Question Video: Graphing Polar Coordinates Mathematics

Video Transcript

Identify which of the points plotted on the graph has polar coordinates one, 𝜋 by four.

We’re given a graph with five points plotted, points 𝐴, 𝐵, 𝐶, 𝐷, and 𝐸. We need to determine which of these points has the polar coordinates one, 𝜋 by four. Let’s start by recalling what we mean by polar coordinates. Let’s say the point 𝑃 has the polar coordinates 𝑟, 𝜃. Then 𝑟 is the distance from the origin of the point 𝑃. And 𝜃 will be the angle the ray 𝑂𝑃 makes with the 𝑥-axis measured counterclockwise. So let’s look at the polar coordinates given to us in the question: one, 𝜋 by four. We can see the point with these as polar coordinates will have 𝑟 equal to one and 𝜃 equal to 𝜋 by four.

Since our value of 𝑟 is one and 𝑟 measures the distance from the point to the origin, our point must lie on the circle of radius one centered at the origin. And since our diagram is given with polar coordinates, we can sketch the graph of the circle of radius one. Our point must lie on this circle, so we can see our point can’t be option 𝐵 or option 𝐸. They don’t lie on this circle. In fact, we can see both of these points lie on the circle of radius two centered at the origin. So both of these will have an 𝑟-value of two in their polar coordinates.

Next, the point one, 𝜋 by four will have a value of 𝜃 of 𝜋 by four. Remember, the value of 𝜃 will be the angle the ray 𝑂𝑃 makes with the 𝑥-axis if we measure this counterclockwise. We want this angle to be 𝜋 by four radians. There’s a few different ways of finding this. The easiest way of using this example is we know 𝜋 by two will be a right angle when we measure it. In other words, our point 𝐷 will have a value of 𝜃 equal to 𝜋 by two. We want an angle of 𝜋 by four. This is one-half of 𝜋 by two, so we just need to split this angle in half.

Doing this, we get the following ray. Every point on this ray will have a value of 𝜃 equal to 𝜋 by four. For example, both our points 𝐴 and 𝐵 have an angle of 𝜋 by four. We can see that our point 𝐴 is the only point which lies both on our circle of radius one and our ray with an angle of 𝜋 by four. Therefore, our answer must be point 𝐴. For completion’s sake, however, we notice the point 𝐶 lies on the ray which is listed as five 𝜋 by three. In other words, the polar coordinates of the point 𝐶 will be one, five 𝜋 by three.

Therefore, we were given the following plot of the points 𝐴, 𝐵, 𝐶, 𝐷, and 𝐸. And we were able to show that only the point 𝐴 has the polar coordinates one, 𝜋 by four.