 So are there any questions or issues that people want to address at this point? No. All right. So let me move along then. So I'm going to move on to the next topic unless people want me to go over words some more. I think work is pretty straightforward. It's just a little force over the distance or whatever it is. It's just a little hard thing. As is always the case, it needs to be setting up what the little element is that you're moving. So the next topic that we're moving on, that we're doing is something that's been a little before. It's the ideal polar coordinates. We will move this in calculus and polar coordinates, so it's fun. So this is, I think, appendix H. And so just to remind you how polar coordinates work, you think of having, so if I want to describe the point here, playing, rather than saying how far over I go in X and how high up I go in Y, instead I indicate what angle I make with the x-axis and then how far over I go to walk that angle. So this is the angle of theta equals theta is the angle of the, let's call it the x-axis. It's actually the positive x-axis. Then R is usually the variable that we use for the distance to the origin. So this is something I think you've all seen before. And in the context, I mean, we can do this more generally, but typically in the context of this course we're going to, well, let's just do some standard stuff here first, yes. So we can convert back and forth from one coordinate system to the other just by observing that we always have a little right triangle here and in the right triangle which I will redraw, there's theta, this distance is x, this distance is y, this distance is r, that we can just use trigonometry to see that r, well, so we can see that the cosine of theta is y over r and the sine of theta is x over r. So that can give us, or if you prefer the tangent of theta is y over x. So we can certainly get theta out of this just by taking the arc tangent of y over x, although we have to pay attention to whether it's positive or negative. We have to pay attention to which quadrant it's in. So we might have to look at the sine, s, a, g, and either go sine of the sine and figure out whether it's here or the tangent is positive or here. I'll do an example in a minute. Yeah, tangent's still right. So we have that. And we also have that r squared is x squared plus y squared from the value of zero. So this should be enough to allow us to convert points from one quadrant system to the other. And I suspect that you all did this before. Is there anyone who has never done this before? Really? And you actually went to high school? OK. Shocking. Shocking. OK. So the polar coordinates are really moving. Wow. OK. So that makes me have a little more practice to do than the other people. So maybe I should do an example or two here, just thinking of points so that if we have, OK. So suppose we have the point r equals 2 theta equals, I don't know, 4 in polar. Obviously it's in polar because I'm, actually before I say that, r. So then what is it in a rectangular coordinates? Or this is also sometimes called Cartesian. And after Descartes, we came up with this. So in Cartesian coordinates or rectangular coordinates to do the conversion, it's pretty straightforward. We just look at the formula. Well, I suppose I did write it quite down. So this means that x is r cosine theta and y is r sine theta. So this is pretty easy. We just plug in. Right? So x is 2 cosine pi over 4 and y is 2 sine pi over 4. And the cosine pi over 4 is, well, we could almost just draw the triangle to figure it out. I mean, this is the same in terms of the picture. That's a bad picture already. Pi over 4 is 45 degrees or a quarter of a half of the right angle. And so this way is 2 and this way to the square root of 2 is the square root of 2. So I kind of don't even need the formula. You just look at the picture and see that this is root 2. Or I can remember that the cosine of root 4, pi over 4 is 1 over root 2 and 2 over root 3 is the same as the square root of 2. Either way, it's fine. So that's pretty straightforward. Any questions about that? Yeah. No? So, okay. So cosine pi over 4 is 1 over root 2. 1 over root 2. And so 2 cosine pi over 4 is 2 over root 2. But I can divide this root 2. This is root 2 times root 2. That's another way to write 2. And I'm dividing by root 2. Sometimes, you know, people insist on writing this instead of 1 over root 2. root 2 over 2. So then I have 2 root 2 over root 2 and anything else on it? No. Square root of 2 squared plus square root of 2 squared is 2 plus 2 is 4. And the square root of 4 is 2. The hypotenuse is 1 without the 2. With the 2. All right, so really easy. And going the other way is also really easy if we take a point like in Cartesian coordinates let's say I don't know, I'm just going to make something up. And we want it, so this is not really a nice angle. So we're saying I did 1, 3, this point here. And I want to know what this is in polar. Well, certainly the angle will be between b between 90 degrees and 180 degrees or between pi over 2 