 Welcome back everyone. This is actually the start of lecture 33 in our lecture series, Math 1220, Calculus II for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misalign. We're going to continue that discussion of polar functions that we started in lecture 32 and actually want to switch over to Desmos again to look at some examples, some more examples of polar graphs. Right? This time we're going to focus on a very special family of polar functions that we're going to play around with a lot. So for our first example, let's graph the function r equals 2 sine theta. So what you see in front of you looks like a circle and you are right in thinking it's a circle, but better yet you should be thinking this as a one petal flower. Kind of like a, like, I actually can't think of a flower with only one petal. I know they exist but clearly I'm not a florist. This is a circle. This is a circle and if we play around with this thing a little bit more, as we allow the angle to go from zero to pi halves, we get this first, we get a semi circle. As it goes from pi halves to pi, you actually get the whole circle and then after this, what's going to happen is this is going to repeat itself because of the negative radius, the radius that happens for sine, it's actually going to reflect and give you the same picture over and over again. It gives you this circle, which again I like to think of as a one petal flower. If you switch this to like cosine, the orientation will be a little bit different. You can stick negative signs to do some reflections right there. We'll go back to sine. You can reflect it as well. Move it downward. You can also adjust this parameter right here, the a. This will allow it to actually have a larger radius. You can enlarge it. Notice right here, as you enlarge it, it's going to grow from the x-axis upward, bigger, bigger, bigger, bigger. This is going to be a circle where the a in front actually will be the diameter of the circle, not the radius. You see right now a circle, which is centered on the y-axis. It's diameter three and it does pass through the origin right there. This does match up with the type of functions we saw previously, but it also is in relation to some new functions I want to play around with. Take the function r equals one plus sine theta. What this looks like is the following little thing right here. To me, this kind of looks like a lima bean or something, but to others, this looks like a heart, right? This is often referred to in the literature as a cardioid, which is meaning heart-like object. It's not perfectly a heart, but the idea is the following. It kind of does resemble a heart with the right modifications, right? If you put a negative sign right here, I can reflect it upwards right here. Doesn't that look like a heart? Maybe we make it pink or is that purple? I can't tell right now. It's like, oh, this is a sweet function to send to your girl on Valentine's Day. r equals negative one plus sine theta. She'll love it. It'll be adorable. We'll switch it back to the way it was, our little cardioid, this heart-like object. This is a fun little graph here. Notice what happens with the angle as we draw it. As you start with the origin, you're actually going to be at one, right? Because you take one plus sine, sine here is zero, because the angle is zero. It's going to start off at r equals one. I think it's bigger, bigger, bigger, because in the first quadrant, sine will get bigger, bigger, bigger until you eventually get to the top, which will be a distance of two from the origin, one plus one, one being sine's largest value. In the second quadrant, it'll then fall, fall, fall, fall, fall, fall, because sine's getting smaller, smaller, smaller, going to zero. And so now you're going to be one unit to the left of the origin. In the third quadrant, sine's actually negative. So you actually start gobbling away pieces of that plus one. And so you go from one radius all the way down to zero right here. And then at zero, then here, now you're into the fourth quadrant, you're going to start, sine's going to get bigger again. It's still negative. So you don't gobble as much as the one there because you're getting a one minus one right here at three pi halves. It will turn back to its starting location. So this function has this so-called dimple on it, the short little point that comes here at the origin that happens at three pi halves. That's a fun little thing about this graph. Of course, we can modify this graph. We can throw an A in front of it, which then enlarges it. You see in blueish green there, the original one. As we allow A to get bigger, we get a larger cardioid. We can get a smaller cardioid. If we take a negative cardioid, it will reflect on the other side like so. We can replace sine with cosine. And this will give us a horizontal cardioid, one that's centered on the x-axis. And you can reflect this as well. So you can look at some modifications of the cardioid right there. All right, turn those ones off. Another modification we're going to do here is, what if we kind of put these two together? We had the two sine and we had the one plus sine. What if we kind of put them together? We're going to take R to be one plus two sine theta. And when you do that, you get the following type object. This right here, to some people, resembles a snail shell. And actually, in geometry, these are often referred to as limousons, which is essentially a French word for snail, right? So the idea is this kind of looks like a snail. This is very similar to the cardioids we saw before, with a critical difference. There's this inner loop that appears inside of the limouson. To give you some idea of what's going on here, let's trace this thing out, back to the start. So much like the other one, like the cardioid we did earlier, when you start off this thing at angle zero, you're going to be at one because sine at zero gives you zero. But you start adding to it some sine. As sine, as theta goes from zero to pi half, sine will get bigger, bigger, bigger, bigger, bigger, bigger until you reach the top of the limouson. You're going to get one plus two, which is three, because sine contributes a maximum amount of there. Then as you go from pi half to pi, you're going to backtrack and mirror image what you just did. Now, when you enter the third quadrant, you're going to take one and you're going to start subtracting because sine's negative here now. But look, the coefficient in front of sine is now a two. So we potentially are going to start gobbling up more than the plus one has to offer. And so at some point, the radius becomes negative. And so it actually starts drawing in the first quadrant, which is antipodal to the third quadrant, it's kitty corner there. Eventually, you're going to get this point of negative one, radius, because you take one minus two. And in the fourth quadrant, it's going to mirror image what it did before. So this inner loop comes from the fact that you're subtracting more than you're adding to the function. And you get this limouson. Cardioids are special types of limousons where the coefficient in front of the sine is equal to the constant you're adding. On the other extreme though, what if we were to take the graph of r equals two plus sine theta? What if we add more than the coefficient or the amplitude in front of the sine? You get a limouson that looks something like the following. Here, you see that it looks kind of flat at the bottom. There's no cusp that's pointing at the origin. There's no inner loop. This one actually has a dimple, a cute little dimple, in which case, to kind of show you this dimple, we have to exaggerate it a little bit. If we stick an A in front of this thing, that will change the amplitude. But actually what I'm going to do is put an A and B right here. So at the moment, you can see that if I make A get bigger, bigger, bigger, this thing starts to look more and more like a circle. As I make it A is smaller, smaller, smaller, and when it collapses down, this gets us something similar to the object we had before. A is bigger than B, looking at our formula here, r equals A plus B sine theta. A is bigger than B, because again, as B gets bigger, you start getting this inner loop going on here, this inner loop. The bigger B is, that is the farther away B is from A, because if we make it A get bigger, bigger, bigger, you can get a limouson with a large outer range but a smaller inner loop. You can get one with a small outer radius compared to the inner radius. That is the inner loops and outer loop are almost the same thing. We can also distort it, again, like so, we want the inner loop to be really, really small compared to the outer loop. And then like we said before, if these things are exactly the same, so if you actually get one in one, that's exactly where the cardioid's going to happen. And you're going to get a sharp corner right here at the origin, but then if we allow A to get, if B is bigger than A, we get an inner loop. If A is bigger than B, you're actually going to get a dimple right here, right? And so this is kind of like in the movie Coco, you get no dimple, dimple, no dimple, dimple, kind of like that. And so these different limousons have these different flavors based upon the difference between these coefficients A and B right here. Larger B versus larger A, you can make inner loops or dimples on these limousons. And this is really a fun polar type graph one can get. And one can get a lot more creativity in this as well. Like if we were to plug in a two here, you get something like this. And so this isn't a limouson, but you can still get like, whoa, it kind of looks like to come together, you get this limouscate. If you allow A to get bigger than B, you get sort of like this peanut shape. But then as A gets smaller than B, right, you're going to get this four-petal flower, which two of the petals are smaller than the other. And that's kind of fun. You can switch this to a three, right. Now you're going to get this flux capacitor with that some inner loops. That's because B is bigger than A. If we allow A to get bigger, you're going to get some dimples on this. The dimples will go away as you get bigger, bigger, bigger. This will get you to the circle. But again, as you get A close to B, but A is bigger, you're going to get some dimples. When they're perfect, they'll be the same. They'll touch each other. And then when A gets smaller than B, you get some inner loops, just like the limousons. And this would also be true if you try other powers like this. So you can get lots of these little flowers with inner loops and dimples and things like that. So we have some dimples there. And so you can see a lot of variety of functions you can get using signs and cosines. And you can also play around with other trigonometric functions. What I would encourage you to do is actually to play around with like a software, a graphing calculator, like Desimos yourself, type in some trigonometric functions. R equals some trigonometric expression of theta. And you'll see some really cool graphs. This right here is sort of like the chemistry lab of a calculus class. This is when we start throwing the kimbals together and see if we blow up stuff or not. I certainly don't want anyone to blow up any things. But that's kind of the basic idea. So in the next video, we're going to pick up where we left off right here. And we're going to start doing some calculus with these polar functions.