100 = 10 Step 2: Find the reference angle using tan(θ) = y/x tan(θ) = (-5√3)/5 = -√3 Step 3: Determine the correct quadrant Since x = 5 (positive) and y = -5√3 (negative), the point is in Quadrant IV Step 4: Find the angle in Quadrant IV The reference angle where tan(θ) = √3 is π/3 In Quadrant IV, θ = 2π - π/3 = 6π/3 - π/3 = 5π/3 Step 5: Write the polar coordinates The polar coordinates are (r, θ) = (10, 5π/3) The answer is (10, 5π/3).

2. An engineer is designing a robotic arm that moves in a rectangular coordinate system. The arm's end effector needs to reach a point at coordinates (-5, 5√3) centimeters from the base. To program the arm's rotational movement, she needs to convert these rectangular coordinates to polar coordinates (r, θ) where r ≥ 0 and 0 ≤ θ < 2π. What polar coordinates should the engineer use? Answer: (10, 2π/3) Solution: Calculate the radius r using the formula r = √(x² + y²) r = √((-5)² + (5√3)²) = √(25 + 75) = √100 = 10 Find the reference angle using the formula tan⁻¹(|y/x|) Reference angle = tan⁻¹(|5√3/5|) = tan⁻¹(√3) = π/3 Determine the actual angle θ based on the quadrant Since x = -5 (negative) and y = 5√3…
   Full step-by-step solution

   Step 1: Calculate the radius r using the formula r = √(x² + y²) r = √((-5)² + (5√3)²) = √(25 + 75) = √100 = 10 Step 2: Find the reference angle using the formula tan⁻¹(|y/x|) Reference angle = tan⁻¹(|5√3/5|) = tan⁻¹(√3) = π/3 Step 3: Determine the actual angle θ based on the quadrant Since x = -5 (negative) and y = 5√3 (positive), the point is in Quadrant II In Quadrant II: θ = π - reference angle = π - π/3 = 2π/3 Step 4: Verify the angle is in the correct range 2π/3 is between 0 and 2π, so it's valid Step 5: Write the final polar coordinates (r, θ) = (10, 2π/3) The answer is (10, 2π/3).

3. Liam is designing a drone navigation system that uses polar coordinates. His drone is currently at the rectangular coordinates (4, -4√3) meters relative to its home base. To program the drone's return 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, 5π/3) Solution: We start with rectangular coordinates (x, y) = (4, -4√3). r = √(x² + y²) x² = 4² = 16 y² = (-4√3)² = (16 × 3) = 48 r = √(16 + 48) = √64 = 8 Thus r = 8.
   Full step-by-step solution

   Let's go step-by-step. We start with rectangular coordinates (x, y) = (4, -4√3). --- **Step 1: Find r (the radial distance)** The formula is: r = √(x² + y²) Substitute: x² = 4² = 16 y² = (-4√3)² = (16 × 3) = 48 So: r = √(16 + 48) = √64 = 8 Thus r = 8. --- **Step 2: Find θ using the tangent ratio** The formula is: tan θ = y / x Here: y / x = (-4√3) / 4 = -√3 So: tan θ = -√3 --- **Step 3: Determine the correct quadrant** x = 4 (positive) y = -4√3 (negative) So the point is in Quadrant IV. --- **Step 4: Find the reference angle** We know tan θ = -√3 in magnitude means the reference angle is π/3 because tan(π/3) = √3. So in Quadrant IV, the angle is given by: θ = 2π - π/3 = 6π/3 - π/3 = 5π/3 --- **Step 5: Check the range** We have 0 ≤ θ < 2π, and 5π/3 is about 300°, which is in Quadrant IV, matching our coordinates. --- **Step 6: Verify coordinates** Polar coordinates (r, θ) = (8, 5π/3) should give rectangular coordinates: 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 (4, -4√3). --- **Final answer:** (8, 5π/3)

4. A point is located at coordinates (3, 4) 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. Answer: (5, 0.9273) Solution: To convert the rectangular coordinates (3, 4) to polar form (r, θ), we follow these steps: Calculate the radial distance r.
   Full step-by-step solution

   To convert the rectangular coordinates (3, 4) to polar form (r, θ), we follow these steps: Step 1: Calculate the radial distance r. The formula for r is: r = sqrt(x^2 + y^2) Substitute x = 3 and y = 4: r = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 So, r = 5. Step 2: Calculate the angle θ. The formula for θ is: θ = arctan(y/x) Substitute x = 3 and y = 4: θ = arctan(4/3) Using a calculator, arctan(4/3) ≈ 0.9273 radians. This value is between 0 and π/2, which is correct for a point in the first quadrant (since both x and y are positive). Step 3: Verify the quadrant. Our point (3, 4) has x > 0 and y > 0, so it is in the first quadrant. The angle we calculated (0.9273 radians) is already in the correct range of 0 to 2π and lies in the first quadrant, so no adjustment is needed. Therefore, the polar coordinates are (r, θ) = (5, 0.9273). Final Answer: (5, 0.9273)

5. 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).