pi. And so here r squared is x squared plus y squared is 1 plus 3 or 1 plus 3 squared which is 9. So r is 3. Yeah, 10. One of those numbers is where we were 10. And theta is whatever angle it is to get that angle here. So we have to take the arc tangent. So I know that the tangent of theta is pi over x and so theta is the arc tangent of negative 3 which is not really a nice angle. I mean it's just whatever it is. But if the arc tangent of negative 3 that is, there are two angles that give you negative 3 as their arc tangent. There's this one and we want this one. So it is the arc tangent of negative 3 which is between pi over 2. Any questions on that? So you convert back and forth between these things. This is something that you should have done in high school. I'm really annoyed that you didn't do this in high school. It's not your fault. So you'll have to play with these a little bit. A couple of things to notice about polar coordinates. So they're very useful in many applications. So for example, imagine if you are an air traffic controller everything is done with polar coordinates. And often a lot of navigation and those kinds of things are done with polar coordinates as we say things over there a mile and a half. So that's naturally polar coordinates because when we say over there that's an angle and then how far a mile and a half. And I don't really know that this is a mile and a half. Let's say it's that way one mile. So I don't really know whether this is root 2 miles west and root 2 miles north or whatever and I don't have a compass with me to measure rectangles. So a lot of times things are more naturally described with polar coordinates. So some issues with polar coordinates is that there's not really a unique theta. Unlike with x and y, I tell you x, I tell you y, I know the place. So in the sense that theta equals to pi is the same angle equals zero. It just means I wrap around once. And theta equals negative pi of let's say pi over 2. Negative pi over 2. So again this is just for those of you that are still familiar with this. That's in my writing. So 360 degrees is the same as zero degrees. Works a lot better in calculus if we use radians. But okay, so this is negative 90 degrees is the same as equals 3 pi over 2. 270 degrees. And any multiple of 360 degrees as it were subtracted doesn't change the point. It's useful, however, because we can think of if we have something rotating around we can measure how many times it went around. It's the source of the history. So it's both good and bad that theta is not unique. That is if I want to describe something that's starting here and ended there it might be more natural we'll put it down here. We'll put it here. We'll describe the angle of this as being 6 pi because I started at zero. Sorry, 4 pi. Because I started at zero, I went around. I went one full term. That's 2 pi. And then I went around again and I went another full term. That's 4 pi. So while I end at the same place so this place is not uniquely described by a given theta. There's sort of a history built in. It is still y equals zero and maybe x equals like 2. But if I describe this place as 2 zero I don't have any sense of this thing turning around twice before I got there. So there's sort of a difference between describing this in polar and r equals 2, theta equals 4 pi because r equals 2 theta equals zero. They describe the same point but this one sort of implies a history. So there's some advantage to this lack of uniqueness for describing points because it lets you know about how things turn around. Okay, so that's one issue. Another issue another issue is r equals zero has every angle. So r is zero, theta is sort of not well defined. I could be distance zero and angle 270 or distance zero and angle zero or whatever. If I don't move you can't tell which way I'm facing. So all of the things are at the same point. There's a little bit of a singularity at the place you are. So usually but not always theta between zero and 2 pi and we take r bigger than zero r equal to zero to describe the origin but we don't have to. Sometimes it's useful to allow r to be gone, to be negative and so taking theta outside can be useful and also r less than zero. Naturally if I'm facing this way and I want to describe one foot behind me then I would say r is negative one. Right? So this point here the same point r equals 2, theta equals 4 pi which is the same point as r equals 2, theta equals zero is also the same point as r equals negative 2 and theta is hot. Again it's the same point it means face this way put back up two well put back up two units. So negative r needs to be behind you. This means that theta transforms you have 5 to theta. Does that make sense? If I want to transform something like so let's go back to this conversion problem here so if I want to understand what is the point so what is the point in the coordinates it's called rectangular in terms of a picture this is saying 3 pi over 4 is go a half turn right this is pi what am I doing? 