Polar coordinates are coordinates based on an angle and a radius.

Polar coordinates

Polar coordinates are great for certain situations. However, there is a price to pay. Every point in the plane has more than one of description in polar coordinates.

Which of the following represent the origin, , in -coordinates?

It is useful to recognize both the rectangular (or, Cartesian) coordinates of a point in the plane and its polar coordinates.

Let be a point in polar coordinates. Describe in rectangular coordinates.
Let be a point in polar coordinates. Describe in rectangular coordinates.
Let be a point in rectangular coordinates. Describe in polar coordinates.
Let be a point in rectangular coordinates. Describe in polar coordinates.
We’ll tell you the angle, you think about the radius.

Polar graphs

Let’s talk about how to plot polar functions. A polar function corresponds to the parametric function:

However, if you are sketching a polar function by hand, there are some tricks that can help. If you want to sketch , it is often useful to first set , and plot in rectangular coordinates. Let’s just work examples. It is my belief that “doing things” is better than “describing.”

Converting to and from polar coordinates

It is sometimes desirable to refer to a graph via a polar equation, and other times by a rectangular equation. Therefore it is necessary to be able to convert between polar and rectangular functions. Here is the basic idea:

Given a function in rectangular coordinates, polar coordinates are given by setting and solving for .

Given a function in polar coordinates, rectangular coordinates harder to find. The basic idea is to “find” and and write: Sometimes it is useful to remember that: