In this video, we're going to talk about how to graph polar equations. These include circles, lemmisons, rose curves, and lemniscates. So let's start with a circle.
The first equation you may see is r is equal to a cosine theta. Now, if A is positive, this is going to be a circle directed towards the right. Now, granted, my circle is not perfect, so bear with me. A is basically the diameter of the circle.
And this is going to be the center of the circle, so if you go up to find this point here, it's half of A. So, if A is greater than 0, if A is positive, it's going to open towards the right. And if A is negative, you're going to get a circle.
Let me draw a good looking circle this time. You're going to get a circle that is directed on the left. So keep in mind, this is going to be A, and this is half of A. So let's try some examples.
Let's say if we have the graph R is equal to 4 cosine theta. If you want to, feel free to pause the video and try it yourself. So we're going to have a circle on the right side. Now a is 4, and half of a, well 4 divided by 2 is 2, so that's half of a.
So what we're going to do is travel 4 units to the right, and then up 2 units, and down 2 units. So the circle is going to start at the origin. And it ends at 4 on the x-axis.
From the center, which is at 2, we need to go up 2 units and down 2 units. And then simply just connect it. So that's how you can graph r equals 4 cosine theta.
Let's try another one. Try this one. Let's say that r is equal to negative 6 sine theta. I mean, not sine, but cosine theta.
We can get to sine later. Now, because A is negative, the circle is going to be on the left side on the x-axis. So, let's travel 6 units to the right, since A is negative 6. One half of A is negative 3. Now, don't worry about the negative sign too much. The negative sign just tells you if the circle opens to the left. Now, the center is going to be at negative 3. which is here.
So we need to go up three units and down three units. So the graph is going to be at the origin at negative 6, at negative 3, 3, and at negative 3, negative 3. And so that's how you can plot. the circle.
So keep in mind this is equal to a, that distance, and this is also equal to a as well, which means this part is one half of a. So half of a is basically the radius of the circle. So if for some reason you need to find the area of the circle, you can use this equation pi r squared.
The radius is 3, so it's pi times 3 squared. which is 9 pi. So now, the next form we need to know is r equals a sine theta. Cosine is associated with the x values.
So as you can see, the circle was associated with the x axis. Sine is associated with the y values. And so the circle is going to be centered on the y axis.
So let's say if a is positive, then we're going to have a circle. that goes above the x-axis centered on the y-axis. So once again the diameter will still be equal to a and this portion the radius is half of a.
So that's when a is positive or when a is greater than 0. Now in the other case if a is negative Or, if a is less than 0, the circle is still going to be centered around the y-axis, but it's going to open in a negative y direction. So it's going to be below the x-axis. And so the radius, as you mentioned before, is just 1 half of a.
And the diameter is equal to a. So let's try some examples. Let's say if r is 2 sine theta.
Go ahead and graph that. So first we need to travel up two units. A is 2. Half of A, which is the radius, is 1. So we're going to have these two points.
Now let's travel one unit to the right and one unit to the left. So the green dot is the center of the circle. So we're going to travel one unit to the right and one unit to the left from it, and so this is going to be the graph.
Try this one. Let's say r is negative 8 sine theta. Go ahead and work on that example. Now, the majority of the graph will be below the x-axis, so I'm going to focus on that. So let's travel 8 units down.
And then half of a, or 4 units to the right, and 4 units to the left. So the point is going to be at the origin and 8 units down. The center is 4 units down. So if A is 8, the radius is 1 half of A, which is 4. So we've got to travel 4 units to the right from the center and 4 to the left. And so this is going to be the graph.
So now you know how to graph circles when you're given a polar equation. Now the next type of graph that we need to go over is the Lima song. And the equation is r is equal to a plus or minus b sine theta.
Now, if you have positive sine, it opens towards the positive y-axis. That is in the upward direction. Negative sine opens in a downward direction, in the negative y direction.
You could also have a plus or minus. B cosine theta. So if cosine is positive, it's going to open towards the right in the positive x-axis direction, and if cosine is negative, it's going to open towards the left.
So let's draw the general shape if it opens towards the right. So this is the limousine with the inner loop, and you get this particular shape if A divided by B. is less than 1. Now both a and b represent positive numbers. a and b are both greater than 0. So if you get the graph 3 minus 4 sine theta, b is not negative 4, b is positive 4, and a is positive 3. So let's say if it was 3 plus 4 cosine theta, both a and b would still be 3 and 4. positive 3 and positive 4. So A and B are not negative. Now the next shape that we have if A divided by B is equal to 1 is the heart shape limousine, also known as the cardioid.
And here's the generic shape for it. So it has like this dimple. So it looks something like that.
Maybe I can draw that better. So that's the cardioid. Now the next one is the dimpled limousine with no inner loop.
So that occurs if A over B is between 1 and 2. So let's start with the x-axis. It's a small dimple. Sometimes it's hard to notice.
So that's the dimpled Limassol with no inner loop. The next we need to know is if A divided by B is equal to or greater than 2. So this limousin looks almost like a circle, but it's not. There's no dimple and there's no inner loop.
So I'm going to start from the left. I'm going to draw it straight up. And then it looks like this. But it's not exactly a circle because, as you can see, the right side is bigger than the left. But it almost looks like a circle.
So that's the Lima song without a dimple or an inner loop. So those are the four shapes you need to be familiar with. Let's graph this equation.
Let's say r is equal to 3 plus 5 cosine. What do you think we need to do here? We know this is a type of Lemur salt. It's in the form a plus or minus b cosine theta.
