Polar Coordinates - Calculus 2

Up to this point in calculus and really every other math class you have probably taken, you have only been working within the rectangular coordinate system. This is the coordinate system in which you measure a horizontal distance (x) and a vertical distance (y) to create (x, y) coordinate points. And we could form sets of these points that create graphs by using rectangular equations (whether that be in the regular form or parametric form).

However, now we are going to work in an entirely new coordinate system called the polar coordinate system.

Now, it is possible that some students have seen the polar coordinate system before in a precalculus or trigonometry course depending on the pace or length of the course, but for the majority of students in calculus 2, the polar coordinate system is a brand new concept, so I am going to treat it as such in this course.

The polar coordinate system differs from the rectangular coordinate system in that a point in the polar coordinate system is represent by a radius and an angle instead of an x-coordinate and a y-coordinate. As such, the polar coordinate system will look different than the rectangular coordinate system. Instead of graphing in a grid of squares, we will graph in a circular grid (as seen below).

(You can download some free blank polar graphs that I made at this link: https://www.jkmathematics.com/blank-polar-coordinates-worksheet)

Plotting points in this polar coordinate system is not all too difficult, it just involves measuring an angle, and then finding the appropriate location of the radius.

What is neat about the polar coordinate system however, is that unlike the rectangular coordinate system, there is not one unique pair of coordinates to represent a point. There are going to be an infinite number of ways to represent the same point since a full circle is measured by an angle of 2π. What this means is that you can add an angle of 2π to the angle of any pair of polar coordinates and end up with an equivalent pair that represents the same point!

There is also another way to find equivalent polar coordinates, but I will leave that for you to find out in the lesson video!

One more thing that is important to note about polar coordinates is that they can be converted to rectangular coordinates (and vice versa!). This can be particularly helpful when graphing polar equations, but for now we will just focus on developing the skill of converting points between the different coordinate systems. 

We discuss all of this and more in this new lesson of Calculus 2. Let's get to it!

In this lesson, you will learn:

• How to plot polar coordinates in the polar coordinate system
• How to find equivalent polar coordinates that represent the same point in the polar coordinate system
• How to convert from polar coordinates to rectangular coordinates and vice versa

I hope you find this video to be helpful!

-Josh