Which of the following best describes the graph of the polar curve $r=2+5\sin \left(\theta \right)$?

• A. A limaçon with an inner loop
• B. A rose curve with five petals
• C. A cardioid
• D. A circle centered at the origin

Answer: (A)

Verified step by step guidance

1. Identify the general form of the polar equation given: \(r = 2 + 5 \sin(\theta)\). This matches the form \(r = a + b \sin(\theta)\), which is a standard form for limaçons.
2. Recall the classification of limaçons based on the relationship between \(a\) and \(b\): if \(|b| > |a|\), the limaçon has an inner loop; if \(|b| = |a|\), it is a cardioid; if \(|b| < |a|\), it is a dimpled or convex limaçon without an inner loop.
3. Compare the values: here, \(a = 2\) and \(b = 5\). Since \$5 > 2\(, this means \)|b| > |a|$, indicating the curve has an inner loop.
4. Eliminate other options by recognizing their forms: a rose curve with five petals would have an equation like \(r = \sin(5\theta)\) or \(r = \cos(5\theta)\); a cardioid requires \(|a| = |b|\); a circle centered at the origin would have a constant \(r\) or a form like \(r = a\).
5. Conclude that the graph of \(r = 2 + 5 \sin(\theta)\) is best described as a limaçon with an inner loop.