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. 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 (0,0). 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 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. As we get further away from that pole, r increases. 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 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 π2 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 θ=π2.

Now, 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 π6 radians away from that polar axis, so I have a theta value of π6. And ordered pairs in polar coordinates are always going to be written in this order, r, theta. 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, π3). Here I have an r value of 4, and theta is equal to π3. 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 π3, I want to come over to my polar coordinate system and locate that angle measured from the polar axis, π3 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 to plot this first point here located at (4, π3).

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, -π3). Here we have an r value of 5, and theta is -π3. 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 -π3, I'm going to measure this angle clockwise from this polar axis. So clockwise from this polar axis, π3 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, to plot the second point located at (5, -π3).

Now whenever you have a negative value for theta, you can also think of this as being a reflection over the polar axis from which we measure that angle. 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, π6). I have an r value of -3, and theta here is π6. Now remember, we want to locate that angle theta first, so here my angle π6 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. 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, π6). Now, whenever faced with a negative r value, we can also think of this as being a reflection over the pole.

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.