Identify and graph the limaçon described by the polar equation \( r = 1 + 2\cos{\theta} \).
Answer
Step 3: Graph the limaçon. Plot points for various angles and connect the points smoothly, making sure to include the inner loop.
Step 1 ：Step 1: Identify the type of limaçon. Here, the equation is of the form \( r = a + b\cos{\theta} \) or \( r = a + b\sin{\theta} \) where a = 1 and b = 2. Because b > a, this is a limaçon with an inner loop.
Step 2 ：Step 2: Identify the characteristics of the limaçon. The limaçon makes a loop inside the circle \( r = 1 \) and crosses the pole at \( r = 1 + 2 = 3 \) when \( \cos{\theta} = 1 \) and at \( r = 1 - 2 = -1 \) when \( \cos{\theta} = -1 \).
Step 3 ：Step 3: Graph the limaçon. Plot points for various angles and connect the points smoothly, making sure to include the inner loop.