3 quarters of pi so this is theta equals 3 pi over 4 here and so I'm facing this way so this is my positive axis and this is my negative axis here in terms of a picture I face at theta equals 3 pi over 4 and then I back up to you so this should have both x and y negative it's clear? How many of you have never seen this notion before of a negative radius? Well now that's interesting because only 4 people have never seen the idea of a negative radius before but like 15 people have never seen a radius before so something is strange but maybe your arms are broken ok so then what are the rectangular how do I figure out the rectangular if we're going to this I just draw this triangle or I take the cosine right so my point in rectangular is going to be negative 2 times the cosine negative 4 and my y coordinate is negative 2 times the sine which is x equals so the cosine of 3 pi over 4 you know what that is negative 2 so this is negative 2 we like as negative 1 over 2 negative and the sine of 3 pi over 4 negative 2 or 1 over 2 whatever you want so this is 2's cancel and this gives me a negative root 2 my y coordinate and my x coordinate which is what we see here in the picture this is root 2 this is root 2 but it's negative and this is 2 really helps a lot to think about where you are rather than just whining and plugging in so you don't make little mistakes well I can't do what I was planning to do because I'll have to fix that so I guess maybe I won't it's kind of boring we can just convert back and forth from the points but calculus is not about coordinates but more about functions so really what we want to do here is not think about how to describe locations in the plane but how to describe things which vary which gives us locations so really what we want to think about and this is really where polar coordinates is useful is to describe a further collection of functions that we weren't able to describe before for example we can think about a radius a function of the angle we can also think about the angle being a function of the radius but let's for the moment stick to radius being given by the angle that is if I tell you the angle is pi then you can calculate how far away it is so what application of this is if I have say something in orbit and I want to describe the orbit then I can describe where it is in terms of what I look at it might be more natural to say that at a certain angle the thing is a thousand miles away at a different angle it's three thousand miles away I can describe this in terms of depth the specific one let's try something easier so let's take the easiest possible the easiest possible thing is the radius is the constant that's a very simple function what is the graph of this function it's a circle this is just all the points that are tuned in two ways from the origin this is a much simpler graph than say x squared plus y squared which is 4 so a much simpler equation a much simpler way to describe the circle this is really easy and this is a little more complicated or even worse y is plus minus square root of 4 minus x squared but to describe this circle it's a natural polar function the radius is the constant and theta is whatever it wants to be for every theta I get a radius and it holds the same one this is sort of the natural way to describe a circle just like what you learn about functions with x's and y's the first things that you learn about are say linear functions so a linear function here is saying that the radius is the same what kind of thing is this how would you figure it out well we can just start plotting points and then realize what the circles are telling us so when theta is zero but I mean I'm not saying this is the best way but it's a way to get some idea so when theta is zero the radius is zero so when I look this way and now as I start to increase so I take here theta being pi over 6 the radius is pi over 6 so when I'm looking this direction I must feel kind of big I'm here which is about the hat so I've gone from looking this way and being at the center and looking that way and as I look up my angle increases and as I continue to look on higher and higher angles I go further and further out until when theta is pi over 2 I get pi over 2 which is about 2 thirds and then I continue to grow my angle to when I'm looking this way an angle theta equals 5 I'm 3 units away 3.1415 blah blah blah and then I continue I'm looking now this way and I'm even further out and now I'm looking this way and I'm even further out and when I come all the way back I'm at 2 pi so it grows and grows and grows and it will continue to grow so here this is where the non-uniqueness of the angle helps us more and more around and around and as theta grows my distance from the origin gets bigger and bigger it's like I have a string I have a fishing line and I just have it infinitely long and I just let the line out and out and out and out and it goes very far away until it hits the movement that I'm done but it just keeps spiraling around and around is this the whole graph so nobody knows yeah theta can also be negative so I get a spiral for