6. Convert the rectangular coordinates (-9, 9√3) to polar form (r, θ) where r > 0 and 0 ≤ θ < 2π. Answer: (18, 2π/3) Solution: Calculate r using r = sqrt(x^2 + y^2). Here x = -9 and y = 9√3. So r = sqrt((-9)^2 + (9√3)^2) = sqrt(81 + 81*3) = sqrt(81 + 243) = sqrt(324) = 18.
   Full step-by-step solution

   Step 1: Calculate r using r = sqrt(x^2 + y^2). Here x = -9 and y = 9√3. So r = sqrt((-9)^2 + (9√3)^2) = sqrt(81 + 81*3) = sqrt(81 + 243) = sqrt(324) = 18. Step 2: Find the reference angle using tan θ = y/x = (9√3)/(-9) = -√3. The reference angle whose tangent is √3 is π/3. Step 3: Since x is negative and y is positive, the point lies in Quadrant II. In Quadrant II, the polar angle θ = π - reference angle = π - π/3 = 2π/3. Step 4: Write the polar coordinates: (r, θ) = (18, 2π/3).

7. A complex number is represented on the complex plane at the point (3, 4). Convert this rectangular coordinate representation to polar form (r, θ), where r is the magnitude and θ is the angle in radians measured counterclockwise from the positive real axis. Express your answer in exact form. Answer: (5, arctan(4/3)) Solution: We are given the complex number at the point (3, 4) in rectangular coordinates.
   Full step-by-step solution

   We are given the complex number at the point (3, 4) in rectangular coordinates. We want to convert this to polar form (r, θ), where r is the magnitude and θ is the angle in radians measured counterclockwise from the positive real axis. --- **Step 1: Find the magnitude r** The magnitude r is the distance from the origin to the point (3, 4). We use the Pythagorean theorem: r = sqrt(3^2 + 4^2) r = sqrt(9 + 16) r = sqrt(25) r = 5 So the magnitude is 5. --- **Step 2: Find the angle θ** The angle θ is given by the arctangent of the ratio of the imaginary part to the real part: θ = arctan(y / x) = arctan(4 / 3) We must check the quadrant: The point (3, 4) is in the first quadrant (both x and y positive), so arctan(4/3) is already the correct principal value between 0 and π/2. No adjustment to the angle is needed. --- **Step 3: Write the polar form** The polar form is (r, θ) = (5, arctan(4/3)). Since the problem says "exact form", we leave it as arctan(4/3) rather than a decimal. --- **Final answer:** (5, arctan(4/3))

8. Convert (8, -6) to polar coordinates (r, θ) where r > 0 and 0 ≤ θ < 2π. Answer: (10, 5.8195) Solution: Step 1: Calculate r using the formula r = √(x² + y²) r = √(8² + (-6)²) = √(64 + 36) = √100 = 10 Step 2: Calculate θ using the formula θ = arctan(y/x) θ = arctan(-6/8) = arctan(-0.75) ≈ -0.6435 radians Step 3: Since the point (8, -6) is in Quadrant IV (positive x, negative y), we need to add 2π…
   Full step-by-step solution

   Step 1: Calculate r using the formula r = √(x² + y²) r = √(8² + (-6)²) = √(64 + 36) = √100 = 10 Step 2: Calculate θ using the formula θ = arctan(y/x) θ = arctan(-6/8) = arctan(-0.75) ≈ -0.6435 radians Step 3: Since the point (8, -6) is in Quadrant IV (positive x, negative y), we need to add 2π to get an angle between 0 and 2π θ = -0.6435 + 2π ≈ -0.6435 + 6.2832 = 5.6397 radians Step 4: Verify the angle is in the correct range: 0 ≤ 5.6397 < 2π ✓ Final answer: (10, 5.6397)