Introduction to Polar Coordinates

Key Topics

• Understanding Polar vs Rectangular Coordinates
• Plotting Polar Coordinates
• Handling Negative R Values
• Finding Equivalent Polar Coordinates
• Converting Polar to Rectangular Coordinates
• Converting Rectangular to Polar Coordinates

Polar vs Rectangular Coordinates

• Rectangular Coordinates: Use (x, y) format.
• Polar Coordinates: Use (r, θ) format, where:
  • r = radius (distance from origin)
  • θ = angle from positive x-axis

Plotting Polar Coordinates

1. Example: Plot (3, 45°)

   • Plot circles with radii 1, 2, 3.
   • Draw ray at 45° angle to intersect the third circle.
   • Positive angle measured counterclockwise from x-axis.

2. Example: Plot (2, 3π/4)

   • Convert to degrees: 3π/4 = 135°.
   • 135° places point in second quadrant.
   • Stop at second circle for radius of 2.

Handling Negative R Values

• Example: Plot (-2, 60°)
  • Plot positive (2, 60°) first.
  • For (-2, 60°), move 180° opposite the direction of positive angle.
  • Result is equivalent to (2, 240°).

Finding Equivalent Polar Coordinates

• Example: (2, 30°)
  • Original: (2, 30°)
  • Negative angle: (2, -330°)
  • Negative r, positive angle: (-2, 210°)
  • Negative r, negative angle: (-2, -150°)

Converting Polar to Rectangular Coordinates

• Formulas:
  • x = r * cos(θ)
  • y = r * sin(θ)
• Example: Convert (4, 60°)
  • x = 4 * cos(60°) = 2
  • y = 4 * sin(60°) = 2√3

Converting Rectangular to Polar Coordinates

1. Formulas:
   • r = √(x² + y²)
   • θ = arctan(y/x)
2. Example: Convert (2, -4)
   • r = √(2² + (-4)²) = √20 = 2√5
   • θ (reference) = arctan(4/2) = 63.43°
   • Actual θ for Quadrant 4 = 360 - 63.43 = 296.57°

Summary

• Understanding and converting between polar and rectangular coordinates involves using trigonometric identities and recognizing quadrant positions.
• Negative r values require 180° adjustments.
• There are multiple equivalent representations for any polar coordinate, based on angle and r sign adjustments.