So now for our discussion on polar coordinates. I already handed this out to you guys, but if you are watching this later, get yourself this, a polar coordinate graph. And don't get dizzy.
You're tripping out now. Get yourself one of these. It's polar coordinate graph paper. Just type it into the internet and you can find it.
This will help you learn how to graph polar coordinates. So this is what these people are going to be using for right now. I'm going to just draw it on the board. I won't draw all these lines and stuff because you'll be able to follow along if you have this. So if you don't have this, Get it right now.
Press pause and go polar coordinate graph paper, enter, and then print it, and then watch this video. Okay. That was for them.
Word. No, word. You guys, since I like you guys, I gave you a present, but these people have fun themselves.
So polar coordinates. This is a way to represent curves with something other than rectangular coordinates. Now, we already learned about parametric, but sometimes it's easier, nicer, or more relevant to use polar coordinates.
And we're going to show. I'm going to show you how to do that. So it's just a different way to represent curves without x's and y's with something a little different. Here's how it works. It's going to look really similar to a lot of trigonometry you've done because, hey, polar coordinates deals with angles.
It's going to be based on a lot of trigonometry. So here it goes. Polar coordinates. Polar coordinates look like this. We start with something called the polar axis.
Which takes the place of the x-axis. And then we have this point O, which is our origin. So we have O, the origin, or the pole.
We got the polar axis. This is always going to be our initial side, like where we start our angle, we start our motion. So we start here at the polar axis, and then we're going to go up, or we're going to go down, angularly around the pole.
sense. The pole is our origin. It should be very similar to drawing any angle that you have ever drawn. We're going to start at the polar axis and we're going to go up or we're going to go down with an angle theta. We're going to go up and we'll have this.
Other side. Now, what's different about polar coordinates is it's not just an angle. You don't go forever with a ray, a segment of a line. You actually stop. You have this terminal point.
What that P is, that P is defined by R theta, where R is the length of that segment, and theta is the angle in relation to your polar axis. So theta gives us our angle, less u. P is some point and r is the distance from o, your origin, your pole, and p, your point.
And the most basic sense of it, that's it. Does it look familiar to you at least? We draw lots of angles like this. We have our initial side, we have our terminal side for our angle.
Now this is called the polar axis. It goes around a pole. That's our point O, our origin. It goes for some angle up or down.
We'll talk about positive versus negative in just a moment. The distance of that between O and P is our R. It doesn't go forever.
In this way, you could actually graph a curve. Here's what it says, go up a little bit and go out. And then if I wanted to make a curve, I could vary the r, vary that distance, and vary the angle, it would give me a curve around that. Does that make sense? So the r is some sort of a distance, it doesn't just go on forever and ever and ever.
That doesn't happen with polar coordinates. So, couple of little notes for us. If theta is positive, do you guys remember this? If theta is positive, this means that we're going to have a certain.
What direction are we going if theta is positive? Yeah, that's right. Counterclockwise would be positive. Clockwise would be a negative angle.
Also, it's interesting that your distance r, r can be positive or r can be negative. ...and be negative as well. And I'll talk about that in just a moment.
In fact, let me do it right now. If r is positive, then how your angles look is this. Listen carefully because we're going to do a few examples, but I want you to really get it.
If r is positive, then what we would do is we have our polar axis and we have some sort of angle. theta, let's assume it's a positive angle, it's going counterclockwise. If r is positive, we get that look to it. We just get, okay, you use your angle, and r is positive, that's where your side actually is.
Are you with me on this one? It's just like what we started. Compare that to this. If r, and this would be some point r comma theta.
Notice how your coordinates are no longer in terms of x and y. Your coordinates are given by the length of your line segment and the angle that it's creating with the polar axis. Does that make sense to you? So it's not x and y.
It's how long the segment is and where the angle is in relation to your initial side, your polar axis. Now if. If the r is negative, here's how it works.
You'll still use your theta, you'll still use your angle, if it's positive we'll go clockwise, if it's, sorry, counterclockwise, if it's negative we'll go clockwise, you'll still use that. This would be where your point would be if you had a positive r. If you have a negative r, then here's what you do. You use that angle, you find where that point is, and then you reflect that about the origin.
So basically you go, okay, well, it's not over here. This would be a positive r. A negative r goes 180 degrees in the opposite direction for the same length.
So basically for negative r we use the theta as normal and then we just go in the opposite direction. That's the idea. So I think you'll be okay with this so far.
Now take out that graph paper and we're going to do a couple examples. So just to get really used to this, we'll go through maybe three, four, five examples or whatever so we feel comfortable. Let's start with something real simple. We're going to start with 1, 2 pi over 3. Now just to get started.
We've got to be used to naming what these coordinates are. They're not x's and y's. What's the one giving you, right here?
One unit. Yeah, it's giving the length. So I want you to look at your...
in your graph paper. If you look at it, there's some circles, right? There's some circles in your graph paper. There's one little circle, and then there's another one.
There's another one that keeps going out and out. I see one, two, three, four. I count five concentric circles. that too. So if you have five concentric circles, the distance of that first circle, that would be a unit of one.
And then the second would be a unit of two, and then three, and then four, and then five. So if r is one, you're not going to graph the entire length of your circle. You just go to the first little concentric circle, and that would be your r.
Make sense? Now, I can't show that because I don't have this on the board. But what we do here is that we have a little bit of a If you have this on your circles, you don't need this.
This is a rectangular coordinate system. I'm doing this just for the sake of quadrants, okay? So let's see where the quadrants are.
So really this doesn't have to be here and this doesn't have to be here and neither does this one. The idea is your x-axis is like your polar axis. Your origin is still your origin. Now what's my origin, everybody?
What's my theta? Two pi over two. Is it positive or negative? So positive means I'm going to be going counterclockwise until I reach 2 pi over 3. So on your, it's like a unit circle, just has more units to it.
So go to 2 pi over 3. Is 2 pi over 3 in the first quadrant? No. Is it in the second quadrant? Yes. Yeah.
So we'd say, okay, well, 2 pi over 3 is roughly, let's see, roughly right here, like that. So that's 2 pi over 3. We always, always label our polar coordinates. Always. We label the angle and we label the distance. Are you with me?
Always label it. So, now, we've done the angle, but that's only half the battle. We've just done the angle, now we've got to do the length of our segment.
How long should this r be? One minute. So I'm not going to go forever. I'm going to just go to, for you guys.
down that first little circle. And that's where our point is. Should we be okay with that? And this we label it as 1 comma 2 pi over 3. That way we have an angle of positive 2 pi over 3 counterclockwise.
We have a length of 1. So far so good? Okay, let's do some more. Let's do 3 comma negative pi over 4. Don't label these x and y axes, they're not.
If you label anything, this would be your polar axis, this would be the pi over 2 axis. This would be the polar axis in the negative direction, or the pi axis, 0 over pi. This would be the 3 pi over 2 axis.
They're not x's and y's anymore because we're not even talking about that, ok? We're talking about polar... coordinates. So let's start.
Here's our polar axis. We're going to start at the origin. What's our r, please? R. Three.
Good. What's our angle? Theta. Theta. So should I go clockwise or counterclockwise?
Clockwise. Say it again. Clockwise. Clockwise from here, that would be this way.
If this is negative pi over two, then negative pi over four, and it should be on your graph, right? Negative pi over four, just, well, it doesn't say, it might not say negative pi over four. might say what is it?
Seven pi over four. Seven pi over four, that's right. And if you think about that, here's one, two, three, four, five, six, seven pi over four is coterminal with negative pi over four. Okay, we're going to label it. We got to label it correctly.
The angle is going negatively, and it's negative pi over 4. How long do I make my segment here? How long is that? So I'm going to go to, on you guys, the third concentric. So 1 would be 1, 2 would be 2, 3 would be 3. On a graph like this, this is why it's important for you to label it.
You don't just put a point randomly like this. You say, okay, this angle is negative pi over 4, and we're going to 3. So we have 3 and negative pi over 4. And you all just label that. So here we have one.
So you guys feel okay with the idea of polar quarters? Easy, medium, hard? Once you get the hang of it, not too bad.
It's just an angle. You know about angles. And now that you have that nice, isn't that graph paper nice? Isn't that kind of cool? Oh yeah, it goes out three.
It's kind of nice. So let's continue. We'll do maybe, I don't know, two more. I'll show you what to do with negative r's.
Okay, how about this one? So just get in the habit of it. What's our r, please? Theta 2. What's our theta? Is theta going to move in the counterclockwise direction or the clockwise direction?
Counterclockwise. Counterclockwise. From our polar axis, how much is it going to go? Theta 2. Up or down? Okay, that is counterclockwise.
Pi over 3 is right about there. So we're going to label it positive pi over 3. Now here's the deal. If we have positive pi over 3, we also have an r of? Negative 2. Negative 2. Wait a minute.
So here's what we do. I showed you this up here. If we stop this at positive 2, is that my correct line?
No. What I do is I find the 180 degree opposite ray than that. So for our case, pi over 3, well we're going to go, what is that line down here?
One, two, three, four pi over three. Did you guys find the four pi over three? The graph is kind of nice because you just follow it straight down, but it would be four pi over three. So we go, okay, cool.
So we're going to go down here, we'll have two. And how we label it, we don't label this as four pi over three. What we label it as is negative two.
Pi over 3. We labeled the original angle that we had, and then we labeled the negative distance out of that. And that's the idea. Does that make sense to you?
So we've had positive angles, we've had negative angles, we've had positive R. Now let's do one last one. And you guys went to the second concentric circle, correct? Okay. What direction am I going?
Am I going to go in the clockwise direction or the counterclockwise direction? Quickly. Counterclockwise.
Say it again, what is it? Clockwise. Clockwise, okay. How far am I going? One.
Three. Okay, that was a very ambiguous question. On my angle, how far am I going?
Three. So this is one pi. This is two pi. This is three.
So when I do this, you're going to have to show that. You go, okay, here's... So here's one pi, two pi, three pi.
We label that three pi in the negative direction. So negative three pi. The direction matters here.
Now, how far out are we gonna go? One. So you're gonna go to your first concentric circle and label that one. And this is the point one comma negative three pi in terms of polar coordinates. I wanna make sure that you guys are okay with that so far.
