 Welcome back to our lecture series math 1220 calculus 2 for students at Southern Utah University. As usual, I'm the professor today, Dr. Andrew Misseldine. This lecture represents this video, I should say is the first in our series for lecture 32, and it does put me in a festive mood because I want to talk about polar coordinates today. Now, before we jump into polar coordinates, maybe we should make it very clear what we mean by coordinate system. A coordinate system is some way of representing a point in the plane as an ordered pair of numbers. We call these the coordinates of that number, and we're actually quite used to coordinate systems. That is, we use the x and y-axis to represent points in the plane all the time. We have the x-coordinate, we have the y-coordinate, and so if we take a point in the plane, say this is the point P, then what we can do is we can represent the distance that the point is to the right of the y-axis, which is the same distance you see down here. This is known as the x-coordinate. How far from the y-axis are we? That x-coordinate is measuring that. We can also measure the distance above the x-axis that the point is, or you can run along the y-axis as well, and you get this point right here, y. We call this the y-coordinate, and so this point can be represented by the coordinates x, y. This is a way of trying to represent where is the point looking in space, and we can give it an address or we can give directions to get to that point. Curious enough, this is exactly how people in Utah do street addresses. Maybe if you're not from Utah, if you haven't lived here for a long time, you're not familiar with this, but Southern Utah University is located in Cedar City, and like most Utah towns, addresses are based by the following system. There is one major road that travels east to west, and typically in a Utah town, this is called Center Street. Now in Cedar City, this is actually called University Boulevard, because it travels through Southern Utah University's campus, but there's always some Center Street, some reference of going east to west. There's also going to be a street that goes north to south, it's like the most important one. It's typically referred to as Main Street. This is true for Cedar City as well. If you go say four blocks to the right, or I should say east of Main Street, we refer to this road right here as 300 east, or if you went like two blocks to the west of Main Street, we'd call that road 200 west, super clever there. On the other hand, if you went say three blocks north of Center Street, that road we would call 300 north. I just noticed we went four blocks over here and we called it 300, 400 east and 300 north, right? And then therefore you get this point right here, which has now an address. This is 300 north, 400 east. You can describe every point in town by referencing from these two major roads, Center Street and Main Street. This is just like having the x-axis and the y-axis. And so for Cartesian coordinates, this is something that people in Utah deal with all the time. Now I was not raised in Utah, I'm a Utah transplant here. And I was raised in the gym state that is Idaho. And like other places in the United States, streets are not named by numbers, they're named by interesting things that people come up with by geography, like, oh, this is Stony Brook, oh, this is Quaking Aspen's Drive, this is Whispering Cliffs Avenue or things like that. This is Emerald Road, like we're going to the Wizard of Oz or something. And you know, Overland, McMillan, you know, they may have been named after someone. There's all these names for things. And so if you're trying to navigate a town that has street names like that, they're easy to remember, but it's not always obvious how they're associated to each other. So you have to kind of remember directions in the following way. So let's say that I have a friend who's right here at some point, which we'll call the origin, that's where they're starting from. And I want them to get to point P, following the roads, I might say following. It's like, okay, go for your house, I want you to drive three blocks going east until you find the crooked old tree. This tree, it's dead, right? Once you find the crooked old tree, then you're gonna start going north by five blocks until you see the haunted mansion. And so this is a ghost. Ooh, at the haunted mansion. At the haunted mansion, then go east for one more block until you hear the sound of silence. And then you go up towards, my house will be right there. We give direction in this manner. It's like, okay, from this point, go in this direction such and such a distance. Then from the next point, go in this direction by such and such a distance. This way of describing directions actually leads to an alternative coordinate system, which is known as polar coordinates. So if we have a point in the plane, like you do right here, here's our point P, the X and Y coordinates, again, it's X is this horizontal distance, Y is this vertical distance there. Instead, what we're gonna do is the following. We're gonna take a point that's gonna be the center of our space. This is where we start. It's called the origin, the origin O or sometimes called the pole. This is where we're gonna start our journey and take the point that we wanna get to and connect the dots. This gives you a line segment OP. The length of this line segment, we're gonna call it R for reasons that'll become clear in a little bit. And then associated to this, there's gonna be an axis, what we call the polar axis. This is just a ray actually, half of a line. The polar axis will point in some direction. So this is some reference point from everyone else. If you were like in an orienteering course using a compass to find your direction, this would be like using north. I want you to