Polar Coordinates Study Notes

Polar Coordinates Overview

  • Use trigonometry to transform xy-coordinates into polar coordinates.
  • Polar coordinates are expressed as (r, θ).

Key Concepts

  • r: Radial distance from the origin.
  • θ: Angle measured from the positive x-axis.

Converting Between Coordinate Systems

Rectangular to Polar

  • Formulas for conversion:
    • r=x2+y2r = \sqrt{x^2 + y^2}
    • θ=tan⁡−1(yx)θ = \tan^{-1}\left( \frac{y}{x} \right)
  • Example: Convert (−3, 3) to polar.
  • Example: Convert (5√3, −5) to polar.

Polar to Rectangular

  • Formulas for conversion:
    • x=rcos⁡(θ)x = r \cos(θ)
    • y=rsin⁡(θ)y = r \sin(θ)
  • Example: Convert (2, π).
  • Example: Convert (−8, 3π/4).