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Multiple Choice
Identify whether the given equation is that of a cardioid, limaçon, rose, or lemniscate. r=1−sinθ
A
Cardioid
B
Limacon
C
Rose
D
Lemniscate
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Verified step by step guidance
1
Start by recognizing the general form of polar equations. The equation given is r = 1 - \(\sin\)\(\theta\).
Understand the characteristics of different polar curves: Cardioids have the form r = a ± b\(\sin\[\theta\) or r = a ± b\(\cos\]\theta\) where a = b.
Compare the given equation r = 1 - \(\sin\[\theta\) with the standard form of a cardioid. Notice that it matches the form r = a - b\(\sin\]\theta\) with a = b = 1.
Recall that a cardioid is a special type of limaçon where the coefficients a and b are equal, resulting in a heart-shaped curve.
Conclude that the given equation r = 1 - \(\sin\)\(\theta\) represents a cardioid, based on its form and the equality of coefficients.