8.2: Graphs of Polar Equations


Simple Polar Graphs


If $r$ or $\theta$ are equal to a number, then it's easy to graph by hand.

If $r$ is a number, then $\theta$ will take on all possible angles!

Sketch a graph of the equation $r = 3$.

Conversely, if $\theta$ is a number, then $r$ can take on all possible distances!

Sketch a graph of the equation $\theta = \dfrac{\pi}{3}$. Convert it to a rectangular equation.

More Polar Graphs


Polar equations can be written in the form $r = f(\theta)$.

In English, this is "$r$ is a function of $\theta$," meaning, plug in an angle $\theta$ to determine how far you need to walk in that direction $r$.

A few examples are \[r = \theta \qquad \qquad r = 2\cos \theta \qquad \qquad r = 1 - \cos \theta\]

Graph $r = \cos \theta$ by hand on the interval $[0, \pi]$.

Here's an animation of the previous problem. Notice how the length $r$ is like an "arm" tracing out the graph:

Use Desmos to graph $r = \cos 2\theta$. Describe it's shape in English.

Most polar graphs in the form $r = f(\theta)$ are difficult to graph by hand. Use Desmos to help you.