Identify whether the given equation is that of a cardioid, limaçon, rose, or lemniscate.
$r=4\sin2\theta$

• A. Cardioid
• B. Limacon
• C. Rose
• D. Lemniscate

Answer: (C)

Verified step by step guidance

1. Recognize the general form of the given polar equation: \( r = 4 \sin 2\theta \). This equation is in the form \( r = a \sin(n\theta) \) or \( r = a \cos(n\theta) \).
2. Identify the value of \( n \) in the equation. Here, \( n = 2 \).
3. Recall that when \( n \) is an integer greater than 1, the equation \( r = a \sin(n\theta) \) or \( r = a \cos(n\theta) \) represents a rose curve.
4. Understand that the number of petals in a rose curve is determined by \( n \). If \( n \) is even, the rose will have \( 2n \) petals. If \( n \) is odd, it will have \( n \) petals.
5. Since \( n = 2 \) is even, the rose curve described by \( r = 4 \sin 2\theta \) will have \( 2 \times 2 = 4 \) petals, confirming it is a rose.