Lecture Notes on Polar Coordinates

Introduction to Polar Coordinates

• Polar coordinate graph paper helps in graphing polar coordinates.
• Polar coordinates represent curves using angles instead of rectangular coordinates (x and y).
• Useful when graphing is easier or more relevant using polar coordinates.

Basics of Polar Coordinates

• Polar Axis: Replaces the x-axis.
• Pole (O): The origin or starting point.
• Angle (θ): The direction measured from the polar axis.
• Distance (R): Length of the segment from the pole to a point P, denoted as (R, θ).

Drawing Angles in Polar Coordinates

• Start at the polar axis, go up or down with an angle θ.
• Terminal Point (P): Defined by (R, θ).

Properties of R and θ

• Positive θ: Counterclockwise direction.
• Negative θ: Clockwise direction.
• R can be positive or negative:
  • Positive R: Segment extends out along the angle.
  • Negative R: Reflects 180 degrees in the opposite direction of the angle.

Examples and Graphing

• Use polar coordinate graph paper for plotting points.
• Example: (1, 2π/3) involves plotting in the second quadrant based on 2π/3 and extending to the first concentric circle for R=1.
• Examples with negative R involve reflecting across the origin.

Transition between Polar and Rectangular Coordinates

Converting Polar to Rectangular Coordinates

• X = R * cos(θ)
• Y = R * sin(θ)
• Use trigonometric identities to convert polar coordinates (R, θ) to rectangular (x, y).

Converting Rectangular to Polar Coordinates

• Use the Pythagorean identity: R² = x² + y²
• Tangent identity: tan(θ) = y/x
• Consider quadrant position to choose appropriate angle θ.

Symmetry in Polar Graphs

• Symmetry about polar axis: f(-θ) = f(θ)
• Symmetry about θ = π/2: f(π - θ) = f(θ)
• Symmetry about origin: f(θ + π) = -f(θ)

Graphing Polar Equations

• Identify symmetry to reduce work.
• Use a table of values for θ to calculate R.
• Account for symmetry to graph the full curve efficiently.

Examples of Polar Graphs

• Cardioid example: R = 1 + cos(θ)
• Use symmetry to graph from 0 to π or 0 to π/2, then reflect based on symmetry.
• Example of more complex polar graphs involving transformations and identities.

Polar Equation Derivatives

• Polar equations can act like parametric equations.
• Derivative formula for dy/dx in polar is derived using parametric approaches.

Finding Tangent Lines

• Derivative dy/dx is found using:
  • dy/dθ / dx/dθ
• Used to find horizontal and vertical tangents.
  • Horizontal tangent: dy/dθ = 0
  • Vertical tangent: dx/dθ = 0

This lecture provided a comprehensive understanding of polar coordinates, transformations between coordinate systems, and the calculus associated with polar equations.