Which of the following best describes the graph of the polar curve $r=4\cos \left(3\theta \right)$?

• A. A rose curve with $6$ petals, each petal extending to a maximum length of $4$
• B. A limaçon with an inner loop
• C. A rose curve with $3$ petals, each petal extending to a maximum length of $4$
• D. A circle centered at the origin with radius $4$

Answer: (C)

Verified step by step guidance

1. Identify the form of the polar equation given: \(r = 4 \cos(3\theta)\). This matches the general form of a rose curve, which is \(r = a \cos(n\theta)\) or \(r = a \sin(n\theta)\).
2. Recall the rule for the number of petals in a rose curve: if \(n\) is odd, the rose has \(n\) petals; if \(n\) is even, the rose has \$2n$ petals.
3. Since \(n = 3\) (which is odd), the rose curve will have exactly 3 petals.
4. The maximum length of each petal corresponds to the amplitude \(a\) in the equation, which is 4 in this case, so each petal extends to a maximum radius of 4.
5. Conclude that the graph is a rose curve with 3 petals, each petal extending to a maximum length of 4.