What are the two main differences between the graphs of r1=3+3sintheta and r2=8+3costheta?

To compare the graphs of the polar curves \( r_1 = 3 + 3\sin\theta \) and \( r_2 = 8 + 3\cos\theta \), we can analyze the following two main differences:

### 1. **Type and Position of the Curve:**
   - **Curve Type:**
     - \( r_1 = 3 + 3\sin\theta \): This equation represents a **limacon** with an inner loop because the coefficient of \( \sin\theta \) is equal to the constant term. The general form \( r = a + b\sin\theta \) with \( a = b \) produces a limacon with an inner loop.
     - \( r_2 = 8 + 3\cos\theta \): This equation represents a **limacon without a loop** because the coefficient of \( \cos\theta \) is less than the constant term (i.e., \( a > b \)). The general form \( r = a + b\cos\theta \) with \( a > b \) produces a limacon without a loop.

   - **Position of the Curve:**
     - \( r_1 = 3 + 3\sin\theta \): The graph is oriented with respect to the vertical axis, because it is based on \( \sin\theta \). The curve will be symmetric about the line \( \theta = \frac{\pi}{2} \) (vertical axis).
     - \( r_2 = 8 + 3\cos\theta \): The graph is oriented with respect to the horizontal axis, because it is based on \( \cos\theta \). The curve will be symmetric about the polar axis (horizontal axis).

### 2. **Size and Shape of the Curve:**
   - **Size:**
     - \( r_1 = 3 + 3\sin\theta \): The maximum value of \( r_1 \) occurs at \( \theta = \frac{\pi}{2} \), where \( r_1 = 6 \), and the minimum value occurs at \( \theta = \frac{3\pi}{2} \), where \( r_1 = 0 \). This means the curve has a smaller overall size compared to \( r_2 \).
     - \( r_2 = 8 + 3\cos\theta \): The maximum value of \( r_2 \) occurs at \( \theta = 0 \), where \( r_2 = 11 \), and the minimum value occurs at \( \theta = \pi \), where \( r_2 = 5 \). This curve is larger and does not touch the origin.

   - **Shape:**
     - \( r_1 = 3 + 3\sin\theta \): Since the curve has an inner loop (due to \( a = b \)), it will have a distinctive loop near the origin.
     - \( r_2 = 8 + 3\cos\theta \): This curve will have a dimpled or slightly elongated shape (but no loop) and will be wider on one side, specifically extending more along the positive \( x \)-axis (polar axis).

### Summary of Differences:
1. **Type and Position:**
   - \( r_1 = 3 + 3\sin\theta \): Limacon with an inner loop, symmetric about the vertical axis.
   - \( r_2 = 8 + 3\cos\theta \): Limacon without a loop, symmetric about the horizontal axis.

2. **Size and Shape:**
   - \( r_1 = 3 + 3\sin\theta \): Smaller, with a loop near the origin.
   - \( r_2 = 8 + 3\cos\theta \): Larger, without a loop, dimpled shape.

Would you like more details or have any questions?

Here are five related questions to explore further:
1. How would the graph of \( r_1 = 3 + 3\cos\theta \) compare to \( r_1 = 3 + 3\sin\theta \)?
2. What happens to the graph of \( r_2 = 8 + 3\cos\theta \) if the coefficient of \( \cos\theta \) increases?
3. Can we classify the curve \( r = 8 + 8\cos\theta \)? What does it look like?
4. How does the choice of \( \sin\theta \) versus \( \cos\theta \) in a polar equation affect the orientation of the graph?
5. What is the significance of the inner loop in the limacon \( r_1 = 3 + 3\sin\theta \)?

**Tip:** When analyzing polar curves, pay attention to the coefficients and their relationships—they directly affect the shape, size, and symmetry of the curve.

Explore the differences between the polar curves r1=3+3sinθ and r2=8+3cosθ. Understand how their shapes, sizes, and orientations vary based on their equations. This analysis covers key concepts in polar coordinates and trigonometry, suitable for undergraduate-level mathematics.

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