We're used to describing points in \(\R^2\) by their \(x\)- and \(y\)-coordinates, but there are times when it's more convenient to describe points some other way. (This will be especially true when we start integrating functions of 2 variables soon.)
Think about how you might give directions to someone; you might say, “Walk 500 feet north and 200 feet east”, or you might point in a particular direction and say, “Walk in that direction for 300 feet.” The first description is like giving \(x\)- and \(y\)-coordinates (also known as Cartesian coordinates); the second is like giving polar coordinates. More precisely, to describe a point in polar coordinates, we describe how to get there from the origin by giving a direction and a distance \(r\) to go. Since \(r\) is a distance, it's always \(\geq 0\text{.}\)
How do we specify a direction? When talking with someone, you might use words like “northwest” to describe the direction shown below:
Mathematicians describe such directions by specifying the angle \(\theta\) the direction makes with the positive \(x\)-axis, measured counter-clockwise starting from the positive \(x\)-axis; that's the red angle shown below, which is \(\frac{3 \pi}{4}\) for this example:
Suppose a point has polar coordinates \((r, \theta) = \left( 2, \frac{\pi}{3} \right)\text{.}\) What are its Cartesian coordinates?
Graphically, the point is located here, 2 units from the origin:
From the right triangle we've drawn in, the point's \(x\)-value is \(2 \cos \frac{\pi}{3} = 1\text{,}\) and its \(y\)-value is \(2 \sin \frac{\pi}{3} = \sqrt{3}\text{.}\) Therefore, in Cartesian coordinates, the point is \((1, \sqrt{3})\text{.}\)
We see that, in general, polar coordinates \(r\) and \(\theta\) are related to Cartesian coordinates \(x\) and \(y\) by \(\boxed{x = r \cos \theta}\) and \(\boxed{y = r \sin \theta}\text{:}\)
In addition, by the Pythagorean Theorem, \(\boxed{r = \sqrt{x^2 + y^2}}\text{.}\)
Note: A point can be expressed in polar coordinates in more than one way. For example, the point \((r, \theta) = \left( 2, \frac{\pi}{3} \right)\) can also be expressed as \((r, \theta) = \left( 2, \frac{7 \pi}{3} \right)\text{.}\) As another example, the origin can be expressed in polar coordinates as \((r, \theta) = (0, \text{anything})\text{.}\)
We are familiar with describing curves in \(\R^2\) using equations like \(y = \sin x\text{.}\) We will often want to do the same thing with polar coordinates.
What do the following equations describe?
\(r = 2\)
\(\theta = - \frac{\pi}{3}\)
Let's look at each separately.
Since \(r\) represents distance to the origin, the equation \(r = 2\) describes all points that are 2 units from the origin. This is the circle of radius 2 centered at the origin.
The set of all points with \(\theta = -\frac{\pi}{3}\) forms half of a line, starting at the origin and extending forever in only one direction. (A “half-line” is called a ray.)
As the above example suggests, polar coordinates will be especially useful to us when we deal with circles. But let's also take a look at some more interesting shapes that have simple descriptions in polar coordinates.
What does the equation \(r = 2 + \sin(3 \theta)\) in polar coordinates describe?
Here's one way to picture such an equation in polar coordinates. First, graph the equation in the \(r \theta\)-plane:
(Although we restrict \(r\) to be \(\geq 0\text{,}\) \(\theta\) can have any value, so we could extend this graph to \(\theta < 0\) and \(\theta > 2 \pi \text{;}\) it simply repeats.) Now, to picture the equation in the regular \(xy\)-plane, picture wrapping the above graph circularly around the origin, so that \(\theta = 0\) stays where it is (along the positive part of the horizontal axis), the lines where \(r\) is a constant become circles, and \(\theta = 2 \pi\) wraps to be on the positive \(x\)-axis; play the video below to see that in action:
What you see at the end of the video is \(r = 2 + \sin(3 \theta)\text{.}\)
As you can imagine, it is quite difficult to write an equation for this shape in Cartesian coordinates.
Can you figure out what the equation \(r = \theta\) looks like?
We can use the same strategy as in the previous example; here's a video that shows that. The final result is \(r = \theta\text{:}\)
What does \(r = 6 \sin \theta\) look like?
We can try the same strategy as in the previous example. Since \(r\) must be non-negative, when we graph \(r = 6 \sin \theta\) in the \(r \theta\)-plane, we only care about \(\theta\) between \(0\) and \(\pi\text{:}\)
If you picture wrapping this around circularly as in the previous two examples, you should see that we get an oval-ish shape. But, unlike in the previous examples, it's actually easy to convert \(r = 6 \sin \theta\) to Cartesian coordinates, and that will give us a more precise picture of the curve.
Recall that polar and Cartesian coordinates are related by \(x = r \cos \theta\) and \(y = r \sin \theta\text{.}\) Although our equation \(r = 6 \sin \theta\) doesn't involve either, we can multiply both sides by \(r\text{:}\)
Now, this is easy to rewrite in polar coordinates:
\begin{align*} x^2 + y^2 \amp = 6y \\ x^2 + y^2 - 6y \amp = 0 \\ \end{align*}The key to figuring out what this looks like is to complete the square by adding 9 to both sides:
\begin{align*} x^2 + y^2 - 6y + 9 \amp = 9 \\ x^2 + (y - 3)^2 \amp = 9 \end{align*}Now, we can see that this describes the circle centered at \((0, 3)\) with radius \(3\text{:}\)
