Lecture Notes: Polar Coordinates

Overview

• Polar coordinates offer a different way to represent curves compared to rectangular coordinates.
• In polar coordinates, points are defined by a distance from the origin (r) and an angle (θ) from the polar axis.
• Useful for curves better represented by angles and radii.
• Often involves trigonometry.

Key Concepts

Polar Coordinate Components

• Polar Axis: Replaces the x-axis as a reference line.
• Origin (Pole): Point O, the center from where measurements start.
• Polar Coordinates: (r, θ) where r is the distance from the pole and θ is the angle.

Characteristics

• θ is positive in counterclockwise direction, negative in clockwise direction.
• r can be positive or negative; negative r involves reflection about the origin.

Graphing Polar Coordinates

• Use polar coordinate graph paper for accurate plotting.
• Plot points by moving outwards from the pole according to r and rotating by angle θ.
• Positive and negative values of r affect the direction of plotting.

Converting Between Coordinates

• Conversion between polar and rectangular coordinates uses trigonometric relationships.
• Rectangular to Polar: Use r² = x² + y² and θ = tan⁻¹(y/x).
• Polar to Rectangular: x = r cos(θ), y = r sin(θ).

Symmetry in Polar Graphs

Types of Symmetry

1. Polar Axis (X-Axis): If f(-θ) = f(θ), symmetric about the polar axis.
2. Pi/2 Axis (Y-Axis): If f(π - θ) = f(θ), symmetric about θ = π/2 axis.
3. Origin: If f(θ + π) = -f(θ), symmetric about the origin.

Graphing Techniques

Steps to Graph Polar Equations

1. Determine the type of equation (polar or rectangular).
2. Check for symmetry to reduce graphing work.
3. Create a table of values for θ and corresponding r.
4. Plot points and connect them smoothly.
5. Use symmetry to complete the graph.

Graph Examples

• Cardioid: r = 1 + cos(θ) or r = 1 + sin(θ), resembling a heart shape.
• Lemniscate: Complex shapes like r² = 2 cos(2θ).

Calculus with Polar Coordinates

Derivatives (dy/dx) in Polar Coordinates

• Use parametric-like approach: dy/dx = (dy/dθ) / (dx/dθ).
• Calculate dy/dθ and dx/dθ using product rule.

Finding Tangents

• Horizontal Tangents: Where dy/dθ = 0.
• Vertical Tangents: Where dx/dθ = 0.

Practical Use

• Calculate slope of tangent lines at given angles.
• Handle horizontal and vertical tangent lines for complete analysis.

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These notes cover the essential elements of polar coordinates, graphing methods, and calculus applications. They provide a structured reference to understand how curves can be represented and analyzed using angles and radii instead of x and y coordinates.