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Question

A polar curve is represented by the equation r₁ = 3 + 3sin θ. (10 points)

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

Problem Description: A polar curve is represented by the equation r₁ = 3 + 3sin θ. (10 points)

(Note: This equation is in the form of r = a + b sin θ)

Part A: What type of limaçon is this curve? Justify your answer using the constants in the equation.

Because a/b = 3/3 = 1, the limacon is a cardiod and is heart shaped.

Part B: Is the curve symmetrical to the polar axis or the line θ = ¶/2? Justify your answer algebraically.

Assume two angles α1 and α2 symmetrical around θ = ¶/2.

Let α1 = ¶/2 + x and α2 = ¶/2 – x

Using sum of angle formula for sine:

sin α1 = (sin ¶/2)(cos x) + (cos ¶/2)(sin x) = cos x → r1 = 3 + 3sin θ = 3 + 3cos θ

sin α2 = (sin ¶/2)(cos x) - (cos ¶/2)(sin x) = cos x → r2 = 3 + 3sin θ = 3 + 3cos θ

Therefore, since r1 = r2, the curve is symmetrical about θ = ¶/2

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

The first equation represents a limacon that is a cardiod (heart shaped). It is symmetrical about θ = ¶/2.

The second equation, a = 8, b = 3; a/b = 2.67 (>2). Therefore , the plot of this equation is a limacon with no dimple and no inner loop. And since the equation is a cosine it’s symmetrical about the polar axis.