Introduction to Polar Coordinates

Overview

• Polar Coordinates differ from Rectangular Coordinates
  • Rectangular: (x, y)
  • Polar: (r, θ) where:
    • r = radius of the circle
    • θ = angle measured from the positive x-axis

Graphing Polar Coordinates

Example 1: Plotting (3, 45 degrees)

• Plot 3 circles with radii: 1, 2, 3
• Draw a ray at a 45-degree angle
• The point is on the circle with radius 3
• Positive angles measured counterclockwise from the x-axis

Example 2: Plotting (2, 3π/4)

• Convert 3π/4 to degrees: 135 degrees
• Understand degree locations:
  • 0, 90, 180, 270 degrees
  • 135 degrees in Quadrant 2
• Angle 135 degrees, radius 2

Example 3: Plotting Negative r

• Plot (-2, 60 degrees)
• Plot (2, 60 degrees) first
  • 60 degrees in Quadrant 1
  • Radius 2
• Negative r: Travel 180 degrees opposite direction
  • (-2, 60 degrees) equivalent to (2, 240 degrees)

Example 4: Finding Equivalent Points

• Plot (-3, 120 degrees)
• Equivalent to (3, -60 degrees)
• Rules for negative angles:
  • Negative angle = clockwise direction
• Calculate other equivalent points
  • (Negative r, Positive θ) and vice versa

Conversion Between Polar and Rectangular Coordinates

Polar to Rectangular

• Equations:
  • x = r cos(θ)
  • y = r sin(θ)
• Example:
  • (4, 60 degrees) -> x = 2, y = 2√3
• Example:
  • (6, 5π/6) -> x = -3√3, y = 3

Rectangular to Polar

• Equations:
  • r = √(x² + y²)
  • θ = arctan(y/x)
• Example:
  • (2, -4) -> r = 2√5, θ ≈ 296.56 degrees
• Example:
  • (-5, 5√3) -> r = 10, θ = 120 degrees or 2π/3 radians

Practice Problems

• Find equivalent polar coordinates for given points
  • Use addition/subtraction of 180 or 360 degrees
• Convert between coordinate systems using trigonometric relationships

Conclusion

• Understanding polar coordinates involves:
  • Graphing points based on radius and angle
  • Converting between polar and rectangular systems
  • Finding equivalent points using angle manipulations