5. Sketch the given polar curve and identify the name of this polar curve.

𝑟=3+2cos(𝜃)

The given polar equation is \( r = 3 + 2\cos(\theta) \). This is a **limacon** with a cosine term.

### Steps to sketch the curve:
1. **Understanding the shape**: The equation represents a limacon. Since it has the form \( r = a + b\cos(\theta) \), the limacon will be symmetric about the polar axis (the horizontal axis in polar coordinates).
   - If \( a > b \), the limacon does not have an inner loop.
   - If \( a = b \), the limacon will have a sharp point (called a cardioid).
   - If \( a < b \), the limacon will have an inner loop.

   Here, \( a = 3 \) and \( b = 2 \), so \( a > b \). This means the limacon does not have an inner loop.

2. **Plotting specific points**:
   - At \( \theta = 0 \): \( r = 3 + 2\cos(0) = 3 + 2 = 5 \). So the point is \( (5, 0) \).
   - At \( \theta = \frac{\pi}{2} \): \( r = 3 + 2\cos\left(\frac{\pi}{2}\right) = 3 + 0 = 3 \). So the point is \( (3, \frac{\pi}{2}) \).
   - At \( \theta = \pi \): \( r = 3 + 2\cos(\pi) = 3 - 2 = 1 \). So the point is \( (1, \pi) \).
   - At \( \theta = \frac{3\pi}{2} \): \( r = 3 + 2\cos\left(\frac{3\pi}{2}\right) = 3 + 0 = 3 \). So the point is \( (3, \frac{3\pi}{2}) \).

3. **Plotting the curve**: 
   - The maximum value of \( r \) is 5, and the minimum value is 1. The curve will be larger on the right side (positive \( r \)) and smaller on the left side (negative \( r \)).

4. **Shape of the curve**: Since \( a > b \), the curve resembles a **limacon without an inner loop**. The curve bulges outward but does not loop back inside.

### Final Answer:
The name of the polar curve is **limacon without an inner loop**.

Would you like me to generate a sketch of this polar curve for you? 

Here are some related questions to expand on this concept:
1. What happens to the shape if \( a = b \) in the limacon equation?
2. How does changing \( \cos(\theta) \) to \( \sin(\theta) \) affect the curve?
3. Can you find the area enclosed by this limacon curve?
4. What type of symmetry does this curve exhibit?
5. How would you convert this polar equation into Cartesian form?

**Tip**: Limacons can have different appearances depending on the relationship between \( a \) and \( b \) (inner loop, dimpled, or no loop).

Sketch the given polar curve and identify the name of this polar curve. r = 3 + 2cos(θ)

This page provides a detailed solution for sketching the polar curve r = 3 + 2cos(θ), which represents a limaçon without an inner loop. Learn about polar coordinates, graphing techniques, and the characteristics of limaçon curves. Suitable for Grades 10-12.

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