Polar Coordinates

I. Introduction: Polar coordinates provide an alternative system for locating points in a plane. Unlike Cartesian coordinates (x, y), polar coordinates use a radial distance (r) and an angle (θ) to represent a point. Understanding polar coordinates is crucial in various fields, including physics, engineering, and mathematics.

II. Representation of Polar Coordinates:

  1. Basic Definitions:

    • Polar Axis: The reference line (usually horizontal) from which the angle is measured.
    • Pole (Origin): The point where the polar axis intersects the plane.
  2. Coordinate Notation:

    • A point in polar coordinates is represented as (r, θ), where r is the radial distance from the pole, and θ is the angle measured in a counterclockwise direction from the polar axis.

III. Conversion between Cartesian and Polar Coordinates:

  1. From Cartesian to Polar:

    • r=x2+y2
    • θ=arctan(yx)
  1. From Polar to Cartesian:

    • x=rcos(θ)
    • y=rsin(θ)

IV. Common Polar Graphs:

  1. Circle:

    • The equation of a circle in polar coordinates is r=a, where a is the radius.
  2. Line:

    • The equation of a line through the pole is θ=α, where α is the angle with the polar axis.
  3. Rose Curves:

    • r=acos(nθ) or r=asin(nθ), where a is a constant and n is an integer.

V. Polar Equations for Conic Sections:

  1. Circle:

    • r=a
  2. Ellipse:

    • r=a(1e2)1ecos(θ), where e is the eccentricity.
  3. Hyperbola:

    • r=a(1+e2)1+ecos(θ), where e is the eccentricity.

VI. Area and Length in Polar Coordinates:

  1. Area:

    • The area A enclosed by a polar curve r=f(θ) between two angles α and β is given by A=12αβ[f(θ)]2dθ.
  2. Arc Length:

    • The length Lof a polar curve r=f(θ) between two angles α and β is given by L=αβ[r(θ)]2+(drdθ)2dθ.