Question

A polar curve is represented by the equation r1 = 2 + 3cos θ.Part A: What type of limaçon is this curve? Justify your answer using the constants in the equation.Part B: Is the curve symmetrical to the polar axis or the line theta equals pi over 2 question mark Justify your answer algebraically.Part C: What are the two main differences between the graphs of r1 = 2 + 3cos θ and r2 = 7 + 3 cos θ?

Answer 1

Given the equation in polar form:

r1 = 2 + 3 cos θ

Part a

The graph of this curve is called a Limacon. There are four types of Limacons.

Consider a = value of the independent term and b = coefficient of the cosine.

If a < b, the Limacon is looped. We can determine that a = 2 and b = 3. This is a looped Limacon, with the loop on the right side, as shown below:

Part b.

The curve is symmetrical to the polar axis if the points (r, θ) and (r, -θ) are on the graph.

Substituting θ for -θ:

r1 = 2 + 3 cos (-θ)

Since the cosine is an even function, cos (-θ) = cos (θ), thus:

r1 = 2 + 3 cos θ

We get the same equation, so the curve is symmetrical to the polar axis

Part c.

We are given a second equation:

r2 = 7 + 3 cos θ

Below is the graph of both functions:

The second graph represents a dimpled Limacon because a/b > 2.

The graph looks like a circle.

The graph is much bigger than the first graph.