Description down below:

Question

A polar curve is represented by the equation r₁ = 3 + 3sin 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  Justify your answer algebraically.

Part C: What are the two main differences between the graphs of r₁ = 3 + 3sin θ and r₂ = 8 + 3cos θ?

Answer 1

For a polar curve r = a + bsinθ (vertical pole):

If 0 < a/b < 1, the curve is an inner loop limaçon

If a/b = 1, the curve is a cardioid

If 1 < a/b < 2, the curve is a dimpled limaçon

If a/b ≥ 2, the curve is a convex limaçon

Part A

Since a/b=3/3=1, then the curve is a cardioid

Part B

We need to keep in mind two things:

1. Substitute the appropriate combination of components for (r,θ): (−r,−θ) for θ=π/2 symmetry; (r,−θ) for polar axis symmetry; and (−r,θ) for symmetry with respect to the pole.
2. If the resulting equations are equivalent in one or more tests, the graph produces the expected symmetry.

If we are testing for θ=π/2 symmetry, then we substitute (-r,-θ) for (r,θ) in r=3+3sinθ to get -r=3+3sin(-θ) which is equivalent to -r=3-3sinθ or r=-3+3sinθ. Since the original curve, r=3+3sinθ, is not the same as r=-3+3sinθ, then there is no symmetry about the line θ=π/2.

If we are testing for polar axis symmetry, then we substitute (r,-θ) for (r,θ) in r=3+3sinθ to get r=3+3sin(-θ), which is equivalent to r=3-3sinθ. Since the original curve, r=3+3sinθ, is not the same as r=3-3sinθ, then there is no polar axis symmetry.

So, neither symmetry works.

Part C

The difference between r₁ = 3 + 3sin θ and r₂ = 8 + 3cos θ is that r₁ is a cardioid given a/b=3/3=1 and r₂ is a convex limaçon given a/b=8/3≥2.

Hope this helped!