positive theta and I also get a spiral looking the other way so negative theta so really let's try that again here for theta positive looks like this and for theta negative let's just think about where this point is I start for a small negative value I look this way this is a small negative theta but r is behind me so when theta is negative I start at the origin and I move behind and as my angle goes this way I look more and more behind so this point here is r equals negative pi over 2 theta equals negative pi over 2 I'm looking this way my radius is pi over 2 units behind me and then I continue to look and my radius is behind so this point here is r equals negative pi theta equals negative pi and it continues to spiral out in the other direction now I'm sort of so increasing theta from negative infinity these spirals in then I hit 0 and then it spirals out it crosses itself at all of these places where it crosses itself on the y-axis people confused by this is this reasonable? okay so even though this seems like so this very simple equation r equals theta gives rise to something that is extremely hard to describe in this angle of origin so this is a arc of median spiral but it spirals out proportional to the angle obviously r equals 5 theta has the same shape it's just bigger or sometimes it's called the spiral arc of median but the same okay so now let's move along a little more and so it was my plan to give all sorts of quicker questions and show you all sorts of cool graphs and all that neat stuff and it turns out that this computer is still bad at neat so next time I will definitely make sure that it's working a lot of times these things get very hard to do by hand and it's very useful to have technology to try and figure these out so Tony any questions about this so far how many of you are completely lost how many of you are okay it's good one honest person how many of you are completely following okay good I don't know whether you're honest or dishonest okay so let's move along how many of you have done those of you that have heard of polar coordinates I hope it's not that you need to have it how many of you have studied functions of polar coordinates along these lines okay so let's move on to something else in fact I mean well in rectangular coordinates the graphs and polynomials are sort of the next simplest object beyond the linear ones this is not true for polar coordinates polar coordinates are sort of naturally and intimately related to trigonometric functions so really the next simplest thing would be something like r equals the sine of theta or r equals the cosine of theta let's do the cosine so if you think about this for a minute because the cosine is 2pi we only have to pay attention to theta between 0 and 2pi because the radius will repeat when theta is bigger than 2pi or less than 0 so we don't have to worry about theta outside of 0 or 2pi for this function and again let's you know rather than trying to analyze because we can analyze this via calculus and all sorts of things but let's just sort of do it so this is I don't know this is our polar plane so just to emphasize that it's polar put in 45 degree angle lines and then to remember what the cosine of theta looks like stop it go away remember what the cosine looks like the cosine starts out at 1 2pi here and the relevant bit of cosine is what we need to think about this is i for 2 this is i this is 3pi so those are really the points that we need to think about so the first point that we get easily on the graph is here polar coordinates 1 r equals 1 and as theta increases so this is if we had a polar graph paper it would look sort of like a series of concentric circles with rays running so let's just think in terms of a polar graph case so I get point on my graph here in fact let me change my scale a little bit you get a point on my graph here so this is 0 r equals 1 that's at x is 1 y is 0 that's the cosine of r 0 is 1 and then as the angle increases the radius decreases so instead of following this I want to move in as the radius as the angle increases my radius increases so the thing moves in number 2 it's in at the origin so that means that as I move up here I want to get so my picture is not exactly right but pretty close so this point here is I'm looking this way and my angle my radius is 0 when I'm looking up and in between so say at 4 r is 2 over 2 over 2 or whatever you want to call it 0.7 whatever it is so we do that and then how will it continue back to the beginning we made it up on this point right I would give a circle now is this really a circle well you have a 50-50 shot well I basically said it's 45-45 and then there's 10% of major back to maybe it's maybe the best answer at this point I don't know how to read Jack's but how do we know so so this is a circle it works out to be a circle in fact sometimes trigonometric functions are also called circular functions it's not an accident so this describes a circle I've done the circle twice so one thing to notice is that this piece of graph gives us the whole circle down