That's pretty much everybody. You guys over here, are we okay with it? So you know how angles work?
Positive angles mean counterclockwise, negative angles mean clockwise. Negative, positive r's mean you just go wherever you're at that certain distance, so one or two or three or four. Negative r's mean that you go the correct angle and then you just make it opposite on the opposite line from that. Have I explained this one enough for you guys to understand it? Okay, now I want to give this up a little bit different.
I want to start switching between polar coordinates and rectangular coordinates. and between rectangular coordinates and polar coordinates. So I'm gonna show you how to do it super not that hard, okay, not hard at all, provided you understand what's going on on these polar coordinates.
So if we're to draw one of these, if we just draw some general polar coordinate, we'll have an r, we'll have a point, we have a polar axis, and we've got an angle. Now our idea is let's translate between polar coordinates and rectangular coordinates. If we do it, well every rectangular coordinate in the world has an x component and a y component, doesn't it?
Well, if we want to make an x component, we want to make a x component. Here's our origin, where we start. Our x component's going to be in which direction?
Up, down, or left, right? And it's going to go over to however far that point is right there. Does that make sense? And that would be our x component.
And our y component would be, well, how far we rise from here to here. Well, that would be... That would be our y component.
Does that look familiar to you? What angle is that? 93. So if you draw the perpendicular... Well, now can we come up with some relationships with this angle and the triangle? Hello, what class is that all about?
You've taken trigonometry, so some simple, basic trigonometry lets us convert between polar and rectangular coordinates, and here's how it works. We have a couple relationships with this. We have cosine theta.
We have sine theta. We have tangent theta, and then we have the Pythagorean relationship, r squared. Let's go over cosine. We're going to go through it very quickly because it's a complete review. Cosine theta is whatever what?
Adjacent over high. Sine is? Lines.
I'm sorry. Tangent is? Lines.
Lines. R squared. Pythagorean.
R squared. x squared. If we do just a little bit of manipulation with this, well, what we can find is, could you solve this for x?
How do you solve it for x? Well, x equals r times cosine theta. Could you solve that for y? Then y equals r times sine theta. Do you guys see it?
We just have our x-coordinate and our y-coordinate. That's really what we need for that. That's it.
Now. A couple other things we're going to use. We're going to use tangent equals y over x.
We're also going to use the Pythagorean identity a lot. So I'm going to write those, well, we already have them written down, but this is another thing that we have. So I'll write it twice, r squared equals x squared plus y squared.
These things are what we use to go between polar and rectangular coordinates. Now let's give this a try. So let's say I give you 4, pi over 6. What have I given you? Have I given you rectangular coordinates? Have I given you polar coordinates?
Polar. What gives it away? The pi.
Yeah, the 4 doesn't because that could be an x, but that's definitely a polar coordinate because we got the pi over 6. Now, here's what I want you to do every time. No matter whether you're going... between polar and rectangular, or rectangular and polar.
I want you to graph these first. Graph the point, because what can happen, especially when you go between rectangular and polar, you can mess up the quadrant. And so I want you to make sure these points end up in the correct quadrants.
Okey dokey. So let's graph that. We're gonna do it very quickly.
Graph all these really quick. You can do it on your graph paper, you can do it just right now like I'm doing. Pi over 6, you guys know where pi over 6 is? Is it up or down?
How up, up, up, up, up, up? That's right about there. How far out do we go? So you're going to go four concentric circles out.
So four at an angle of pi over 6. Here's the point. No pun intended. That is the point. But the point is, what quadrant is it in? One.
So we better make sure that when we convert this to rectangular coordinates, that our rectangular coordinates end up in quadrant one. It's got to be the same. It needs to represent the same point. Okay, are you sure you're okay with this? Yes.
Got to end up with the same thing. So, do you know what your r is? What's your r? Four. Do you know what your theta is?
Can you convert that using that formula? Yes. It's really easy.
So the x-coordinate will equal, well, r is 4 times cosine of theta is pi over 6. Might be useful to have a unit circle handy that gives you those relationships for sine and cosine of pi over 6 and whatever the other ones are that we're going to use. y. y is going to equal, well, look up y.
y says take your r, multiply by sine of your theta. And if you If you do that, you automatically get your rectangle four. It's really, really, really nice.
X is four times, can you tell me what's cosine of pi over six, do you know? Three over two. That's what it is, three over two.
Cosine counts down, sine counts up. Four times sine of pi over six. One.
So that means that we're gonna get x equals 2 root 3, y equals 2. Can you make a point out of that? So write this as a point. Our point is, what's that square root of? And your y coordinate? 2. So what we're saying here is that this point represented as polar coordinates, pi over 6, I'm sorry, 4 comma pi over 6, where we have an angle of pi over 6 and a length of 4, is also represented by the rectangular coordinate 2 root 3 comma 2. Is it in the same quadrant?
Yes. X is positive, y is positive, it's got to be up there, and that's exactly how you represent that. Make sense? Kind of neat, right?
Interesting at least. Let's do a few more to get the hang of this. All right. Thank you.
Thank Ok, we're going to change this. What have I given you? Have I given you rectangular, have I given you polar, what have I given you?
Polar. Sure, what tells you? The one with the pile of poles.
Yeah, it's an angle. Ok, so let's go ahead and we're going to change these into rectangular. I want you guys to do it.
So all we need to do is identify the r, identify the theta, and use these two formulas. X is given by this, that's x coordinate. Y is given by that, that's a y coordinate.
Go for it. What should the first thing you be, well, what should you do first? Graph everything. Why do we graph things?
To make sure it's in the right quadrant. Because I told you to. But yeah, to make sure it's in the right quadrant.
Leonard said. Leonard said. Forget Simon. Uh, yeah. What quadrant did we end up in?
We ended up in 3. So our point better be in quadrant 3. So we go, okay, well, wait a minute. X is r cosine theta, so x is negative square root 2 times cosine pi over 4. Don't get these two things confused. X is with cosine.
Y is with sine. Well, I guess that's not, that's pretty obvious, right? Because when you have a unit circle, it's cosine sine, isn't it? That's x and y, x and y, same thing. So when we have this, x is negative square root 2 times square root of 2 over 2, and y is negative square root 2 times square root of 2 over 2. Wait a minute, that's interesting.
What's this going to give you? Maybe one. Why is negative 1?
Interesting. Can you come up with a point with these two? Yeah.
Does that fit well with where we're at on our full importance? Yes. Yeah, that's a 45 degree angle from here. That's right in the middle of our 90 degree angle. And it's down square root of 2. Well, it's down square root of 2. That's 1 squared plus 1 squared and then we...
take a square root. So that's exactly what that is, is negative one, negative one. So if you guys feel okay with our changing. Now, we can also go back.
We can change between rectangular and polar. So we did from polar to rectangular, now rectangular to polar. This is where the quadrant really does come into play and you're gonna see that on a lot of these examples.
So let's say that I give you this one. Negative one comma one. Is it apparent to you that that right there is in rectangular coordinates?
It doesn't have an obvious angle. Now usually I'll be a little bit more explicit here and I'll say change this rectangular coordinate, or coordinates pair. into a ordered pair, a rectangular ordered pair, into polar coordinates. I'll tell you explicitly. Right now I'm just gonna tell you in words.
That's rectangular, we wanna change it into polar. Well, what do we need to know in order to change something from rectangular to polar? Right now we have x and y. What do we want to have for polar? We want r, we've got to have r.
What else do we want? An angle. An angle, theta, that's right.
Well you know what, two of these things are going to give that to us. And it's not going to be very nice to use these, because this has both of our unknowns in it, doesn't it? That has both of our unknowns in it. Oh, but this one, this one has both of the things that we know in it, doesn't it?
This one has both of the things that we know in it. So to go from polar to r, we're going to have to have r. polar to rectangular, we use these two.
That's why you have them listed. To go from rectangular to polar, we use the other two. Does that make sense to you?
I'll show you exactly how that works right now. So, R squared. R squared equals X squared plus Y squared. So in our case, we know X and we know Y, so let's do it. R squared's gonna be what, please?
Square. Which equals? One. Or two.
Okay. Okay, how about r? r equals what?
Square root of two. That's right, square root of two. Now, kind of, kind of, here's how it works.
You see how when we do a square root, you typically put a plus or a minus? We're going to omit the plus or the minus. but that's what can mess up your quadrant is because we get that right there.
So we're going to take care of the correct quadrant with our angle. We're always going to try to make a positive angle. We're always going to try to make it be in the correct quadrant.
But why this screws up is because... because when you take a square root, we typically omit the negative here. Does that make sense to you?
You understand why we can mess this up? Okay, so r is square root of 2. That's nice. We know how long this line segment's going to be. What's the other thing we've got to find?
Data. So here's part number 1. Here's part number 2. If tangent theta equals y over x, and again, the reason why I'm using tangent right now is because it relates the two pieces we know. We know x.
and we know why we didn't know artists started with we didn't know artists started with so If we know x and y, then tan theta equals, what's our y? Yeah, let's get these ones right. 1 over, so that's going to be negative, what's tan theta equals negative 1? What's the angle going to be there, do you know?
You actually have choices here. There's a lot of angles that are going to give you. tan theta equals negative 1. A lot of them.
An infinite number of them. We're going to pick the right one. What we're looking for is a positive angle between 0 and 2 pi. We don't want like 77 pi over whatever.
I don't want that. Okay? I want...
Something between 0 and 2 pi. I want it to be positive. So let's look at what happened.
This is what we should have done first. This was a rectangular system. So if we plot these points, what quadrant is it in? 2. Yeah, it's right there.
And what we want is we want to associate the angle to this and the length to this. We want to convert that rectangular point into a polar coordinate. So far so good? We've already done part of it. We know that this distance is going to be the square root root of 2. But we got to pick the angle correctly so that we get the positive angle out of it.
What you could do, you could say well you know what this is negative pi over 4. That would give that to you, but then you'd have to use the negative square root of 2 to get there. Do you guys see what we're talking about? Since we avoided the negative square root of 2, we can't use the negative pi over 4. That's not going to do that.
So if ever you get the negative angle, like, man, that's in the wrong quadrant, just add pi to it for tangent, and it's going to make it in the correct quadrant. So for us, we think about this first. This is why we graph it first, to make sure you have the correct quadrant.
So if you understand that idea, you've got to have the correct quadrant. So plot your rectangulars first. and then start looking for your angle. So if we're going to be in quadrant two, and we have to have tangent of our angle equal negative one, we're going to have to be at three pi over four. Using your circle or think back to your trigonometry, this is going to have to be three pi over four.
Ok, I need to make sure that this makes sense to you. This to me is a little bit harder than this one. This one is very straight forward.
This is a little bit different. Are you guys ok with that one? Yes. But you've seen how important it is to graph your point first so you know what quadrant you're in.
Ok, now let's put it together. What polar coordinates do we have? Yeah, what's our point?
Root two. And? Three pi over four. That's it. So we know we're gonna go three pi over four, and we have a length of square root of two, and that's it.
So far so good? Are you sure? Okay, let me show you one more, and then we'll go on to symmetry and how to graph these things. So I'll give you a rectangular, rectangular coordinate system.
Order of pair. Three comma negative four. What quadrant is that going to be in? Do you know?
Four. So that's going to be one, two, three, one, two, three, four. So we're going to be down here somewhere.
What we want is that distance and this angle. So far so good? Let's do it.
Why don't you guys go through it? I want you to find at least the R. Go ahead and find the R right now.
And then get down to the trigonometric relationship that's going to allow us to find the theta. So go for it. Thank you.
Let's see, x squared. Y squared, r is equal to the square root of what? 25. 25, which is 5. Oh, perfect.
R is 5. I like it. We already have half of our work done. Did you find r is 5?
Yes. Show me if you got it. Yes. Perfect.
Okay. Now, do you notice we're omitting? We're getting the negative 5. Okay. Now, what else do we need?
Tangent, theta. Why tangent? Why are we using tangent?
Not sine or cosine. What's tangent compare? Y.
And that's what we want. We're given an x and y. So tan theta equals negative 4 over. Over 3. Okay.
What angle is going to give you negative 4? This one was easy. It's negative 1. It's somewhere that's actually on our unit circle. If it had been root 3 over 2, it would be easy.
Or negative root 2 over 2, or negative 1 half, or positive 1 half, or any of those nice ones. If it's not nice like this or it's not on your unit circle, don't make it harder than it is. How do you solve for angles?
Do an inverse. Theta equals tan inverse of negative 4 over 3. You with me on that one? Now, here's the deal.
Don't let me mess you up here, but here's the deal. Check it out. This tan inverse is based on negative 4 over 3, negative 4 over 3, or 3 comma negative 4. It's going to be in the correct quadrant.
It has to be there, but it's going to be negative. This is a negative angle and I don't want that. This is how I know it's negative, by the way. Positive angles give you positive tangents. This is tangent.
Negative angles give you negative tangents. Therefore, when I go in reverse, tan inverse of negatives give me negative angles. Tan inverse of positive...
give me positive angles. This will be a negative angle, but it will be in the correct quadrant. Here's how you remedy that.
If you have something that's in the correct quadrant, but it's negative, add a full circle to it. Just add two pi and it will make it positive. So what we're gonna do is we're gonna add two pi, and this is to make positive.
So now we have it. Our r is 5. It's going to be ugly looking, but our angle is this. Listen, this is our theta, right?
This is an angle. It's tan inverse of negative 4 thirds plus 2 pi. It's a stupid looking angle, but that's the angle.
That's exactly what that is. Did that one make sense to you a little bit? Could that be considered like an answer you put down, or do you actually have to use the calculator? This is it. This is fine.
This is exact, okay? calculator radians and you actually do it you can find an approximation for that but if you do this in radians it's going to be negative it's going to be negative angle we want just to add 2 pi to that or 6.28 roughly on your calculator and then you get that whatever that is can I move on a little bit okay what I want to do now This is the last thing we'll do today. I want to show you how to find a rectangular equation from a polar equation.
This is very interesting and very useful for us because we're used to rectangular equations. We're not used to polar equations. We need to get used to them.
We need to get used to looking and going, oh, that's a this. Oh, that's a whatever it is. And this is one way that we do that.
We start by just changing some basic equations. So right now we're going to convert a polar equation into a rectangular equation. Now here's the deal.
When you convert a polar equation to a rectangular equation, we need something that relates these two ideas. It really relates these two ideas to us is this, the Pythagorean identity or the Pythagorean theorem for us. R squared equals x squared plus y squared.
That has x's, that has y's, that has r's. You with me? The theta's are...
going to be tied up into one of these things. So we're not worried so much about that right now. Well actually, I'll show you how that relates. But start with something like the Pythagorean theorem or identity and try to work with it.
If you remember with parametric equations, we actually did this, did something very similar. Remember how we had like, x equals something cosine theta, y equals something sine theta, and we used the Pythagorean Identity to put them together? Really similar idea. So here we go. R squared equals x squared plus y squared.
What we're gonna try to do, we're trying to make an R squared up here somewhere. Now the only way that I see how to do that is, well, what if I just multiplied both sides by R? Well then what this is gonna...
Remember, this is what we're trying to get, is r squared equals something. So, to use a substitution. So, if r squared equals 2 times r times sine theta, well, this is great. If we have an r squared, now we can make a substitution to change the r squared into something in terms of x and y. Remember, a polar equation has r's and theta's, right?
Rectangular equations don't. They have x's and y's. So, if I can... find r squared somewhere in there. How much is r squared equal to?
x squared plus y squared. So then r squared becomes x squared plus y squared. This is r squared. I'll do one more step in a moment.
Quick show of hands if you're OK with that so far. Now, the next part of it. This is kind of fun.
Do you remember that x equals r cosine theta? Do you remember that y equals r sine theta? Remember that?
We had it up here a while back. Which one of these do you see in our equation right now? x and y.
I see x and y. I don't want to change it back, okay? I already had polar equation. I want rectangular.
So what am I going to do? My goal here is, if I, listen, listen, you need to get the goal. The goal is, if you have a polar and I want rectangular, I don't want r's.
I don't want thetas, I want x's, and I want y's. Does that make sense? So with our r, we go, let's just multiply both sides by r. We get r squared.
That's x squared, y squared. Hey, halfway done. On the right hand side we got r sine theta. Let's do it all at once.
How much is r sine theta? Y. Because I asked you.
Never too old to bring back that joke. Love that. Okay, yeah, you're exactly right.
So this is x squared plus y squared. This is 2. Y. Good. 2 is there.
This is? This is 2. Y. Very confidently said. When you're little, when you're like seven.
Why? Why? No.
No. Yes. Because I said why. Hey, is everything in terms of X and Y? Yes.
You're done. That is a rectangular equation, isn't it? It's all in terms of X and Y. You're done.
Now, do you know what it looks like? No. Can you make it into something you know what it looks like?
Yes. Yes. So practically you're done. In actuality, you can make it actually look like something.
Now, I'm going to do this for you. so that you see what's going on. I'm gonna go move kind of quickly through it, but here's the idea. Did I explain it well enough for you guys to see that between polar and rectangular, you wanna get rid of r's and thetas.
You wanna get x's and y's. Square both sides if you have just an r, hey, that's a square y squared. Bless you. R sine theta is y.
R cosine theta, that's x. We can put that in there. So, here we go.
Let's say I get everything, so basically we're done, but. But not done. We can do more.
If I get everything to one side, I end up getting x squared plus y squared minus 2y equals 0. You guys know it? If I complete the square, I get x squared plus y squared minus 2y. Take half this number, square it, add it to both sides.
You ever completed the square? That's now a perfect square trinomial. x squared plus y minus 1. This is half this number, that's how you complete the square. I don't have time to explain how to do that.
You should know how to complete the square. But once we do, we now have x squared plus y minus 1 squared equals 1. Oh my gosh, holy cow. That's something that we should know what that is. What shape is that in terms of a rectangular equation?
It's a circle. Everybody, what is the radius of the circle? Remember this is the radius squared?
Can you find the center of this circle? Zero. Nothing to add, subtract.
It's zero and one. It's opposite of what you want to say it is. Zero and one.
The center is zero one. So this is really a circle. The center is at zero one. The radius is one. Could you graph it?
Just go to zero, one for the center, go up one, left one, right one, down one, make a circle, and that's exactly what that thing is gonna look like. Now, that's a little bit more concise in terms of the equation, but once we get it. a handle on, oh, well, that's what this looks like.
That's a circle. We can start seeing these things a little bit better. We're going to get some really, really cool-looking graphs out of our polar equations that we wouldn't be able to get with rectangular, or that would be really hard to get with rectangular.
Because with rectangular, you have these problems with not being functions. Now, that's not a, obviously not a function. It's a circle.
But we have some problems with rectangular equations that are just making it really hard. So we're going to learn how to graph next time. We'll learn about symmetry, and then we'll learn how to use that to our advantage.
With me? Did I explain it well enough today for you? Yeah.
Okay. We're going to graph some of these polar equations, and what I put on the board here is a way that you can eliminate a lot of the work, a lot of the work here. So we have symmetry. You'll remember from, like, your rectangular equations.
that we can have even functions, we can have odd functions, we have functions that are symmetrical about, well, the x and the y-axis, like I said, and about the origin. We have the same thing with polar equations. So here's how you determine that.
If you can take, so suppose we have some function r, r equals f of theta. So what that means, that means that we have a polar equation here. If you can take and plug in negative theta and it gives you back your original function, that means that this polar equation is gonna be symmetric about the polar axis. The polar axis is the one that goes this way. It's the one where we start.
We talked about that last time. So it's gonna be your x-axis. So it's just gonna be flippy floppy across that axis.
Does that make sense? Now, if we plug in, and this is a little weird, but if you plug in pi minus theta, theta and it gives us back our original function. That means that we're symmetric about the pi equals 2 axis. So theta equals pi over 2. That is this one.
Pi over 2 goes this way. Remember that from our unit circle and from our polar graphs that we did this way. So it's flippy floppy across kind of the y axis here.
That's the idea. If we have this one, which I don't know if we're going to get to use that very often, but if we have when you plug in theta plus pi, it gives you negative of our original function. That means that we're symmetric about the origin. You could take it, rotate it 180 degrees, and that's the same exact symmetry that we're talking about. Show of hands, are you okay with this?
Now I'm gonna show you how to use it right now. So simple example, not a hard one. I just want to show you one idea of symmetry.
I'll show you how to graph these, then I'll kind of extend this concept. So let's say that we have this equation, r equals one plus cosine theta. r equals one. plus cosine theta.
First thing, look for any symmetry, well, actually, you know what the first thing? Make sure it's a polar equation. Is that a polar equation?
How can you tell it's polar? R. R, that's right.
Are there any x's or y's? No, no, no x's or y's. There's r and theta.
That means you've got a polar equation here. After you determine it's a polar equation, because sometimes you're gonna have to convert it into polar equations, and that can be kind of tricky. I'll show you how to do that one.
of these examples. Next thing we check for, check for any symmetry. So here's how you check for symmetry. Check for this one first, go in this order. So here's our symmetry.
Let's suppose I have F of theta, well, f of theta is 1 plus cosine theta. We know this because r is a function of theta, so f of theta equals just that function. So set your f of theta equal to your equation, 1 plus, over the right side of your equation, 1 plus cosine theta. Now, how symmetry works is you take negative theta or you take pi minus theta or you take theta plus pi and you plug it into your... to your equation, you plug it in here.
So we're gonna say, all right, well, f of negative theta then is one plus cosine of negative theta. Click head and if you're okay, you understand the concept here. So here's what we're checking for.
We're saying, hey, if I plug in negative theta, is it gonna give me the same? I think he's going to plug in positive theta, because if it is, that's symmetric about the x-axis, the polar axis here. Make sense? So what we've done is said, okay, what we're trying to see is, does this give me this? If it does, then these two things are equal.
Well, this is going to require you to use some identities. So in our case, 1 plus cosine of negative theta. If you know this identity, cosine of negative theta is the same as cosine positive theta.
If you think of cosine, cosine is an even function. It's symmetric about. about the y, so remember that one, it goes this way, it's symmetric, you fold it over on itself. Well, what that means is that cosine negative theta is the same as cosine theta.
If you want to look up that identity, it's in there. It's there. Well, now check it out. Is this the same as what we started with? Yeah.
So what that says is that f of theta, hey, here, is the same thing as f of negative theta, therefore, this says, and you're gonna state this out, this says that this is symmetric about the polar axis. Thank you. So, if you feel okay with that one. Now, we can stop there or we can check for more symmetry.
I'm going to stop there because I know it's not going to satisfy these other two. And we can deal with this one a little bit differently than finding every possible symmetry. So, here's the idea. Think about this, think about why this helps you out.
If you know that something is symmetric about your x axis, your polar axis, then if we can find this half, we automatically find this half. Does that make sense? So what we're going to do is we're just going to plug in the angles from zero.
to pi. Because if we can plug in from 0 to pi, then we can find out pi to 2 pi. Make sense?
So we use the symmetry to eliminate like half our work, or 3 fourths of our work if we have double symmetry. Like across pi over 2 and the polar axis. It's kind of nice.
So here's how I'm going to show you how to graph it. First thing you do, step 1, plug in points. Step 2, think.
That's it. You've got to think about it. You've got to take those points and think about how about how the symmetry works, think about what the points are doing, and I'll show you that right now.
This is the reason why we learned how to plot polar coordinates is for this right here. So, one option you have, and this is a really good option, start by creating a table. Now, the type of table we're gonna use, it almost looks like a graph of x and y. It's like a rectangular system, but you're using r and theta here.
Now, this is not the graph of our function, it's just a way that. that you can organize your points. Does that make sense to you? Better than a T-table for us.
So here's what we're going to do. We're going to have r on this axis. We're going to have theta on this axis. Use values of theta to find out values of r.
Of course, our r is our dependent variable. Theta is the one that you plug in and stuff to, okay? That's your independent variable. Plug in this.
You get out that. Don't go the other way around. It's really hard to do that. So plug in values of your theta.
Get out values of your r. Also, plug in values of your theta. of your theta that are easy. Don't pick up, let's say pi over 16. That's a good one, don't do that. What are some easy values of theta to plug in?
Think easier. Zero. Think easier. Pi.
Zero. Zero. Let's plug in zero.
Zero. Zero's a great one. A great one.
Now, plug in some other ones. PI over 6, okay. We can do that. We can plug in PI over 4. I'm not going to plug in PI over 6 because we're going to get a good picture without that one. If you need to flesh this thing out more, then you use stuff like PI over 6, PI over 3. Let's plug in 6. Let's do stuff like pi over two.
Let's plug in stuff like pi over four. Let's do stuff like, what's the next one? Three pi over four.
Let's plug in stuff like pi. Again, can you explain to me why I'm stopping at pi right now? Because if we're going to get half of it, we're going to get the other half.
I know I'm symmetric. If I go to zero to pi on this particular graph, because I don't have a 2 theta, I just have a theta right here. If I just have a theta, that means I'm going to go from zero to pi.
That's going to be half my graph. The other half is going to be on the bottom side. It's going to be symmetric about the x axis. So, can you guys feel okay with this so far? Yes, no?
Okay. Now, let's plug them in. We're going to go kind of quickly here because I'm going to. Assume that you're pretty good at plugging these things in. So let's plug in zero.
Where am I plugging in zero? Cosine theta. Yeah, let's plug in zero. What's cosine of zero?
One. Plus one. So when I plug in zero, notice this is not a polar graph right now. All I'm doing is listing out my points.
If I plug in theta equals zero, I get out. 2. Does that make sense to you? Okay, now, let's plug in pi over 4. If I plug in pi over 4, what's cosine pi over 4? Root 2 over 2. Root 2 over 2. Plus 1. Plus 1. How much is that?
1 plus root 2 over 2, wherever that happens to be. So, 1 plus root 2 over 2. It's above 1, it's not to 2. It's like 2.71, I think. Isn't it? Somewhere around there. You mean 1.2.
No, I mean 2, because you're adding 1 to it. So I, I'm just, you're right, 1.2. Figure it out, what is that? I lost track of my math in my head. 1.7 something?
1.7. 1.7, you're right, 1.7. Let's keep going.
Pi over 2, pi over 2. If I plug in pi over 2, let's cosine pi over 2. And then plus 1. So we're just using some trigonometry here. All right. How about 3 pi over 4?
What's cosine of 3 pi over 4? Negative root 2. Negative root 2 over 2. Okay. So 1 minus root 2 over 2. How much is that? 1 minus root 2 over 2? 1 minus root 2. How much is it about?
1 and 3. 3. So we're done. And then lastly we have, so 1 minus, so this is 1, this is 1 minus 2 over 2. And lastly, cosine of pi. Everybody, what's cosine of pi? 1 plus negative 1 is how much?
together in our polar graph right now, but I want to know firstly, are you okay with getting all these points? Show of hands if you are. Is this a polar graph?
No, it's just relating theta to r so that we can follow it almost like a t-table for x and y. Now we're going to translate this to a polar graph. So here's pi over 2, axis here's 0. So here we go.
Hey, when our angle is zero, what's our r when our angle is zero? So where is angle zero? Where is it?
Right on what? Right on the fuller axis, so right here. So when we're at angle zero, remember all the graph when we did last time. That's why we did this. So we can plot these points.
That's the nice thing about it. So we have this at angle zero we have two. So we have one point right there.
So I'm actually okay with this one so far. Now where's pi over four? So that's pi over four. What's our r value at pi over 4? 1 plus 2. About 1.7 or so.
So here's what we're going to do. We're going to imagine this is 1.7. our point should be right there.
So just take that idea and move it up on that line. Does that make sense to you? Are you sure? I don't wanna lose you here. I wanna make sure that you're okay with it.
Do you get how we're taking these values and we're just now putting this on? them on this graph. It's just a different way to think about it. This was angle of zero, you had two. Angle of zero, I have two.
Angle of pi over four, I have 1.7. Angle of pi over four, I have 1.7. It's gonna be less than this distance by just a little bit. So far so good? Now we go to the next one.
At pi over two, what's our r? Well that's an easy one. At pi over two, we're at one.
Ok, now at 3 pi over 4, where's 3 pi over 4? Yeah, it's right here. How much are we at 3 pi over 4? So think about how much that is with a calculator or something. Plug in 1 minus root 2 over 2, get an approximation.
It's about.2, 5 over 3, yeah. So here's, check it out. Here's your scale, correct?
So we have.3. It's about right there. And lastly, lastly, when I plug in pi, so I'm going, when I plug in pi, what's my r? That means I'm at.
The origin. Okay, I really need to make sure that you are okay with this. Do you see that at 0 we're at 2? Angle of 0 we're at 2. Angle of pi over 4 we're at 1.7, roughly.
Pi over 2 we're at about 1. Well, actually, exactly 1. At root 3 pi over 4 we're at... 0.3. At pi, we're at 0. Oh wait, we're at 0. So yeah, it has an angle, but it has no distance there. We're right at the origin, right at the pole.
Honestly, did you guys feel OK with that one? Now, we use... use this idea of, well, r is shrinking as we're going down to graph this.
So does it look like r is going to shrink? The r shrinking means that these points are going to always get closer and closer and closer to our pole because we always start here. So how is this going to look?
It's not going to do weird things. Because if it did weird things, our r would be shrinking and expanding. And it's not doing that.
Our r is constantly shrinking, constantly decreasing. What this looks like... It looks like it's almost like the start of a spiral. Like that.
That's how that looks. Kind of cool, right? Now.
What else do we know about this? Symmetry. That's why we did symmetry.
So we didn't have to graph all the rest of it. So if we know it's symmetrical, then all we gotta do is put these points at the same spots but on the other side of our axis and we get this. That's, does anybody know what that's called actually? What's it look like if you turn your head to the side?
A heart? Actually it kind of looks like a butt, but it's not called the, it's not called the buttioid, it looks like a heart, so they call it the cardioid. This is called the cardioid right here, and it kind of does look like a heart.
like a buttock, doesn't it? It's horrible. I always thought it should be the buddyoid, but it's a cardioid because it looks like a heart. I told you we were gonna drop some pretty pictures today. Should have answered you okay with this one.
That's the idea. Find out if it's polar. equation or not.
If it is, great. If it's not, make it a polar equation. After that, find out your symmetry. You could have three types of symmetry as I've talked about over here.
After you find your symmetry, use it. So don't be doing more work than you have to. Here, if I know it's symmetric about the polar axis, I'm going to go from zero to pi, no problem.
Just flip you flop you right over the polar axis and we have ourselves our graph. So far so good? Okay, we're going to continue this.
Are there any questions before we continue? Am I going too fast or am I explaining well enough for you? Okay. Okay, so let's do it.
Okay, r equals 2 cosine 2 theta. What's the first thing that we look for in this type of graphing that we're doing? What kind of coordinate is it?
Yeah, what kind of equation it is. Is it polar? Is it rectangular? What is it? It's polar right now.
Is it easy to tell? Should be pretty easy. There's no x's and no y's.
It's just r and theta. So that's nice. It's very concise too, right?
I mean, it's kind of weird that that concise of a graph can give such a complex picture. It's kind of cool. So now that we've determined that this is a polar equation, what's the next step? What's the next thing that we're going to do?
Say what? Find symmetry. Are we going to start plugging in points right now?
No. That's a bad idea because we don't even know our symmetry. We don't even know what points to plug in.
So let's start plugging in some points. I'm sorry. That was a bad, I said that the wrong way.
So we can't start plugging in some points. I said plug in some points. We need to find symmetry so we know what points to plug in. Let's do our symmetry.
Let's start with the same one. Let's start with, we know that f of theta is. 2 cosine 2 theta. Let's think about f of negative theta.
f of negative theta, what's the only thing that's going to change? The theta. So this would be 2 cosine 2 times negative theta.
Well, if you look at that, 2 times negative theta is negative 2 theta. Now, you guys tell me, how much is cosine of negative 2? 2 theta equal to, according to the identity that we just used, what is it? It says here that cosine negative theta is the same as cosine theta. Therefore, cosine negative 2 theta is the same thing as cosine 2 theta.
Does that show you any symmetry in this problem right now? Yes. How this, when I plugged in negative theta, gives us the same thing what we started with.
Gives us our original function back. Right now, what's that say? How is that symmetrical? Over the x-axis. Very good.
We're not going to call it the x-axis. We're going to call it the polar axis. So this is symmetric about the polar axis. Honest show of hands if you do understand why.
Guys in the middle are we okay with that one? Yes, no? Or are we at the end? If we plug in negative theta and it works out that when we plug in negative theta it gives us positive the. the same function with positive theta back again, if it gives us the same thing back again, we're symmetric about the polar axis.
So that's what we try. We plug in negative theta. Hey, look at that.
It's the same thing, but with negative theta. Now, the question becomes, is cosine of negative two theta theta the same thing as cosine of positive 2 theta. It is by our identity. We know that about cosine.
If I plug in 1 theta or a negative of that same angle, I'm going to get the same thing out. So we say, all right, well, when I plug in the negative theta, ultimately, it gave us back our original function. What that says is that we're symmetric about the polar axis. That's all we're saying right now.
Now, let's try another one. I want to show you this one. Let's start from the beginning. So f of theta is 2. two cosine two theta.
Let's now plug in pi minus theta. I want to see if this thing is symmetric about the y-axis, only we call it the theta equals pi over two line. We'll see. So if I plug in that, what's this gonna look like? What's going to be there?
2's going to be there. 2's going to be there. Okay. 2 and then what?
Definitely going to be there and then what? 2. 2's still going to be there. Now what's going to be here? Parentheses. Oh, I like how you said parentheses.
You see why the parentheses are so important here? Because if I have 2 theta and I plug in pi minus theta, that 2 is going to do what? Distribute. Distribute. So this is going to become 2 cosine 2 pi minus 2 theta.
Quick head down if you're okay with that so far. Just some trigonometry that we're going through right now. So you okay with a little substitution?
This just goes in place of our theta. So if I go from here to here, the only thing that's changing is my theta becomes pi minus theta. All right, let's distribute that.
Not a problem. Let's use an identity here. We're gonna have to split this up because right now I have no idea what that thing looks like at all.
I don't know what to do, and I don't know if it's the same as that. Remember, I'm ultimately trying to get back to the sequence. whether it's the same as this thing or not. Does that make sense? You sure?
Okay, so let's use an identity. Maybe like the subtraction identity for cosine of two angles getting subtracted. So what that is is two, I'm gonna pull that two out because we're gonna have subtraction identity here.
We have cosine two pi times cosine two theta minus sine two pi times sine of two theta. Where am I coming from? Am I making this up? No. Yes, just making it up randomly.
It's complete magic. What is that? It's an identity that you would use right now. Have you found that identity yet? Have you looked for it?
I hope I have it right. You might want to double check me. Check it out. Do you have that identity down somewhere?
Many chapters ago. Yeah, a long time ago in your trig class. It's in the back of your book too. So we go, right, well, this is the way that the subtraction identity works for cosine.
You say cosine of this times cosine of this minus sine of this times sine of that. That's the way it works. So. So let's keep on going. Can you tell me how much, this is the fun part, because we need to actually use some other identities to make parts of this just disappear.
How much is cosine of two pi? Yeah, same thing as cosine of 0. This whole thing is 1. How much is sine of 2 pi? It's the same thing as sine of 0. How much is it? I hope that you said 0. And we're just mumbling 1 mistakenly.
What's 0 times anything? This whole thing's gone. What's 1 times cosine 2 theta?
So this is 2 times cosine 2 theta minus... Oh, wait a minute. Did it work for us?
What type of symmetry do we have? Symmetry across what? It's like the y axis, but we call it the, yeah, theta equals pi over 2. So this says, soundproof about theta equals pi over 2. Yeah, it's like the y axis. It's basically the horizontal axis from here and the vertical axis from here.
Quick show of hands if you're okay with that one. That was a lot more advanced than the previous ones. Okay, these are easy.
When you just plug in negative theta, it's pretty easy to see. you can see that right away. Either you have it or you don't. This takes a little bit more work because when you do pi minus theta, you're gonna have to have some sort of subtraction identity. If you did, you know what, I erased it, but if you did theta plus pi, you're gonna have some sort of addition identity depending on whether you're doing or would they have sine or would they have cosine about that?
Does that make sense to you? So you're gonna have to have one of these identities. Here, not such a big deal if you know it.
So we distribute our two, okay, then we separate this by our subtraction identity, cosine and cosine, minus sine and sine. Hey, that goes to zero, not a problem. That goes to one, one times anything gives you that, anything back times two, we have two times cosine two theta.
And that, because this is exactly what we started with, says that we're symmetric about theta equals pi over two. So if I should get. Now, we know that this is symmetric about the x axis and the y axis.
Theta equals pi over 2. What that means is that we only need to go from 0 to where? Pi over 2. We're going to do that. We're going to go from 0 to pi over 2. Because if we do that, the rest of it is going to be symmetric, and then symmetric will go flippy floppy. Make sense?
Got my sunscreen, I got my flippy floppy. copies? On a boat?
No? Yeah. Why aren't we checking to see if it's symmetric for the last one? Because we don't need to. Because if we have this and we have this, it's going to take care of both of them for us.
So we don't need to worry about that one. And I'll show you why in just a second. I do want to make sure that you are okay on the symmetry right now. Are you okay with the symmetry? Okay.
Then we're going to do the same thing now. and I just erased over here. We're going to draw our, kind of a graphic organizer, saying here's our theta, here's our r, and finding out what those points are.
It's almost like a T-table for these polar coordinates. So we're going to plug this in. I want you to understand a couple things. I know I've said it already, but I want you to understand it.
Do you understand why we're only going from 0 to pi over 2? Why? Sure, we're getting symmetry here and here. So we only need to go from 0 to pi.
to pi over 2 because symmetry will take care of all the rest of it for us. So far so good? Now, here's the interesting part. You see the 2 theta?
Because we're going from 0 to pi over 2, it doesn't necessarily mean that our graph is only going to be between 0 and pi over 2. Our graph could be in other quadrants, but we get to use that symmetry from 0 to pi over 2 to our advantage. That's kind of nice. I'll show you that right now. So when you're plugging in points, plug in the in nice points for your particular equation. Don't be plugging in always the same points.
I don't want to see always pi over four, pi over two, because sometimes they can be nice, alright? Plug in some nice ones here. I'm going to show you what I'm talking about.
So, here's our theta. Here's our r. How far do we need to go? Where are we going to start?
Now think about what we have. What we're going to have to do here is we're going to have to plug in our theta and multiply it by two and then do cosine of it. So give me a good point to plug in. Zero.
Zero's great, yeah, we got zero. I got pi over 2. Okay. Besides those two ones. Pi over 4 is a good one.
Because if I do pi over 4 times 2, they give me pi over 2. You with me? Do you know what cosine pi over 2 is? Then that's a good one. So we're going to plug in pi over 4. more.
Give me another one. Pi over 6. If I did pi over 6, it's going to be pi over 3. I could do that one. I could also do pi over 8 maybe if I wanted to.
I chose pi over 8 because if I do pi over 8, 2 times pi over 8 gives me pi over 4 and cosine of pi over 4, I can use that as well. You can do either one of those. It's fine, but pick some nice points to plug in. Are you with me on that one?
For instance, I probably wouldn't be doing... things like, probably not pi over three, because out of that quadrant I don't want to deal with that. I want to deal with something that's in this, within this range, okie dokie. So if I did pi over three, that would be two pi over three, that's not within, that's, it's going to be a little weird for our graph, so pick some nice ones. I'm going to pick pi over eight.
I'm going to pick three pi over eight. Let's give it a try. Let's plug in zero. Everybody do this with me.
So remember that our r is equal to 2 cosine of 2 theta. What that means is we're plugging in our independent variable here of our theta. We're getting out our r and it's just going to be our distance.
We're just going to list them out right now. Plug in zero. What's 2 times zero? Zero.
Cosine of zero is? 1 times 2 is? Zero.
Start at 2 again. Let's plug in pi over 8. Remember that before you do the cosine, you've got to multiply it by 2 in this case. What's 2 times pi over 8? Pi over 4. Pi over 4. OK.
So if I have pi over 4, what's cosine of pi over 4? Times 2. Times 2. Square root of 2. Square root of 2, which is 1.7. About. So I've got a square root of I want to make sure that you guys are okay with that.
Are you seeing where these numbers are coming from? Honestly, are you seeing where they're coming from? Okay. How about pi over 4? Let's do that one.
Pi over 4 goes here. What's 2 times pi over 4? Pi over 2. Pi over 2, very good. What's cosine?
Cosine. of pi over 2? 0 times 2 is? Oh, wait.
That's weird. OK. This is weird. Check this out. Let's do 3 pi over 8. 2 times, this is why, it's unavoidable right now, but our graph's going to get a little funky.
So check it out. 2 times 3 pi over 8. What's 2 times 3 pi over 8? 3 pi over 4. OK. 3 pi over 4. 3 pi over 4 is...
what's cosine of 3 pi over 4? Negative square root of 2. Times 2 is negative root of what? Negative root of 2?
Lastly... Let's play pi over 2. Let's see how this one works out for us. Pi over 2, what's 2 times pi over 2 please? Oh this is weak sauce. What's 2 times pi over 2?
Very good. Alright, cosine of pi is? Negative 1. Negative 1 times 2?
Negative 2. Sure enough, it's negative. Is that okay to have a negative r? Yes.
Did I teach you how to graph polar coordinates with negative r? Yes. Sure as heck you did. You know that.
So yeah, here's what's happening. First of all, are you guys all okay with the points? Show of hands if you are. Good.
All right. And so we're just jumping, jumping some tree angles. No problem, just put it in.
Our r is gonna start at a length of two. It's gonna work itself down to zero, and then it's gonna bounce back up and have a length of negative two. If we can graph this, then we can use our symmetry and get a really nice picture out of it.
You guys ready for it? So let's go for it. I'm going to start this one, I'm going to do all these angles first.
So all of our dotted lines so that we get our appropriate angle relationships here. So we'll have pi over 4. If you have pi over 4, pi over 8 is not one that we typically graph with, but what would pi over 8 be in relation to 0 and pi over 4? Yeah, that's why we're going to use this one because it was nice to graph with. That's why I picked it because we go, okay, you know what, pi over 8 is right between them.
That's kind of nice to have that right there. That way you're not worried about, well, where's pi over 6 compared to that? It was a little bit lower than that, right?
Oh, pi over, okay. I'm sorry, a little above that, before you graph. So pi over 8, and then 3 pi over 8 is right between these two.
Are you still so far so good? Yes. Let's graph. So what happens when I have an angle of 0? What's my r, please?
Angle of 0, r is? Perfect. Angle of pi over 8. What's my r?
R root of 2. So when I get to pi over 8, check it out. Here's 0, I'm at 2. Pi over 8, I'm at about 1.7 square root of 2. So we're going to imagine it would be right here and take it up, probably right about there, square root of 2. Okay, seeing that we have the same exact values, just organized in a different way. This graphic organizer just lets you see what the points are. This is your actual points on a polar graph. You with me?
Next up, how about, oh wait a minute, pi over 4. Hmm. So pi over 4, I get. It gives me something right on the pole. Now look what's happening. This says that my R is decreasing to here and then decreasing very rapidly.
So what's going to happen is this thing is going to go boom right there. Now this is what's going to happen. So it's kind of fun. At three power eight, it says when I get to three power eight, what's my r?
Negative. Does negative go here? No.
What do you do with those negatives? 180. Oh, that's right. It's like right down here. Does that make sense to you? No.
Cool. So, here we go, right? So, down this way. And it's gonna be, looks like the same distance as this guy, but just down here.
Negative square root of 2. And last one. If I'm at pi over 2 that says go to your angle pi over 2, hey, we're right here, and then make it r of what please? Negative 2. Negative 2 is not up 2 but down 2. I want to make sure that you are all okay with those points before we move on any further, are you?
Is this weird a little bit? Kind of cool though, right? A little bit cool.
So what happens is you deal with your symmetry first after you determine that it's a polar equation or you make it. a polar equation. Here we have it. We know it's symmetrical about the polar axis this way.
We know it's symmetrical about the theta equals pi over 2 axis, the y-axis, this way. We know it's this way now. Well.
Now, if we graph from zero to pi over two, yeah, some funky things can happen. We can have some negatives. By the way, one reason why we did this and why we didn't do the revolving around the origin is because sometimes those take care of force. We don't have to worry about it.
So in our case, we have zero two, ain't no problem. Zero two, we have pi over eight root two. We have pi over four and zero, that gives our origin back. We have three pi over eight, oh, but it's negative.
That's 180 degrees different. And then we have pi over two, but it's negative. It's 180 degrees different. So here's what this looks like.
It says our r is decreasing really rapidly, then increasing really rapidly, and then increasing a little bit. So we have something that looks like this. Like that. Do you guys see the picture that I'm seeing out of that?
I know that you literally see it, but do you see why we get that picture? Do you understand why we do this and why we don't just list out our points? This gives you a look at what the r is doing.
It's shrinking to zero and then increasing to zero. but negatively and that's why we get that looking graph. You with me?
Now we get to use our symmetry. If we do, I know that this is going to be symmetrical. I know this is going to be symmetrical. I know this is going to be symmetrical. I know that's going to be symmetrical.
We'll put that in the graph. What we end up getting is this, it's going to be kind of sloppy, but there's two, because I'm not the best artist in the world, I wish I was, but we're going to get something that looks almost like a clover leaf. So it's going to go, okay, out this way, and then, ah, crap, down, and then out this way, and then back, and out this way, and back.
It looks a bad, horrible, ruined it, killed it. But that's the big, it should actually all go. go through the origin, like a pretty flower.
But that's the idea. If you were to do this correctly, it goes here, make it symmetrical, here, make it symmetrical, here, make it symmetrical there. I know I just brutalized this graph, but I'm wondering if you get the picture, pun intended.
Do you get the picture here, the idea? Show the fans if you do, if you feel okay with that one. Man, I wish, right side people, I got nothing from you guys. Are you all right with it?
All right, man, I wish that was a better. artist really do they've got to more time but I didn't see what if I did that My picture's not good either. That looks at least a little better.
It should look more like that. Still bad. Still bad. As long as you get the idea, this is the thing that we're trying to do. Have I made it make sense to you?
Explain it well enough for you? Okay, we're going to go ahead and... We're going to do one more.
Our last example. It's better on my paper. You know what?
I'm going to leave this. I'm going to do it over here. Are there any questions on this at all?
Do you understand where our points are coming from and why our graph looks the way it does because we have our symmetry? Yes, sir. Okay, then I've explained it well enough.
This gives us this symmetry. And so from here, You can go boom and you can go boom. So this gives us here and this. This gives us here and this.
This one you can get here or you can flip it. So it flips basically every way possible because we have symmetry on all of our axes here, which is kind of cool. OK, last one.
So I want us to sketch this. x squared plus y squared squared equals 4x squared minus y. Holy crap.
What? What in the world is that? Do you know what that looks like? No.
Me neither. I have no idea. Are you supposed to know what it looks like?
No. Because that's not even a folder right now. It's in XY and you'd have a hard time.
I'm even plugging in points right here, wouldn't you? What should you plug in? Do you even know the points on the graph?
Nope, don't even know them. So how would we even go about graphing this? I think you could do it with some symmetry.
You could probably get some symmetry out of this. but it's not gonna be all that easy to do. I'll tell you right now, that squared, that squared, that squared, that squared. It's gonna be symmetric about both the x and y axes just by looking at it.
Do you guys see what I'm talking about? So whether I'm pulling a positive x or a negative x, I'm gonna get the same thing. Now, we're gonna use that to our advantage. I'm not gonna show you the symmetry right now. Or, sorry, when we convert this.
I'm gonna tell you this is symmetric about the x axis and about the y axis. Do you see why? I mean, yeah, y is right there.
But, like, literally, do you see that whether I plug in positive or negative, it's going to be the same thing? Same output. Okay, and we're going to see that it's going to look, this is why I left this on the board, it's going to look really similar to that when I convert it, and we're going to have the same exact results. It's going to be x-axis means polar axis, y-axis means pi over 2-axis. So it's going to be symmetric about those two axes.
Again, I'll show you that right now. So number one thing, do we have polar or do we have rectangular? What do we have?
Rectangular. Let's convert it. Here's how you convert it.
fast, just to give you a taste. This is kind of like an extra example. You don't really need to see this one, but I wanted to give it to you.
So here's what we know. Here's our two conversions. Do you remember the next one?
X and Y? I know there's an R in there, but is X associated with sine or cosine? Cosine.
Uh-huh. Okay, I'm going to kind of do the wham-bam fancy-pants math, go real fast on this thing. So you might have to slow this down a little bit to do it on your own, no big deal. Do you see an R up here?
I do, right there. I see an R right there. Do you see the R?
That's an r. Use that to your advantage. Don't be changing that to r cosine theta squared and r sine theta squared.
It'll still work, but it'll take you a lot longer. So if you have an r, then this is now going to be r squared. squared.
Did you get that? So this is r squared and then we still have a square. Equals four, well there's not really a good way around this.
So we're going to have to have r cosine theta squared minus r sine theta squared. Well that's x, that's y, no problem. So we got r squared squared equals four inside here we're going to have r So, for r squared, we're gonna have cosine squared theta minus r squared and then sine squared theta.
If we factor out the r squared, we get r squared squared equals four. r squared and then we get cosine squared theta minus sine squared theta. Now is the time where you start using some identities. As soon as you have your cosines and your sines without any other garbage around them, this is r to the 4th, this is 4r squared.
This right here, there's an identity that puts cosine squared theta and sine squared theta together. It's not 1. The plus would be 1. Don't do 1. Please! This is cosine 2 theta.
That's what that is. If you don't believe me, you can look it up. It's there. Now, tell me something I can do with my r squareds.
If I divide, then this takes care of two of these. I get r squared equals 4 cosine 2 theta. Now lastly, I do want to solve this for r.
