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.