Which of the following best describes the graph of the polar equation $r=1+\cos (25\theta )$?

• A. A rose curve with 5 petals
• B. A circle centered at the origin
• C. A rose curve with 10 petals
• D. A limaçon with an inner loop

Answer: (C)

Verified step by step guidance

1. Identify the general form of the polar equation given: \(r = 1 + \cos(2 \times 5 \theta)\), which simplifies to \(r = 1 + \cos(10\theta)\).
2. Recall that polar equations of the form \(r = a + b \cos(n\theta)\) or \(r = a + b \sin(n\theta)\) represent limaçons, and when \(a = 0\), they represent rose curves.
3. Note that rose curves have the form \(r = \cos(k\theta)\) or \(r = \sin(k\theta)\), where the number of petals depends on \(k\): if \(k\) is even, the rose has \$2k\( petals; if \)k\( is odd, it has \)k$ petals.
4. Since the equation is \(r = 1 + \cos(10\theta)\), it is a limaçon because of the constant term 1, but the cosine term has \$10\theta$, which influences the petal count if it were a rose curve.
5. Analyze the graph behavior: because \(a = 1\) and \(b = 1\), and \(n = 10\), the graph is a limaçon with an inner loop or dimple, but the presence of \(\cos(10\theta)\) suggests multiple petals; thus, the graph resembles a rose curve with 10 petals.