Introduction to Polar Coordinates

Overview

• Rectangular Coordinates: Uses X and Y.
• Polar Coordinates: Uses R (radius) and Theta (angle).

Plotting Polar Coordinates

1. Example: Plot (3, 45°)
   • R = 3, Theta = 45°
   • Draw circles with radii 1 to 3.
   • Ray at 45° reaches third circle.
2. Example: Plot (2, 3π/4)
   • Convert to degrees: 3π/4 * (180/π) = 135°.
   • Plot in Quadrant 2 with radius 2.*

Handling Negative R

• Example: (-2, 60°)
  • Plot (2, 60°) first in Quadrant 1.
  • Negative R inverts direction by 180°.

Finding Equivalent Points

• Find alternate polar coordinates that lead to the same point.
• General rule:
  • Given (R, Theta), find:
    1. (R, Theta - 360° or 2π)
    2. (-R, Theta + 180° or π)
    3. (-R, Theta - 360° + 180°)
    4. (R + 360°, Theta + 180°)
• Example: (2, 30°)
  • Alternate points: (2, -330°), (-2, 210°), (-2, -150°)

Conversion Between Coordinate Systems

Polar to Rectangular

• Equations:
  • X = R * cos(Theta)
  • Y = R * sin(Theta)
• Example: (4, 60°)
  • X = 4 * cos(60°) = 2
  • Y = 4 * sin(60°) = 2√3

Rectangular to Polar

• Equations:
  • R = sqrt(X² + Y²)
  • Theta = arctan(Y/X)
• Example: (2, -4)
  • R = sqrt(2² + (-4)²) = 2√5
  • Theta in Quadrant 4: 296.56°
• Example: (-5, 5√3)
  • R = 10
  • Theta = 120° or 2π/3 radians

Key Insights

• Positive angles measured counterclockwise from the positive x-axis.
• Negative R values indicate reflection across the origin.
• Conversion between coordinates requires understanding of both trigonometric functions and angle quadrant positioning.
• Often, a single polar coordinate can be expressed in multiple equivalent forms by adding or subtracting full rotations (360° or 2π).