Textbook Question
In Exercises 64–70, graph each polar equation. Be sure to test for symmetry.r = 2 + cos θ
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9. Polar Equations
Polar Coordinate System
Multiple Choice
Which of the following best describes the graph of the polar curve ?
A
A limaçon with an inner loop
B
A rose curve with five petals
C
A cardioid
D
A circle centered at the origin
Verified step by step guidance1
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.
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.
Compare the values: here, \(a = 2\) and \(b = 5\). Since \$5 > 2\(, this means \)|b| > |a|$, indicating the curve has an inner loop.
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 \(r = a\).
Conclude that the graph of \(r = 2 + 5 \sin(\theta)\) is best described as a limaçon with an inner loop.
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