Graphing Functions in Polar Coordinates: Process & Examples

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  • 0:01 Polar Coordinates
  • 2:08 Cartesian to Polar Coordinates
  • 3:35 Example
  • 4:57 Graphing Polar Equations
  • 6:14 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will be able to convert any Cartesian coordinate point into a polar coordinate point. You will also learn how to plot these points.

Polar Coordinates

Now that you are very familiar with your Cartesian coordinates of (x, y) points on a grid with an x-axis and a y-axis, it's time to introduce you to another way of plotting points. We call this the polar coordinates. Instead of (x, y) points, we now have (r, theta) points. Can you guess what the r and theta stand for?

If you are picturing a circle in your head, then you are on the right track. The r stands for radius, or distance from the origin of our grid, and theta stands for the angle the radius makes with the horizontal axis. On the Cartesian coordinate grid, the horizontal axis is the positive x-axis. The grid for the polar coordinate system looks a bit different. We still have our x- and y-axes. But instead of vertical and horizontal making little squares, we now have circles for each radius step and different lines coming out from the origin marking different angles.

The way to plot the points are a bit different, too. Think back to your Cartesian coordinates and when you were plotting points galore. What do you remember about plotting (x, y) points? Yes, the x tells you how far to go on the x-axis, and y tells you how far to go on the y-axis. In other words, the x tells you how far to the right or left our point is, and the y tells us how far up or down the point is. For our polar coordinate (r, theta) point, however, we go about plotting differently. We begin at the origin of our grid. The r tells us how far away to go from the origin. Now, the theta tells us how far to swing our radius in a counterclockwise direction from the positive axis. For example, the point (1, 20) in our Cartesian coordinate system is plotted by going 1 step to the right on the x-axis and then 20 steps up. However, the point (1, 20 degrees) in our polar coordinate system is found by going a distance of 1 step away from the radius and then swinging this radius 20 degrees from the positive x-axis.

Cartesian to Polar Coordinates

Would you be surprised to find that these two systems are actually related to each other mathematically? Yes, they are. And yes, we have formulas to help us convert from one system to the other. But first, how are they related, you ask?

Well, look at our Cartesian coordinate point (1, 20) below. If we draw a line from this point to our origin, we would get our radius. We can go ahead and draw our vertical and horizontal lines to represent our x and y values. The angle that is formed by the x line and the r line that we drew is now our theta. Now, look at what kind of shape we have below. Isn't it a right triangle? Yes, it is. And this right triangle gives us our conversion formula for turning Cartesian coordinate points to polar coordinate points.

The right triangle gives the conversion formula.
grid with point 1,20 and angle theta

To find the r value from a Cartesian point, we use the Pythagorean theorem, a^2 + b^2 = c^2, where c is the hypotenuse. In the triangle that we drew, side c is equivalent to the radius side, the r side. The a and b sides are then the x and y sides. If we solve the Pythagorean theorem for c and then substitute in our equivalent values, we get r = sqrt(x^2 + y^2). Our theta comes from the definition of tangent as opposite over adjacent. The side opposite our angle is the y side, and the side adjacent to our angle is the x side. So, we have theta = the inverse tangent of y over x (theta = tan^-1 (y/x)).


Let's look at an example to see just how we convert from a Cartesian point to a polar point.

Convert (3, 4) to polar coordinates.

We have our (x, y) point. Our x is 3, and our y is 4. To find our r value, we plug in these values into our formula, r = sqrt(x^2 + y^2). We get r = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. Our r is 5.

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