Hey, everyone. We can graph a limaçon just like we graphed a cardioid. By determining the symmetry of our graph and finding and plotting points at our quadrantal angles, we're just first going to take one additional step and determine whether our limaçon has a dimple or an inner loop, which we can do easily based on our a and b values given in our equation. So let's jump right into graphing our limaçon here. Remember that the equation of a limaçon will always be of the form r=a±bcos(θ) or r=a±bsin(θ), just like a cardioid, except here, a is either greater than or less than b. Now, the equation that we're tasked with graphing is r=3-2sin(θ). I know that this is going to be the graph of a limaçon because I have subtraction happening here, and I have an a value of 3 and a b value of 2, so a is greater than b.
Now, this tells us that our limaçon will have a dimple rather than an inner loop. So, this tells me that my graph will be shaped something like this, and it's just up to us to get more precise with these other steps. Now, in determining whether our graph has a dimple or inner loop, we also learn some more information about our graph, whether or not it will pass through the pole. Since here our graph only has a dimple, it will not pass through the pole. So let's continue on with step 2 and determine the symmetry of our graph. Here our equation contains a sine function, so that tells us that our graph will be symmetric about the line theta=π/2, which again is something that I want to keep in mind as I move on to my next step where we're actually going to find and plot points at our quadrantal angles.
Now, for that first value of theta=0, I'm going to take 3 minus 2 sine of 0, and the sine of 0 is simply 0, so this will end up giving me a value of 3. I can plot that first point at (3, 0) right here. Then plugging in π/2 to my equation, 3 minus 2 times the sine of π/2, will give me a value of 1. So, I can plot this second point at (1, π/2) right here. Then, looking at this next angle, π, since I know that my graph is symmetric about this line, I can go ahead and just reflect this point over that line, in order to get this third point right here at (3, π). Now, for my final point plugging in 3/2π this gives me 3 minus 2 times the sine of 3/2π giving me a value of 5. So, this final point here is going to be located at (5, 3/2π), which will be right here.
Now, all that's left to do here is connect all of these points with a smooth and continuous curve and we know the general shape of the graph of a limaçon and, of course, that this limaçon has a dimple. So, in connecting these points, I want to make sure that I reflect those. Now here I'm going to go ahead and connect these points with a dimple at the top of my graph here, and my graph of this limaçon will end up looking something like this. Now, remember, if you're asked to get more precise or if you just want to get more precise, you can always plot more points and that's totally fine. But now that we have fully graphed this limaçon, let's continue getting some practice. Thanks for watching and I'll see you in the next one.