The polar curve given by $r = 2 + 5\sin(\theta)$ is a type of limaçon, specifically one with an inner loop. Let’s analyze it:

1. Type of Curve: Since it’s in the form $r = a + b\sin(\theta)$, where $a = 2$ and $b = 5$, and $b > a$, this curve is a limaçon with an inner loop.

2. Behavior of $r$:

   • When $\theta = \frac{3\pi}{2}$, $\sin(\theta) = -1$, so $r = 2 + 5(-1) = -3$.
   • When $\theta = \frac{\pi}{2}$, $\sin(\theta) = 1$, so $r = 2 + 5 = 7$.
   • When $\theta = 0$ or $\theta = \pi$, $\sin(\theta) = 0$, so $r = 2$.

3. General Shape: The curve will have:

   • An inner loop, indicating a region where $r$ is negative.
   • Maximum $r$-values of 7 at $\theta = \frac{\pi}{2}$.
   • Symmetry along the vertical axis because the function depends on $\sin(\theta)$.

If you were to match this to a graph, look for a limaçon shape with:

• A pronounced inner loop that crosses the origin,
• A maximum radius of about 7,
• Symmetry along the vertical axis.

Would you like further clarification on how to plot or analyze this polar equation?

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5 Related Questions

1. How would the graph change if the equation were $r = 5 + 2\sin(\theta)$?
2. What are the coordinates of key points on the curve in rectangular form?
3. How do you find the area enclosed by this polar curve?
4. How does changing the angle parameter affect the graph’s shape in polar coordinates?
5. How does this curve compare to one with $r = 2 + 5\cos(\theta)$?

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Tip

When sketching polar curves, it helps to evaluate $r$ at key angles (like $0, \pi/2, \pi, 3\pi/2$) and note symmetries to better understand the shape of the graph.