The domain of r=4sin2theta is restricted to produce the graph shown. A petal lies in quadrant 4. What interval for theta will produce this graph?

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!

Explore how to graph a rose curve r = 4 sin(2θ) and find the theta interval producing a petal in quadrant 4. Understand the symmetry and positioning of petals in polar coordinates. Suitable for advanced high school students and beyond.

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