Polar Coordinates

Polar coordinates

Polar coordinates have each point labelled by the ordered pair , where is the distance from the origin (or pole) and is the angle from the initial line. By convention, the initial line is taken to be in the direction of the positive -axis, and the angle is measured anticlockwise. is typically taken to be between and .

A polar equation of a curve is a relationship between and . Below are two examples of polar curves.

For some curves, there might be values for for which the is not defined. If is negative, then the curve is not defined. Some calculators still plot parts of the curve with negative , but for this course must be positive for the curve to be defined.

Features of polar curves

Minima and maxima
As is a function of , the minimum and maximum values of are when . For curves where is defined in terms of trigonometric functions, it may be easier to consider the graphs or ranges of the functions instead of differentiating.

Tangents at the pole
Consider the graph of below. The value of changes from positive to negative for . As the curve approaches these values of , gets closer to 0, so points on the curve get closer to the pole. Thus, each of the half-lines (in red) is tangent to the curve at the pole.

For a curve with polar equation , the line is a tangent at the pole if and on one side of the line.

Changing between polar and Cartesian

Trigonometry can be used to change between polar and Cartesian coordinates.:

A point with polar coordinates has Cartesian coordinates .
A point with Cartesian coordinates has and satisfying and .
This is effectively the same method as changing between the Cartesian and modulus-argument form of a complex number.

Area enclosed by a polar curve

The area between a polar curve and the half-lines and is given by: