Coordinate Conversion

Grade 12 · geometry · 100 practice problems · read aloud

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🔁 Coordinate Conversion

This operation involves switching between different coordinate systems, primarily between Rectangular (Cartesian) coordinates (x, y) and Polar coordinates (r, θ). It's crucial for simplifying complex problems in calculus, physics, and engineering where circular or rotational motion is involved.

Step-by-Step Guide

  1. Rectangular to Polar:
    • r = √(x² + y²) (Find the hypotenuse)
    • θ = tan⁻¹(y/x) (Find the angle, minding the quadrant!)
  2. Polar to Rectangular:
    • x = r cos θ
    • y = r sin θ

Visual Examples

Example 1: Rectangular to Polar

Convert (3, 4) to polar coordinates.

  1. r = √(3² + 4²) = √(9 + 16) = √25 = 5
  2. θ = tan⁻¹(4/3) ≈ 53.13° or 0.927 rad

Result: (5, 53.13°)

Example 2: Polar to Rectangular

Convert (2, π/3) to rectangular coordinates.

  1. x = 2 * cos(π/3) = 2 * (1/2) = 1
  2. y = 2 * sin(π/3) = 2 * (√3/2) = √3

Result: (1, √3)

⚠️ Common Mistakes

  • Quadrant Blindness: The calculator's tan⁻¹(y/x) only gives correct angles for Quadrant I and IV. Always sketch the point to determine the correct quadrant for θ.
  • Radians vs. Degrees: Ensure your calculator is in the correct mode (usually radians for Grade 12 math).
  • Negative r: While uncommon, a negative 'r' value reflects the point through the origin.

💡 Tips & Tricks

  • Memory Aid: For Polar to Rectangular, remember "Cos = Adjacent / Hypotenuse" so x = r cos θ.
  • Quick Check: After conversion, verify your point is in the correct quadrant on a sketch.
  • Simplify First: If θ is a common angle (π/6, π/4, π/3), use exact trigonometric values instead of decimals.

Practice Suggestions

  • Start by converting points in all four quadrants to master quadrant adjustment.
  • Practice converting equations between forms (e.g., the circle x² + y² = 9 becomes the simple r = 3 in polar form).
  • Use online graphing tools to plot your original and converted coordinates to visually confirm they are the same point.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 Convert (8, 6) to polar coordinates (r, θ) where r > 0 and 0 ≤ θ < 2π.

Hint: Recall that polar coordinates use distance from origin and angle from positive x-axis. The formulas involve the Pythagorean theorem and inverse trigonometric functions.

Show the answer

Answer: (10, 0.6435)

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

2 Convert (5, -5) to polar coordinates (r, θ) where r > 0 and 0 ≤ θ < 2π.

Hint: To convert rectangular coordinates (x, y) to polar coordinates (r, θ), use the formulas r = √(x² + y²) and θ = arctan(y/x), but be careful to determine the correct quadrant for θ.

Show the answer

Answer: (5√2, 7π/4)

  1. Calculate r using r = √(x² + y²) r = √(5² + (-5)²) = √(25 + 25) = √50 = 5√2
  2. Calculate θ using θ = arctan(y/x) θ = arctan(-5/5) = arctan(-1) = -π/4
  3. Adjust θ to be in the range 0 ≤ θ < 2π Since the point (5, -5) is in Quadrant IV, we add 2π to the negative angle: θ = -π/4 + 2π = 7π/4
  4. Write the final polar coordinates (r, θ) = (5√2, 7π/4)

3 Convert (8, -6) to polar coordinates (r, θ) where r > 0 and 0 ≤ θ < 2π.

Hint: Remember that polar coordinates use distance from origin and angle from positive x-axis. Consider which quadrant the point lies in when determining the correct angle.

Show the answer

Answer: (10, 5.8195)

  1. Calculate r using the formula r = √(x² + y²) r = √(8² + (-6)²) = √(64 + 36) = √100 = 10
  2. Calculate θ using the formula θ = arctan(y/x) θ = arctan(-6/8) = arctan(-0.75) ≈ -0.6435 radians
  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
  4. Verify the angle is in the correct range: 0 ≤ 5.6397 < 2π ✓ Final answer: (10, 5.6397)

4 Convert (3, -4) to polar coordinates (r, θ) where r > 0 and 0 ≤ θ < 2π.

Hint: To convert rectangular coordinates to polar form, first calculate the radius using the Pythagorean theorem, then determine the angle using inverse trigonometric functions, adjusting for the correct quadrant.

Show the answer

Answer: (5, 5.3559)

  1. Calculate r using r = sqrt(x² + y²) r = sqrt(3² + (-4)²) = sqrt(9 + 16) = sqrt(25) = 5
  2. Calculate θ using θ = arctan(y/x) θ = arctan(-4/3) ≈ -0.9273 radians
  3. Adjust θ to be in the range 0 ≤ θ < 2π Since the point (3, -4) is in Quadrant IV, we add 2π to the negative angle: θ = -0.9273 + 2π ≈ -0.9273 + 6.2832 = 5.3559 radians
  4. Write the final polar coordinates (r, θ) = (5, 5.3559)

5 Convert (4, -4√3) to polar coordinates (r, θ) where r > 0 and 0 ≤ θ < 2π.

Hint: Remember that polar coordinates use distance from origin and angle from positive x-axis. Consider which quadrant the point lies in to determine the correct angle.

Show the answer

Answer: (8, 5π/3)

  1. Calculate r using the formula r = √(x² + y²) r = √(4² + (-4√3)²) = √(16 + 48) = √64 = 8
  2. Calculate θ using the formula θ = arctan(y/x) θ = arctan((-4√3)/4) = arctan(-√3)
  3. Determine the correct quadrant Since x = 4 (positive) and y = -4√3 (negative), the point is in Quadrant IV
  4. Find the reference angle arctan(√3) = π/3, so the reference angle is π/3
  5. Adjust for the correct quadrant In Quadrant IV, θ = 2π - π/3 = 6π/3 - π/3 = 5π/3
  6. Final polar coordinates (r, θ) = (8, 5π/3)

6 Convert (4√3, -4) to polar coordinates (r, θ) where r > 0 and 0 ≤ θ < 2π.

Hint: Recall that polar coordinates use distance from origin and angle from positive x-axis. The formulas involve the Pythagorean theorem and inverse trigonometric functions, paying attention to the quadrant to determine the correct angle.

Show the answer

Answer: (8, 11π/6)

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