3.1 - Introduction to Polar Coordinates
Contents
Objectives:
- Convert from polar coordinates to Cartesian coordinates in either direction.
- Describe (simple) regions in the plane using polar coordinate inequalities.
- Definitions
- polar coordinates
Lesson Guide
What are Polar Coordinates?
Today we will be introducing polar coordinates. Polar coordinates are really just formalizing a concept we have worked with since the beginning of class - namely, identifying points in the plane based on an angle $\theta$, and a distance $r$ from the origin (i.e., the radius of the circle they lie on).
Cartesian coordinates identify a point by how far left or right it lies from the origin (the $x$-coordinate), and how far above or below the origin it is (the $y$-coordinate). To contrast, polar coordinates are given as an ordered pair, $(r,\theta)$, and identify a point as being a certain distance from the origin ($r$), and corresponding to an angle from the positive $x$-axis ($\theta$, assumed to be in radians unless otherwise specified).
Give an example, having students help you find the Cartesian coordinates of a point given its polar coordinates, such as the one below. A picture will be helpful for students. You may want to hold off till Problem 6 to show conversion the other way.
The figure below shows the point $P$, whose polar coordinates are given by $(2.5,\theta)$. To find the Cartesian coordinates, we draw a right triangle as shown, and note that $\sin(pi/3)=y/2.5$ and $\cos(pi/3)=x/2.5$. Thus, the Cartesian coordinates of the point $P$ are given by $(1.25,1.25\sqrt{3})$.
Have students work through Problems 1 and 2 on the worksheet. Problem 2 emphasizes that there are multiple names for a given polar coordinate -- this will be useful to refer back to in the inverse trig sections.
Curves Described Using Polar Equations
We will not focus on this, but you can tell students that, as with Cartesian coordinates, we can plot curves using polar coordinates. While many curves can be achieved through altering trigonometric functions, we will focus here only on lines and circles, as a lead-up to the sorts of polar regions we wish to describe using inequalities (see below). Give several examples, such as:
-Example 1: The polar equation $r=1$ produces a circle of radius $1$:
-Example 2: The polar equation $\theta=\pi/8$ produces a ray through the origin (since $r$ must be nonnegative), making an angle of size $\pi/8$ with the positive $x$-axis:
Describing Regions with Polar Inequalities
We can also write polar inequalities to describe regions in the plane: sometimes, this is not easily done with Cartesian coordinates. You might show students some examples such as the following (similar to Problems 3 and 4) to illustrate this.
- In the shaded region below, $\theta$ is restricted to be between $\pi/2$ and $2\pi/3$ (inclusive), while within that $\theta$ range, all positive $r$ values are allowed. Thus, we can define the shaded region with the polar inequalities:
$\pi/2 \leq \theta \leq 2\pi/3$ and $0\leq r <\infty$.
- In the shaded region below, $\theta$ is unrestricted, but $r$ is (strictly) less than $4$. One possible set of polar inequalities describing the region is given below (note that we could have different $\theta$ ranges -- it's worth discussing this with your students).
$0 \leq \theta \leq 2\pi$ and $0\leq r \leq 4$.
Have students work through Problems 3 and 4. Problems 5 and 6 walks students through converting from Cartesian coordinates back to polar coordinates. Make sure to emphasize that they now know how to convert in both directions.
Comments
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.
One way to introduce the idea of polar coordinates to students is to use the analogy of "city blocks" (using Cartesian coordinates to specify walking so many city blocks left/right and up/down) vs. "walking through a cornfield" (using Polar coordinates to turn at a certain angle and walk straight for a certain distance to reach a certain point).
If you go over Problem 5 as a class, it is important to emphasize the relationship between the $(x,y)$ point $(\sqrt{3},1)$ which has a radius of 2 to the $(x,y)$ point $(\sqrt{3}/2, 1/2)$ on the unit circle which has a radius of 1. It might be helpful to explain that we can "scale" the point $(\sqrt{3},1)$ down by dividing the $x$ and $y$ coordinates by the radius of 2 to get the point $(\sqrt{3}/2, 1/2)$. We can then use this point to figure out what $\theta$ is by referencing the unit circle. (Or it might be more natural to go the opposite way and think about what point on the unit circle we would need to "scale" up by a factor of 2 to get to the point $(\sqrt{3}, 1)$ and then go from there. Leave a comment below if you find that one way is more helpful than another in your class.)
The reason why this is important is because we want students to recognize that they can find exact angles from points that are scaled from unit circle points. Students have the tendency to use the inverse tangent function and their calculators to find all angles and forget to think about if there might be a way to find $\theta$ exactly.