So what am I going to do? Square root. So r equals the square root of 4 cosine 2 theta.
There's one more thing I'm going to do. It happens with this 4. Square root of 4 is? 2. Yeah, that's what I'm looking for.
Right now that's converted to a polar equation. Now here's the point that I was trying to make earlier. This is what I really wanted you to see. From here on out, you guys are going to be fine, because you know how to plug in points, right? And we can find out symmetry, right?
Okay, now here's what I was trying to show you. From the rectangular equation, did you grasp the concept that this is symmetrical about the x-axis, sorry, the y-axis and the x-axis? Did you get that? It's all being squared? It has to be.
Now check this out. See how we have the cosine 2 theta? See how we have the cosine 2 theta? That one worked out to be symmetric about the polar axis and symmetric about this theta equals pi over 2 axis.
I'm not going to show you, but do you understand that that's going to work out to be exactly the same thing. It's going to be symmetric about the polar axis, and it's going to be symmetric about the pi equals 2 axis. You can show it if you want to.
The only difference is that's got a square root around it. It's going to work out to be the same. It takes a little bit more manipulation, but it's going to work out to be exactly the same.
So if you guys feel okay that this is going to be symmetric about these same two axes, you'll be okay with that one. I want to show you something interesting that happens here. So we're going to have symmetric about polar and theta equals pi over 2 axes. This is what gets just a little bit weird. You know what?
If I get theta and r, we'll do this again. If I plug in zero, check it. If I plug in zero, what's two times zero?
Cosine zero? One. Square root of 1?
Pi over 4. Pi over 4, yes. Cosine of pi over 4? 2. 2 times pi over 4 is?
Square root of root 2 over 2. It's like the square root of 2, square root of the square root of 2 over the square root of 2 times 2. It's a little awkward to do. What I want you to get out of this is that it's positive. Do you understand it's positive? It's up here somewhere. Alright.
Alright. Alright. Yes.
Let's do pi over four. Two times pi over four. I don't want to go through the math. I don't want to approximate it. I just want to get you this positive and it's up there a little bit.
Two times pi over four is what? Cosine of pi over two? Nice. It's gonna be zero.
So far it looks really similar to the last one that we did, yeah? But check this out. This is what's important for you.
If I try to plug in the next one, specifically if I try to plug in 3 pi over 8, or I try to plug in pi over 2, I want to show you just the pi over 2 because you can probably follow a little bit better. Plug in pi over 2. What's 2 times pi over 2? Now listen carefully. What's cosine of pi?
What's square root of negative 1? These are undefined. So I... I don't have this, it's outside of my domain.
Does that make sense to you? So you can show that with domain, you can actually find the domain, start it, I used to do it that way, but if you plug in the appropriate points, you're gonna find your domain. So our graph does this. We don't have some of the symmetry that we had in the last example. So what happens here is we're just gonna graph from zero to pi over four.
At zero, We were at 2. At pi over 8. We're about here at pi over 4. We're at 0. But that's all we have to go to. Can I go further? No, it's undefined. So really, this one doesn't have that extra little loop here.
which means that all I can do, instead of making this part symmetrical as well, it's just this part that gets symmetrical. What it ends up being is like an infinity sign. Now I want to stop and make sure that you really do grasp what I just said. So Jake, you okay getting down to that far, yeah? Yeah.
Are you okay that when I plug in points, these two things are going to give me a negative inside of my square root? That's a problem. That means it's undefined.
It's outside of my domain. So I can only go from 0 to pi over 4, and then I'm going to use this idea of I'm symmetric about the polar and symmetric about the pi over 2 axis. So hey, here's 0 to pi over 4. At 0, I was at 2. At pi over 8, I'm at something positive, but just a little bit less than 2. At pi over 8, I'm at something negative.
to have a form of 0. Now we need to use our symmetry. I'm symmetry this way and then this way. But I don't have up here. I don't have that.
So I'm going to have, oh, almost. I'm so close. Dang it. Come on, Leonard. That's the best I can do.
That's it. So this part gave a symmetrical here, and these two parts gave a symmetrical this way. Show of hands if that one made sense to you. Okay, this is the idea.
Have I explained it well enough for you guys to get it? Yes. Okay, so find out your...
your symmetry, plot your points to where you need it. If it's undefined, well you don't have to plot those points. And then just use your symmetry.
Alright, we're going to do the last little topic on section, section we're working on, on finding out how to deal with these polar... polar equations. In the next section, we're going to figure out the calculus of polar equations.
So right now, last thing we've got to deal with is how to find tangent lines of these things. So I want you to think of this, it's not exactly like parametric, but it acts a lot like parametric. You see here's the reason why. By the way, this says for x equals r cosine theta and y equals r sine theta, that basically just means this is for use with polar equations, that's all that means.
But now look at this. You see how... So, x is defined as a function of theta and y is defined as a function of theta. It's not x is defined as a function of y. It's not y is defined as a function of x.
So, really, there are a lot like parametric equations. Do you see that? So, if I want to find a derivative, if I want to find a derivative of y with respect to x, because these guys are not functions of each other, the formula for our derivative is going to start out really similar to our formula for...
parametric equations really similar because X is defined as a function of theta, Y is defined as a function of theta. You following the thought process here? Okay so let me show you what that's going to be. So remember with parametrics we had well if we want to find the derivative of dy, firstly you know what I don't want to ever forget what this is.
What is a derivative? What are we doing here? Derivative of slope.
That's right so when we're finding the derivative we're finding the slope. Now what's interesting is that with these polar equations they make some weird curves, right, weird shapes. We're still finding the slope as we remember the slope. That's why it's dy dx. It's going to be a slope in terms of rectangular idea.
So a rise over run idea. Imagine if you understand that concept. So that's why we're not doing, well, let's just find the slope with respect to this or with respect to this.
It's dy dx because this is going to give us the actual rise over run. Click here now if you're okay with that. So now when we had parametrics, we did stuff like.
like this we did, do you remember it? How it was dy dt over dx dt, do you remember that? Only now we don't have a t here, we have a what?
Theta. So instead of dy dt, it's really just dy d theta. Instead of dx dt, it's just dx d theta.
So what we're gonna do right now is we're gonna look at how to find the derivative of dy d theta and dx d theta. So we're gonna take this off to the side for a little bit. So remember this idea, this is where we're gonna So we're going to come back to, let's find out if we want to do dy d theta, what that is going to be.
So notice how dy d theta says, okay, well, that says the derivative of y with respect to theta. That's the derivative of y with respect to theta. Do you all see where that's coming from? Let's really focus up here for a second. So dy d theta says take the derivative of y, that is y, with respect to theta.
respect to theta. Now here's the deal. What do we need to do with this? If I want to find the derivative of this side, we're good to go.
What's this do for us? How can we do that? Tell me something that you know about r.
Constant. Constant. Is it a constant?
Does r stay the same for all those functions that we drew? No. No, r changed and it was based on the angle.
Do you remember that? How r started at 2 a lot of the times for us on our examples by coincidence and ended up going to 0. So is r a constant? No. Definitely not a constant.
But r also was based on theta, wasn't it? For a certain angle, we had a certain r, right? And then we changed for a different angle. What that means is that, and you should know this because we had this the whole time, r is a function of...
of theta. R is a function of theta. Okay now listen, I'll put this together with your calculus ideas. We've got R times sine theta. Sine theta is clearly a function of theta, yeah?
What's going on between R and sine? Product. And what's R? Function of theta.
So we have a function of theta times a function of theta. What do I need here? Product rule. We need the product rule.
You with me? Go ahead and let's do the product rule together, alright? So our product rule says we're going to do the derivative with the derivative.
with respect to theta of r, so derivative of the first times the second plus the first times the derivative of the second. Well, let's think about what this is. Now, now, listen carefully because we're using like three different ideas here, okay?
We're using the idea that it's similar to parametric because they're not defined as a function of one another. We can treat them as a function of theta and that's pretty easy. We're using that, well. R is a function of theta, therefore, have a product rule going on. Do you understand why we have a product rule going on?
It's a function of theta times a function of theta. Also, R is a function of theta, but it's not given explicitly. It doesn't say R equals this.
It doesn't say that. So if r is a function of theta that's not given explicitly, what type of differentiation do I need to use? Implicit? Implicit differentiation, which means that every time we take a derivative of r, look at this, look at this.
It says, hey, take the derivative of r. What's the derivative of r? Dvr? I don't know, because it's with respect to theta, not with respect to r.
So you don't just go, oh, well, it's one, because it's implicit. It's a function of theta, therefore we say, well. Well, the derivative of r with respect to theta is dr d theta. It's exactly what that is. Now, sine theta, it says we don't touch the second one, so times sine theta plus, we don't touch the first one, that's r, and we multiply by the derivative of, now this one's easy, it's a function of theta, it even has theta up there, d theta.
So what's the derivative of sine of theta? Curve of theta. Sure. So, if you feel okay with that so far. Do you understand the idea that we had to use implicit?
Because r is a function of theta. Well, we're taking a derivative with respect to theta. We have to have dr d theta. Do you understand the idea of the product rule here?
Okay, the next one, well, it says, well, look at that. All we needed to do to find our derivative, dy dx, we needed dy d theta. We've got it. This is dy d theta. I'm going to kind of do a shortcut here.
I'm just going to state it. Do you understand that you can do exactly the same thing with dxd theta? Do you guys get that?
With dxd theta, you can even do it from here. We'd say derivative of the first times the second plus the first. Times the derivative of the second, which is negative sign, hence we get a minus. Theta. Okay, I want to click your hands if you feel okay with that so far.
You sure? You sure you're sure? All right, let's put it together.
So, we have this. We said, all right, no problem. dy d theta, this stands for a slope, just like we always think of a slope. So even though we have these weird shapes that we're getting on these polar equations, equations, the slope is still the slope.
It's still rise over run. Hey, look at that. Change in y over change in x. Rise over run.
Only, we don't have y as a function of x, we don't have x as a function of y. We have them both as functions of theta. No problem. So we say, well, let's just do that. dy d theta, dx d theta.
Dy d theta, because of our product rule, because of our implicit differentiation, we get this. Dx d theta, same thing, we get this. If we create that ratio that we want, dy d theta, that's our formula for a derivative with respect to our polar equations. Have I explained this well enough for you guys to get it?
