Hey, everyone. At this point, we're used to plotting points using a rectangular or Cartesian coordinate system where we're given ordered pairs \(x\), \(y\). But here we're going to take a look at a different coordinate system called the polar coordinate system where instead of \(x\) and \(y\) we are now going to plot points in terms of \(r\) and \(\theta\). Now we've been working with circles and angles a lot throughout this course and polar coordinates are no different, so here I'm going to walk you through exactly what polar coordinates are using a lot of what we've already learned and then we'll practice plotting some points on our polar coordinate system together. So let's go ahead and get started.
Now when working in rectangular coordinates we know that we have our \(x\) and our \(y\) axis and our origin located at the point 0. Then when plotting an ordered pair, I would go \(x\) units over and \(y\) units up or down, like here we have the point \(3, 2\). So how does this work in polar coordinates? Well, remember that I said that we're going to work with polar coordinates in terms of \(r\) and \(\theta\). Now \(r\) is going to be the distance from the pole, which is what we think of as being our origin, but here in polar coordinates this center point is referred to as the pole located at \(r = 0\). Then as we get further and further away from that pole, \(r\) increases. So here \(r\) would be 1. Here \(r\) would be 2 and so on. \(R\) is the radius of each of these circles. Then we have \(\theta\), and \(\theta\) is going to be the angle from the polar axis, which is what we think of as being our positive \(x\) axis. Now we're going to measure \(\theta\) counterclockwise from this polar axis, the same way that we would on the unit circle. So if I go \(\frac{\pi}{2}\) radians away from that polar axis, we reach what we think of as being our positive \(y\) axis. But here in polar coordinates, this is just the line \(\theta = \frac{\pi}{2}\).
So what about ordered pairs? Well, let's take a look at this point here. We see that this point is 1, 2, 3, 4, 5 units away from that center pole. So I have an \(r\) value of 5. Then this is \(\frac{\pi}{6}\) radians away from that polar axis, so I have a \(\theta\) value of \(\frac{\pi}{6}\). And ordered pairs in polar coordinates are always going to be written in this order, \(r, \theta\). So here that we see in rectangular coordinates, we would go over and up or down based on our \(x\) and \(y\) values. And now in polar coordinates, we're instead going to go around and out based on our values for \(\theta\) and \(r\). Now here we just identified an ordered pair but more often you'll be given an ordered pair and asked to plot it on your polar coordinate system. So let's go ahead and get some practice with that together.
Now the first point that we're asked to plot here is \(4, \frac{\pi}{3}\). So here I have an \(r\) value of 4, and \(\theta\) is equal to \(\frac{\pi}{3}\). Now when plotting points in polar coordinates, even though \(r\) comes first in that ordered pair, we actually want to locate \(\theta\) first. So here, since \(\theta\) is \(\frac{\pi}{3}\), I want to come over to my polar coordinate system and locate that angle measured from the polar axis, \(\frac{\pi}{3}\) radians. Now once we have located \(\theta\), we can then count \(r\) units away from the pole. Here, since \(r\) is 4, I'm going to count 1, 2, 3, 4 units away from that pole in order to plot this first point here located at \(4, \frac{\pi}{3}\).
Now here our values for \(r\) and \(\theta\) were both positive, but that won't always be the case. So let's go ahead and take a look at another point here. Now here, the second point that we're asked to plot is \(5, -\frac{\pi}{3}\). So here we have an \(r\) value of 5 and \(\theta\) is \(-\frac{\pi}{3}\). Now again we want to locate \(\theta\) first, but here \(\theta\) is negative. So how are we going to deal with that? Well, remember that when working with our unit circle, whenever we were faced with a negative angle, we would simply measure that angle clockwise instead of counterclockwise. And we're going to do the exact same thing here. So here since I have a \(\theta\) value of a negative \(\frac{\pi}{3}\), I'm going to measure this angle clockwise from this polar axis. So clockwise from this polar axis, \(\frac{\pi}{3}\) radians, I end up right along this line. Now I can just use that \(r\) value to plot this point, counting 5 units away from that center point, my pole, in order to plot the second point located at \(5, -\frac{\pi}{3}\). Now whenever you have a negative value for \(\theta\), you can also think of this as being a reflection over the polar axis where we measure that angle from.
Now here our \(r\) value was still positive but again this won't always be the case. So let's take a look at one final point here. Here we're asked to plot the point \(-3, \frac{\pi}{6}\). So I have an \(r\) value of \(-3\) and \(\theta\) here is \(\frac{\pi}{6}\). Now remember we want to locate that angle \(\theta\) first, so here my angle \(\frac{\pi}{6}\) measuring that counterclockwise from that polar axis. I am along this line here. Now since this \(r\) value is negative, what exactly are we going to do here? Well, whenever we're faced with a negative \(r\) value we're simply going to count from our pole in the opposite direction. So typically, if this was a positive 3, I would go ahead and start counting out this way towards my angle. But now I'm going to count in the opposite direction because this \(r\) value is negative. So I'm going to count 1, 2, 3 units away from the pole in that opposite direction for this negative \(r\) value in order to plot that final point at \(-3, \frac{\pi}{6}\). Now whenever faced with a negative \(r\) value, we can also think of this as being a reflection over the pole.
Now as we saw here, it's going to be really important to pay attention to the signs of both your \(r\) and \(\theta\) values. So let's keep this in mind as we continue to practice. Thanks for watching, and I'll see you in the next one.