Polar Coordinates

Understanding Polar Coordinates

  • Polar coordinates, composed of a radius and angle, represent points in a two-dimensional plane.
  • They offer a unique way of mapping the plane using distance from the origin (the radius, r) and the angle measured anti-clockwise from the positive x-axis (the angle, θ, expressed in radians).
  • The origin in polar coordinates is represented as (0,θ) for all θ.
  • To convert from Cartesian coordinates (x,y) to Polar coordinates (r,θ), use the formulas:
    • r = √(x² + y²)
    • θ = atan(y/x)
  • To convert from Polar coordinates (r,θ) to Cartesian coordinates (x,y), use the formulas:
    • x = rcos(θ)
    • y = rsin(θ)

Polar Equations

  • Polar equations represent curves in polar coordinates by expressing r as a function of θ.
  • The polar equation r = a represents a circle of radius ‘a’ units centred at the origin.
  • The polar equation r = aθ represents a spiral with the pole as the initial point where the radius ‘r’ increases by ‘a’ units for each complete turn of the spiral.
  • The polar equation r = a/cos(θ - α) represents a straight line, where ‘α’ represents the inclination of the line and ‘a’ is the distance of the nearest point on the line from the pole.

Graphing Polar Equations

  • Graphing polar equations involves plotting different values of θ to find the corresponding radius, r. Each pair (r,θ) represents a point on the graph.
  • Tools like technology or graphing calculators can be very beneficial for graphing polar equations.

Polar Forms of Complex Numbers

  • Polar form of complex numbers connects polar coordinates with complex numbers using Euler’s formula.
  • The polar representation of a complex number z is: z = r(cosθ + isinθ) = re^(iθ).

Polar Calculus

  • Polar calculus involves differentiating and integrating polar functions.
  • To differentiate r with respect to θ, use the chain rule and product rule of differentiation.
  • To integrate polar functions, use the double-angle formula to convert the expression into a form that’s easier to work with.