Which of the following best describes the graph of the polar equation $r=2-2\cdot \sin \left(\theta \right)$?

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

Answer: (A)

Verified step by step guidance

1. Identify the general form of the given polar equation: \(r = 2 - 2 \cdot \sin(\theta)\). This matches the form \(r = a - b \sin(\theta)\), where \(a = 2\) and \(b = 2\).
2. Recall that the shape of the graph depends on the relationship between \(a\) and \(b\): if \(a = b\), the graph is a cardioid; if \(a > b\), it is a limacon without an inner loop; if \(a < b\), it is a limacon with an inner loop.
3. Since \(a = b = 2\), the graph corresponds to a cardioid, which is a special type of limacon where the inner loop just touches the pole (origin).
4. Review the options: a cardioid is a limacon with no inner loop, but since \(a = b\), the graph is exactly a cardioid, not a limacon with an inner loop.
5. Therefore, the best description of the graph is a cardioid.