The equation $r = 4 \sin(2\theta)$ represents a rose curve, which typically has 4 petals when the coefficient of $\theta$ inside the sine function is 2. Each petal lies symmetrically across the origin, with the petals being centered at specific angles.

For the given problem, where a petal lies in quadrant 4, we need to find the interval for $\theta$ that places $r$ in quadrant 4.

Step-by-Step Solution

1. Understand the symmetry and pattern:

   • The equation $r = 4 \sin(2\theta)$ generates petals at intervals of $\frac{\pi}{4}$ radians due to the $2\theta$ factor.
   • The petals are positioned at angles $\theta = \frac{k\pi}{2}$, where $k$ is an integer.

2. Determine the quadrant location:

   • In quadrant 4, $\theta$ should be between $-\frac{\pi}{2}$ and $0$ radians, but since $r$ is positive (as $r = 4 \sin(2\theta)$ is positive for a petal in quadrant 4), $2\theta$ must correspond to an angle where sine is positive, which occurs between $0$ and $\pi$.

3. Find the specific interval:

   • For quadrant 4, we want $\theta$ to result in $r$ pointing in the correct direction. We need $2\theta$ to be an angle where the sine function corresponds to the petal in quadrant 4.
   • This happens when $2\theta$ is around $\frac{7\pi}{4}$ (because $\sin \left( \frac{7\pi}{4} \right)$ is negative, making $r$ negative and hence flipping the petal into quadrant 4).

   However, the petal in quadrant 4 is produced when $\theta$ is slightly less than $\frac{\pi}{4}$, as the sine wave passes through positive and enters quadrant 4 when $\theta$ is in the range: $\frac{3\pi}{4} \leq 2\theta \leq \pi$ Dividing by 2: $\frac{3\pi}{8} \leq \theta \leq \frac{\pi}{2}$

4. Final Answer: The interval for $\theta$ that will produce the petal lying in quadrant 4 is: $\theta \in \left[ \frac{3\pi}{8}, \frac{\pi}{2} \right]$

Would you like more details on rose curves or related topics? Here are some follow-up questions to consider:

1. How is the general form of a rose curve equation defined?
2. What is the significance of the number of petals in a rose curve?
3. How do different coefficients inside the trigonometric function affect the graph?
4. Can rose curves have more than 4 petals?
5. What are the applications of polar equations like this one in real-world contexts?

Tip: When analyzing rose curves, remember that the number of petals is directly related to the coefficient of $\theta$ inside the sine or cosine function!