So first, we need to identify a and b. a is equal to 3, and b is equal to 5. Now, we need to see if a over b, if it's less than 1, if it's between 1 and 2. if it's equal to 1 or greater than or equal to 2. So a over b that's 3 over 5 and 3 over 5 as a decimal is 0.6 which is less than 1. Now because it's less than 1 we know we have the lemurs on with the inner loop. Now there's four types.
The first type is if it's positive cosine. This graph will open towards the right. The next type is if we have negative cosine, and in that case this graph would open towards the left.
If it's positive sine, then it's going to open in the positive y direction. And if we have negative sine, it's going to open towards the negative y direction. So it's going to look something like that. So just keep that in mind. That's the first thing that you look for.
So we have positive cosine, which means it should open towards the right side. Now, when graphing this type of limousine, you want to make sure you get four points. 2x intercepts and 2y intercepts.
So let's draw a rough sketch of this graph. This point is actually positive a. It's a units relative to the center. And this other y-intercept is negative a units from the center.
The first x-intercept, which is associated with the inner loop, it's the difference between a minus b. So it's the absolute value difference of a minus b. Or you could just say it's b minus a because b is going to be bigger.
Now, the second intercept is the sum of a and b. And that's all you need to get a good, decent graph. If you can plot those four intercepts, then you should be fine. So let's go ahead and do that. So in this case, we can see that a is equal to 3. So we need to go up 3 units and down 3 units.
so those are the y intercepts now b minus a that's going to give us the first intercept that's 5 minus 3 that's 2 so here's the first intercept and then a plus b that will give us the second intercept that's 3 plus 5 which is 8 so that's how you can find the 2x intercepts Now let's go ahead and graph it. So first, let's start with the inner loop, and then let's go towards the first y-intercept, and then the second x-intercept, and then towards the other y-intercept. So that's a rough sketch of this graph.
So the points that you need is 3 and negative 3 on the y-axis, and 2 and 8 on the x-axis. Let's try another example. So let's say r is equal to 2 minus 5 sine theta.
So try this one. The first thing I would keep in mind is, what type of, in what direction will it open? We know that a over b, which is 2 over 5, it's less than 1, so this is a limosome with an inner loop. But notice that we have a negative sign, so therefore it has to open in a negative y direction. So we can see that a is 2, that's going to give us the y-intercepts, and b is 5. So let's go ahead and graph it.
Well, in this case, because it opens downward, A is actually going to be associated with the X intercepts this time, instead of the Y intercepts. So it's going to switch roles. So we need to travel 2 units to the right, and 2 to the left. If we're dealing with cosine, then a would be associated with the y-intercepts. But because we're dealing with sine, the rules are reversed.
Now, a plus b, that's going to be 2 plus 5, that's 7. And b minus a, 5 minus 2, is 3. So we're going to travel 3 units down, and also 7 units. So we're going to have 2 y-intercepts. So let's start with the inner loop.
And then let's... Let me do that again. And then let's get the x-intercept. And that's it. So make sure you get these two x-intercepts, negative 2 and 2, and the y-intercepts, negative 3, negative 7. And as we mentioned before, because we have negative sign, it has to open in a negative y direction.
Now let's try this problem. Let's say that r is 3 minus 7 cosine theta. So go ahead and pause the video and work on that example. So let's find the value of a over b. a is 3, b is 7. And 3 over 7 is less than 1, because 7 over 7 is 1. So what we have is the inner loop limousine.
Now, we're dealing with negative cosine, which means it's going to open towards the left. So the majority of the graph is going to be on the left side. Now, a is associated with the y-intercepts when dealing with cosine. When dealing with sine, as we saw in the last example, a is associated with the x-intercepts.
So we're going to travel three units up and three units down to get the y-intercepts when dealing with cosine. For sine, you need to know that a is associated with the x-intercepts. So next, we'll find the first x-intercept, which is going to be b minus a, or 7 minus 3, and that's 4. And then a plus b, 3 plus 7, is 10. So because we're going towards the left, we need to travel 4 units to the left.
That's going to give us the first x-intercept. And then 10 units to the left, relative to the origin. So that's the second x-intercept. Now let's go ahead and graph it. So let's start with the origin, and let's draw the first inner loop.
And then let's focus on the outer loop. And so that's it. That's how you can graph it. My graph is not perfect, but at least that's the general shape. You get the picture.
Now let's try this one. Let's say that R is 3 plus 3 cosine theta. What do we need to do here? Well, first, we need to determine what type of limousine we have.
A and B are the same. When A and B are the same, A over B is equal to 1. And in that situation, we have the heart-shaped limousine, also known as the cardiort. And because cosine is positive, it's going to open towards the right. Now I'm just going to draw the general shape of the cardioid, which it looks like this. So there is no inner loop.
Now when dealing with cosine, the x and y intercepts are going to be a again, a and negative a. Now, the first x-intercept is just going to be the origin. And so we don't have to do anything, it's just going to start from the origin. Now the second x-intercept is a plus b.
We don't have the inner loop, which was a minus b for the inner loop lemur cell. So we don't have to worry about a minus b or b minus a. So now let's go ahead and graph it.
So in this example, a is 3. So we're going to go up 3 units and down 3 units. And a plus b, that's 3 plus 3, so that's equal to 6. So the x-intercept is going to be 6, 0. And the y-ness hips are 0, 3 and 0, negative 3. So now that's, we're going to have to graph like this. And that's it.
That's how you can graph the heart-shaped limousine.