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Mathematical Expression Editor

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

Polar coordinates

An ordered pair consisting of a radius and an angle can be graphed as
meaning:

Coordinates of this type are called 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?

All of these represent the origin, since represents the origin for any angle .

Plot the following points in polar coordinates:

It helps to use a “polar grid” to plot
these points:

To place the point , go out unit along the horizontal axis (putting you on the inner
circle shown on the grid), then rotate clockwisecounterclockwise radians (or ).

To plot , go out units along the horizontal axis and rotate radians ().

To plot , go out 2 units along the initial ray then rotate clockwisecounterclockwise radians, as the angle given is negative.

To plot , move along the initial ray “” units, in other words, “back up” unit, then
rotate clockwisecounterclockwise by .

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

Given a point in polar coordinates, rectangular coordinates are given by
Given a point in rectangular coordinates, polar coordinates are given by

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.”

Sketch the polar function on .

While one could make a table of values and plot them
in polar coordinates, it is often more useful to first set and then plot . This is what
we’ll do, starting with :

Sketch the polar function on .

While one could make a table of values and plot them
in polar coordinates, it is often more useful to first set and then plot . This is what
we’ll do, starting with :

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:

Convert from rectangular coordinates to polar coordinates.

Replace with and
replace with , giving:
We have found that . The domain of this polar function is . Plot a few points to see
how the familiar parabola is traced out by the polar equation.

Convert from rectangular coordinates to polar coordinates.

We again replace and
using the standard identities and work to solve for :
This function is valid only when the product of is positive. This occurs in the first
and third quadrants, meaning the domain of this polar function is with
.

We can rewrite the original rectangular equation as . Note it only exists in the first
and third quadrants.

Convert from polar coordinates to rectangular coordinates.

There is no set way to
convert from polar to rectangular; in general, we look to form the products and , and
then replace these with and , respectively. We start in this problem by multiplying
both sides by :
The original polar equation, does not easily reveal that its graph is simply a line.
However, our conversion shows that it is.

Convert from polar coordinates to rectangular coordinates.

By multiplying both
sides by , we obtain both an term and an term, which we replace with and ,
respectively.
We recognize this as a circle. By completing the square we can find its radius and
center.
The circle is centered at and has radius .