here so let me go through this argument I skipped it so to get these points down here that means I'm looking this way looking this way but my r is negative so I land there it's a little bit of changing your brain to think about these things but if I'm looking this way so when theta is let's say 3pi over 4 then my radius is the cosine of 3pi over 4 which is negative root 2 over 2 so when I'm looking here at a 45 degree angle this way the 35 degree angle back up to get my point on the circle and as I continue to look through this quadrant I sweep out this part of the circle until when I'm looking east the point is negative 1 unit behind me you're so confused I'm confused too so so those of you that are so confused it's okay right? part of what is supposed to go on when you are in college is you're supposed to be exposed to new things and they're supposed to make your head hurt until you understand how they work and then your head goes oh that's cool or you go yeah okay but in either way it's confusing until you get it so it should be a good feeling because it'll get better it's a bad feeling when you go yeah I know it all and then you do the problems and you discover that's worse and I know that happens to a lot of people okay so let's carry on a little bit more so let me just go through let me do this one it's exactly the same but different what do you think this will be a circle good will it be this circle it'll be up why will it be up so the sign isn't always positive oh really nice good it's fine so let's think about what the graph of the sign looks like between 0 and 2 height looks like this the sign is positive I probably should have done the sign first so that's a little easier for an angle between 0 and pi we start out at 0 our radius increases until when we're looking this way our radius is going all the way up to 1 so this is so here where theta is pi over 2 or 90 degrees the radius is 1 so the radius starts at 0 it grows and grows and grows but it grows quickly and then slower so like here it grows quickly and then slower until at 1 it gets as far away as it's going to go and then the radius decreases back to 0 so this is a two quadrant it's this piece of graph the radius is negative so down here we're looking down this way but the point is behind us the radius goes from 0 to negative 1 and then from negative 1 back to 0 as I do this way so it's not that these angles down here don't exist it's that when we look this way the same thing is happening here for the cosine so for the cosine let's describe the cosine again maybe in easier coordinates for those of you who are not confused by the negative angle so I'm going to do r is the cosine where theta instead of going from 0 to 2 pi let's go from negative 5 from negative actually let's go from negative 5 over 2 pi over 2 let's go from negative 90 degrees to positive 90 degrees to describe the cosine maybe it's a little easier so in that case again we already did this positive path we start out here at the radius of 1 for theta equals 0 and it pulls into the oracle but let's do the negative angles now when theta is negative I take the cosine to this negative angle so think of theta going from 0 to negative pi over 2 look at the graph of the cosine in rectangular coordinates it looks like this but this is negative pi over 2 and this is positive pi over 2 so in the positive part I start here at 1 and my radius decreases and if I go backwards I start here at 1 and my radius decreases so when I go backwards I sweep out this part of the circle so here this is theta equals 0 and I start at 1 and if I go positive angles my radius decreases so how can we tell that this is really a circle? how would it be? it looks like a circle it really is a circle in my usual sense of knowing that it's a circle what's one way I can know? how can I recognize a circle? but the radius from the center the radius from the center is constant so how would I know that? but what's true in this case what do I need to be true? when you say the radius implies it's measured from the center but our equation is measured from the origin so how can I convert this? so my belief my claim is that r equals sine theta is a circle with the center x equals 0 y equals 1 and radius not r, radius with this picture of planes sine theta we believe to be true based on the picture how can we check? it has a clue you can switch it to cartesian coordinates and it should look like x squared plus y minus 1 half square equals 1 so if you switch to cartesian coordinates you should see this there's another way I can tell that'll be done check that the distance of every point here to this is 1 half that is I mean you can do it entirely in polar by noticing that the distance between this and the rack up okay so next time I will do a little more with polar graphs and the computer I hope you're working but there are still planning to move down to math 131 or math 126 the form needs to be signed I can sign it but it should be signed by the instructor from the class you're moving into or by the math department