go 15 degrees east of north or go 20 degrees west of north. You have sort of like this fixed reference point, the polar axis, and you're gonna measure an angle associated to that polar axis. Now in order to translate between polar coordinates and Cartesian coordinates, the pole will always correspond to the origin, the intersection of the X and Y axis. And the polar axis will always be the positive X axis. And so with this interpretation, we can give directions based upon, that is we can give an address to every point in the plane based upon a direction of which way should you go and how far should you go. And so this is always given as R comma theta. And with our convention, we'll always mention the distance first, then the angle theta. Some people swap it around, but for consistency, this is what we're gonna do in our lecture series. So if you take the polar coordinate one comma five pi over four, what this means is you're gonna go one unit away from the origin in the direction of five pi fourth. We're gonna use radians to measure our angles here. Now a radius of, so going a distance of one is associated to the radius of a circle. That's actually why we call R itself R. We think of it as radius. Imagine some criminal escape from prison, right? It's like, oh no, this person's been missing for 30 minutes, right? How far could they have gotten in 30 minutes? Like, okay, we're gonna set up a five mile radius. We don't know which direction they went, but we know that the distance they traveled couldn't have been more than five miles. So we'll send out the dogs in a five mile radius in all the different directions, right? It can be anywhere in the circle, but five radius is what we do know. But our person who escaped, they went in the direction of five pi fourths, which is this direction right here. So if you take theta, which is this yellow ray and the yellow ray, and if you take R, the radius of the circle, these two places, the circle and the ray will intersect a unique point. And this is our point P given by the polar coordinates one comma five pi over four. Now I should mention that in order to have a more robust coordinate system, we do allow for the possibility of different polar coordinates representing the same point in the plane. For example, our theta will be the same thing as R comma theta plus two pi. That is we can replace with any angle, something that is co-terminal to it. They stop at the same place in the plane. That'll describe the same point. So as an example, let's move down to this one. You have the points two comma three pi. What this means is you're gonna rotate three pi. So two pi is a full spin and then you do another half spin. That's the direction you wanna go. And then you're gonna go in this direction for two units of distance. And so you find this point right here. But to find that point, we equally could just done a half spin. We could have skipped the whole first spin, right? That way we don't have to get dizzy. And so doing a half spin, we go in that direction two spaces. We find that point as well. And so this polar coordinate two comma three pi is the same thing as two comma pi. We could use a co-terminal angle and that would give us the exact same point right there. Another convention we wanna be aware of, again, for a robust coordinate system is that typically by default, we want our angle to be a positive. And this is associated to a counterclockwise rotation. But in the other direction, if we take a theta to be negative, this will give us a clockwise rotation. And we do wanna allow for negative angle measures like we see here. This just means instead of spinning counterclockwise, you spin clockwise. So two comma negative two pi thirds means you're going to go clockwise two pi thirds radians are 120 degrees if you prefer. And so this is an counterpoint to a clockwise, the counterclockwise rotation. Now, if ever you wanna switch from negative to a positive, just add multiples of two pi until you get positive, right? So if we take the same point two comma, let's just add two pi to the negative two pi thirds. This will give us a four pi thirds, which is an angle terminating in the third quadrant there. So we could either do clockwise or counterclockwise but just distinguish between these things. Another thing to point out is again, for a robust coordinate system, we wanna allow for a negative radius as well. So a negative radius would mean something like this. If you were to rotate two pi thirds, like we said before, that's 120 degrees. That puts you in the second quadrant. And you're facing this direction. What I want you to do is to step back by two steps. So you get this right here, negative two comma two pi thirds. So a negative radius just means you move backwards in that direction. So you're facing one way, but you move backwards. And it's gonna be useful for us as we talk about polar functions to allow for negative radii right here. Now, if you ever have a negative radius, you can always switch it to be a positive radius with the following convention. Negative r theta is actually equal to r comma theta plus pi. Plus pi just makes it go in the other direction because pi is a half spin there. So just puts it in the other direction so you can correct it if you want a positive radius. So we could write this as the point negative two comma two pi thirds can be written as two comma. We wanna go this direction now. So add pi to it, we get five pi thirds. So that's the same point. And so that gives us polar coordinates, right? They're very similar to Cartesian coordinates, but they have some uses. They can do some things a lot better than Cartesian coordinates, much of the same way that Cartesian coordinates can do things that polar coordinates can. So we wanna be versed in both of them as we go forward with some calculus problems.