How to Graph Limacons

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Graphing limacons will require you to look at the polar coordinates by the analytic equation. Graph limacons with help from an experienced mathematics professional in this free video clip.

Part of the Video Series: Trigonometry, Graphs, & Other Math Tips
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Video Transcript

Hello, my name is Walter Unglaub, and this is how to graph limacons. So a limacon is described in polar coordinates by the analytic equation r being the radius is equal to a plus minus b plus sine of theta where a and b are constants. Or alternatively, a plus minus b cosine of theta. And there's essentially three types of limacons. The first type considers the case where the magnitude of the ratio a over b is less than one and this is going to be a limacon with an inner loop which is this case right here. So if I consider that my case A I notice that there's an inner loop inside of this limacon and there are two crossings at the origin. The second type of limacon is the case where the magnitude of the ratio a over b is greater than one but less than two. And this type of limacon has a dimple in it. So this second case, case B is depicted here as an example where it almost looks like a circle but it has a dimple in it. Finally, if the ratio of a over b is greater than two, then we call that figure a convex limacon. Which is this third case, case C. And it looks like a stretched circle here where the radius isn't necessarily constant throughout all angles. Finally we can consider the case of a limacon where the frequency term in the trigonometric function is not equal to one. So if I have r is equal to b sine of n times theta for example, n can be either even or odd. It's an integer and if it is odd then my figure will have n petals. And if it's even it will have two times n petals. So as an example, we can consider r is equal to sine of three theta where we notice that n is equal to three so it's odd so it's going to be three petals. And in this case our limacon is going to look something like this. So we notice that there are various crossings at the origin and that this value cannot exceed one which is the value of b and that we have three equal sized petals symmetrically placed about the origin. My name is Walter Unglaub and this is how to graph limacons.

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