Polar Coordinates

Polar coordinates

Polar coordinates have each point labelled by the ordered pair $(r,\theta )$, where $r$ is the distance from the origin (or pole) and $\theta$ is the angle from the initial line. By convention, the initial line is taken to be in the direction of the positive $x$-axis, and the angle is measured anticlockwise. $\theta$ is typically taken to be between $0$ and $2\pi$.

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

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

Features of polar curves

Minima and maxima As $r$ is a function of $\theta$, the minimum and maximum values of $r$ are when $\frac{\text{d}r}{\text{d}\theta }=0$. For curves where $r$ 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 $r=3\sin 2\theta$ below. The value of $r$ changes from positive to negative for $\theta =0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}$. As the curve approaches these values of $\theta$, $r$ gets closer to 0, so points on the curve get closer to the pole. Thus, each of the half-lines (in red) $\theta =0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}$ is tangent to the curve at the pole.

For a curve with polar equation $r=f(\theta )$, the line $\theta =a$ is a tangent at the pole if $f(a)=0$ and $f(a)>0$ 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 $(r,\theta )$ has Cartesian coordinates $(r\cos \theta ,r\sin \theta )$.
• A point with Cartesian coordinates $(x,y)$ has $r$ and $\theta$ satisfying $\sqrt{x^2+y^2}=r$ and $\tan \theta =\frac{y}{x}$. 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 $\theta =\alpha$ and $\theta =\beta$ is given by:

$${\int }_{\alpha }^{\beta }\frac{1}{2}{r}^2\text{d}\theta$$