Problem

Identify and graph the limaçon described by the polar equation \( r = 1 + 2\cos{\theta} \).

Solution

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.

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Source: https://solvelyapp.com/problems/SsiZTbTFCw/

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