
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, $(0,0)$, in $(x,y)$-coordinates?
$(0,0)$ $(0,\pi )$ $(0,-\pi )$

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

Let $P=(2,2\pi /3)$ be a point in polar coordinates. Describe $P$ in rectangular coordinates.
Let $Q=(-1,5\pi /4)$ be a point in polar coordinates. Describe $Q$ in rectangular coordinates.
Let $P=(1,2)$ be a point in rectangular coordinates. Describe $P$ in polar coordinates.
Let $Q=(-1,1)$ be a point in rectangular coordinates. Describe $Q$ in polar coordinates.

### Polar graphs

Let’s talk about how to plot polar functions. A polar function $r(\theta )$ 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 $r(\theta )$, it is often useful to first set $\theta = x$, and plot $y=r(x)$ 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 $y=f(x)$ in rectangular coordinates, polar coordinates are given by setting and solving for $r$.

Given a function $r(\theta )$ in polar coordinates, rectangular coordinates harder to find. The basic idea is to “find” $r\cdot \cos (\theta )$ and $r\cdot \sin (\theta )$ and write: Sometimes it is useful to remember that: