To determine the interval for $\theta$ that will produce the specific graph from the equation $r = 4\sin(2\theta)$, we need to analyze the behavior of the rose curve generated by this equation.

Understanding the Rose Curve

The equation $r = 4\sin(2\theta)$ describes a rose curve with 4 petals because the coefficient of $\theta$ inside the sine function is 2. The petals of a rose curve are symmetrically distributed around the origin, and the curve is periodic.

Step-by-Step Analysis

1. Identify the general behavior:

   • The rose curve $r = 4\sin(2\theta)$ will complete one full cycle over an interval of $0 \leq \theta < \pi$, as $r$ repeats its values every $\pi$ radians.
   • The curve has 4 petals, and each petal corresponds to a specific range of $\theta$.

2. Petal locations:

   • The petals occur when $\sin(2\theta)$ reaches its maximum and minimum values.
   • $r > 0$ produces petals in the first and third quadrants for $0 \leq 2\theta < \pi$, while $r < 0$ (due to the sine function) will reflect the petals into the second and fourth quadrants.

3. Determine the interval for a specific petal:

   • To find the interval for a specific petal, consider when the sine function produces a specific petal location.
   • Suppose the desired petal lies in the fourth quadrant. This will occur when $\theta$ results in $r$ producing the correct directional angle.

Specific Interval

• If a petal lies in the fourth quadrant, we need to find where $r = 4\sin(2\theta)$ is positive or negative, depending on the reflection and symmetry of the curve.
• For quadrant 4, this petal will occur when: $\frac{3\pi}{4} \leq 2\theta \leq \pi$ Dividing by 2 gives: $\frac{3\pi}{8} \leq \theta \leq \frac{\pi}{2}$

Final Answer

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

Would you like more information on how this interval was derived or additional related topics? Here are some questions to think about:

1. What is the relationship between the sine function and the positioning of petals in a rose curve?
2. How does the coefficient of $\theta$ inside the sine function determine the number of petals?
3. Can the rose curve have petals that overlap in the same quadrant?
4. What happens to the curve if you change the function to cosine, i.e., $r = 4\cos(2\theta)$?
5. How does changing the amplitude (the number 4) affect the size of the petals?

Tip: To better understand the formation of rose curves, try graphing the equation $r = a\sin(k\theta)$ for different values of $a$ and $k$ and observe how the curve changes!