Question

What are limacons and cardioids?

Answer 1

Hint: Many equations can be graphed either manually or using any computer software. Mostly complex equations are being graphed using any computer software.
When an equation is being graphed, depending on the equation and the resulting shape of the equation, there are different graphs that can be produced.

Complete step by step solution:
Limacons and Cardioids are the type of graphs produced. Limacons and Cardioids are produced for a particular kind of equation.
Limacons are produced by equations of the form $r=a+b\sin \theta$, $r=a-b\sin \theta$,$r=a+b\cos \theta$ and $r=a-b\cos \theta$. The kind of equations which produce limacons have trigonometric equations in them, which are the functions sine and cosine.

The graph is a limacon with an inner loop when the value of $a$ is less than the value of $b$.
The graph is a dimpled limacon where the value of $a$ is greater than the value of $b$.
The graph is a convex limacon when the value of $a$ is greater than or equal to the value of $2b$.
The graph is a special case of the limacon where the value of $a$ equals the value of $b$ and is known as a Cardioid.

Note: It is worth noting that moving the sine to the cosine in both kinds of liamsons' graphs has little effect on the form of the graph, just its orientation. The vertical axis is symmetric in equations using sine, while the horizontal axis is symmetric in equations using cosine. Their orientation can be affected by the sign of $b$ .