Are you sure? So, I'll just write the last thing out here. So dy dx, therefore, is this.
You do dr d theta, you multiply by sine theta, you add r cosine theta over dr d theta, you get the same thing, times cosine theta minus r sine theta. That is what we worked for. Now again, we're working towards finding our tangent lines, okay?
So one part of finding tangent lines is being able to find the slope because we know that where we have horizontal tangent lines, I'll write this down at the very end. We're not actually going to do an example with this, but. Where we have horizontal tangent lines is where the slope is zero, or where the numerator would equal zero. Does that make sense to you?
Where we have vertical tangent lines is where we're undefined, where the denominator would equal zero. You with me? Okay, so we're going to talk about that in just a little bit, but our first step is actually finding the slope.
So let's go ahead, let's do one quick example. I'm going to give you a polar equation here. R equals that.
You know what? Just out of curiosity, does anybody remember what that shape is? It's a cardioid. It's a cardioid. That's right.
That's the cardioid. Remember that one? The butt.
No. I should have said that. It's not the buttioid. Yeah, it's the cardioid, and it's this way. It's to the side.
One plus sine theta is a cardioid, but upward. but it's very similar, it's just in a different direction. So that's a cardioid. I want to find the derivative, and I want to find the slope of the tangent line at a certain point. So here's how it might be worded.
Find dyx, and the slope of the tangent line At theta equals pi over 6. Okay, let's do it. If you look back at this, check this out. Look, I want to show this to you before we actually get right into our example. What you need to know in order to do your derivative is pretty straightforward.
It says, can you find dr d theta? Yes, I can. Here's r.
Could you take a derivative of this? Yes. That's dr d theta. So dr d theta is just a derivative of this with respect to theta. Do you get me?
So you find that first. Let's do that first. So dr d theta. Equals, what's the derivative of 1?
What's the derivative of cosine theta? So the derivative of r with respect to theta is negative sine theta. So here's what's going to happen when we do our ratio. we're gonna have that thing, the equal sign theta here, and we're gonna have it here.
Are you with me? Sign theta, that's not gonna change. Cosine theta, that's not gonna change. The plus doesn't change, the minus doesn't change, this cosine doesn't change, that sine doesn't change, but the r's... There's an issue.
So the r, how much is r? Do you know how much r is? You're just going to plug that in.
So what we do is we plug in, we do this little derivative first. It's usually pretty easy. Do this derivative, we're going to put it here, we're here, then here. Sine theta, cosine theta, plus, minus.
Cosine theta, sine theta, and then we're going to input whatever our r is into our formula. Why don't you guys go ahead and try to do that on your own. Just create our dy, dx right now. I'll do it on the board, we'll see if we can come up with the same thing. We have O over the same thing.
We put a plus, then we do R, sine of F. This test. Now I have a test for you guys. For those of you who like to make me want to drink at night. Can I simplify these?
Can I simplify these? Don't ever freaking do that, okay? Ever, ever, ever.
Unless you're really drunk, you're gonna have to drink. pluses and minuses are in parentheses, you can't simplify anything. If they're in parentheses, it means you're multiplying by everything, all your factors.
Okay? So we can't touch a thing. Let's see if we got it right. Do we have our dr d theta? Yes.
Here and here. We got our sine. We got our cosine. We got our plus.
We got our minus. We got our r. Here and here. Then we got our cosine.
We got our sine. Did you get the same thing that I got? Yes.
Now, you could, what is this? What do we have right here? What does this mean? What is it? What did you just find?
The derivative. The derivative stands for? Slope. Slope.
That's it. That's what it is. This represents the slope of that cardioid.
Remember the cardioid? What's the cardioid? Well, this represents the slope of that cardioid.
slope of that shape at any point that you get it. But our points are now angles. So you say at angle of 0, what is it? Well, it's probably going to be undefined. It's probably going to be a vertical tangent.
At pi over 6, what is it? I don't know. Probably going to be negative.
negative, probably, just a hunts, because you go like that, cardio looks like that. Make sense? Go at this angle.
We can find out what the slope is at any point, any angle along that coordinate. That's really actually interesting. So don't forget what we're doing.
We're finding the slope of things. That's what a derivative is. So since we find dy dx, all you have to do, just plug in pi over 6. Now it's probably going to be a little bit nicer if you simplify this a little bit, because I don't want to have to plug it in once. One, two, three, four, five, six. I don't have to plug it in eight times, okay?
So maybe work with it just as a little bit of exercise right now, we're gonna work through this thing and simplify this. So if we were to do that, could you plug it in? Yes.
Would it be a little bit easier to find horizontal and vertical tangents if you simplified it? Oh yeah. So we're gonna have negative sine squared theta. We're gonna have plus cosine theta plus cosine squared theta. We're just going to distribute some things.
We're going to have negative sine theta cosine theta minus sine theta minus sine theta cosine theta. Can you all check my algebra too just to make sure I'm doing that right? Would you double check me, please? sine squared, you distribute, bam bam, get this, distribute, get this, or multiply, distribute, we get that. With me so far?
Let's start reorganizing some things here because we're gonna see some identities pop out of this. So first identity that we see. What I'm looking for are things like cosine squared plus sine squared or cosine squared minus sine squared.
Don't use the wrong identity. I'm looking for common terms or like terms. Are you guys okay with that so far? Yes, no, yes. Okay, we'll continue.
Does anybody know, man, I hope that you know. Anybody know what cosine squared theta minus sine squared theta is? And don't you dare tell me one.
Cosine 2 theta. Cosine 2 theta. Cosine 2 theta, that's right.
That's right. This thing right here, this piece, is cosine two theta. I know that's too small to see probably, but it's cosine two theta plus cosine theta.
Another question. Can I add cosine two theta plus cosine theta? What's cosine theta, cosine three theta? You passed, very good.
You just passed trigonometry. Okay, next one, how about this? How about, ignore the negative for a second. Just look at two sine theta cosine theta, there's another identity, right? That's sine two theta, that's right.
So we're gonna have this piece right here, negative sine two theta minus sine theta. So far so good. Let's do one more thing just to make it a little bit nicer. If you want to, you can do this. Let's factor out this negative.
That way we don't have to worry about subtraction on our denominator here. So the way I would probably write this, I'd write this as negative cosine 2 theta plus cosine theta over sine 2 theta plus sine theta. That is our derivative.
Ok, I want to show hands if you're ok with that one so far. Ok now, we've answered part of our question. So far we've answered this part. Find dy dx. We've just found it.
Again, I'm going to keep on saying this. What is this? slope.
What's it the slope of? The tangent line. To what? You're right, the slope of the tangent line.
To what? To that. Yeah, whatever shape that is. That happens to be a cardioid.
So this says, you know what's also interesting? It's the slope based on angle. So that's why I'm giving you, I'm not giving you x, I'm not giving you y.
These are polar. So it says, hey, tell me what the slope is of your function, or the slope of the tangent line to your function at a certain angle. So at pi over 6. So it says, okay, start at 0, go to pi over 6. Tell me what the slope is.
Well, can you do it? What are you going to do to find out what my slope is? Plug it in. Go for it.
There's four places to plug it in. So plug in here, do not neglect that that is a negative. Plug in pi over 6 here, here, here, here. Let's figure that out.
I'm going to expect that you can do it on your own. So dy dx evaluated at theta equals pi over 6, or in other words, m, the slope of our tangent line evaluated at pi over 6. I really did run out of room, didn't I? Should I go smaller? I'm going to do most of it in my head. So if we plug in pi over 6 here, what's 2 times pi over 6?
What's cosine of pi over 3? It's 1 half. Did you remember your negative?
Yes. One half plus. What's cosine of pi over six? One over three over one.
Okay, keep going. Remember we're plugging in pi over six, that's why we're evaluating this. Plugging in, what's two times pi over six? Pi over three.
What's sine of pi over three? Three over zero. You're going to see why we factored out the negative in just a hot second.
What's sine of pi over 6? What is that? Now you don't need to go further than this and here's why. You have 1 half plus root 3 over 2, you have root 3 over 2 plus 1 half. What is that?
That's 1. Actually, it's negative 1. Didn't we talk about the cardioid shape like this? You have this heart shape on its side. And we said that at pi over 6, so it's kind of like this. Kind of like that. Here's zero.
At pi over six, you're about right there. What we're finding is a slope at that point. What's the slope going to be?
It should be negative because at pi over six, our tangent line is going like that. It turns out our slope is negative. It's just negative 1. So what we found, which is really kind of cool, we found the slope of a polar equation, a polar graph at a certain angle.
Show your hands if you understand the concept here. It's a really cool idea. Now, I'm just going to wrap this thing up here real quick. I'm going to write down here.
If you wanted to find horizontal tangents, I mentioned it earlier. But I'll state it for the record. Horizontal tangents are going to happen where the top of your derivative, your numerator, your main numerator, where the main numerator of your derivative is equal to zero. Vertical tangents happen where your derivative is undefined. or where the main denominator of your derivative is equal to zero.
So in other words, look at the main numerator is just dy d theta. The main denominator is just dx d theta. So if you want to find horizontal tangents, what would we set equal to zero?
Zero. D1. D1. Yeah, we set the top. So for horizontal tangents, we just do dy, d theta equals zero.
For vertical tangents, we do dx, d theta equals zero. And that's going to give you the angle, the angle, the theta, at which you are going to have either vertical or horizontal tangents. Have I explained this well enough for you? It's very similar to parametric. If you think back to parametric, do you remember that?
Yes. We had dx, dt for vertical. We had dy, dt for horizontal. And now our t is. is really just a theta. So far so good?
Okay, we just finished section 10 point whatever. Four. The next thing that we're gonna do, we're gonna talk about using these polar equations in calculus, so finding out the area, we kinda did that here, but we're gonna find out the area under a curve, or between our curve, contained within it, and that's gonna be kinda interesting, so we're gonna start that right now.
Maybe last. Ask many questions on this stuff. All that to find negative one. All that to find negative one. Well that's just like when you did all that work to find out there was divergent.
Good point. It's like man. You showed nothing.