Question

identify and graph the polar equation.
r = 2 + \sin \theta
what type of graph does the equation represent?
a. a circle
b. a line
c. a limaçon with an inner loop
d. a lemniscate
e. a limaçon without an inner loop
f. a cardioid

Explanation:

Step1: Recall the general form of polar equations

The general form of a limacon is \(r = a\pm b\sin\theta\) or \(r=a\pm b\cos\theta\). When \(a > b\), the limacon has no inner loop.

Step2: Analyze the given equation \(r = 2+\sin\theta\)

Here \(a = 2\) and \(b = 1\). Since \(a=2>b = 1\), it is a limacon without an inner loop.

Answer:

E. a limacon without an inner loop