https://file.aiquickdraw.com/mathbot/file/17245895018790q7fffa0.jpg

The equation in the image appears to be \( \sin \theta = \frac{m}{2} \) with the range \( 45^\circ < \theta < 108^\circ \). To solve for \( m \), we need to understand that the sine function in this range has a positive value, and we will find the maximum and minimum possible values for \( \sin \theta \) within this interval.

### Step 1: Determine the range of \( \sin \theta \)
The range for \( \theta \) is \( 45^\circ < \theta < 108^\circ \). Let's calculate the sine values at the boundary points:
- \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.707 \)
- \( \sin(108^\circ) \approx \sin(72^\circ) \approx 0.951 \)

### Step 2: Express \( \sin \theta \) in terms of \( m \)
Given \( \sin \theta = \frac{m}{2} \), we can write:
\[
\frac{\sqrt{2}}{2} < \frac{m}{2} < 0.951
\]

### Step 3: Solve for \( m \)
Multiply the inequality by 2 to isolate \( m \):
\[
\sqrt{2} < m < 1.902
\]
This simplifies to:
\[
1.414 < m < 1.902
\]

So, the value of \( m \) must lie between approximately 1.414 and 1.902.

Would you like more details or have any other questions?

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Here are 5 relative questions you might explore next:
1. What is the graphical representation of the sine function within a specific interval?
2. How do trigonometric identities help solve equations like this?
3. What would change if the interval for \( \theta \) was different?
4. Can \( m \) be determined exactly if we know \( \theta \) is a specific value?
5. How does the sine function behave as \( \theta \) approaches 90°?

**Tip:** When working with trigonometric functions, always consider the function's periodicity and the specific range of the angle to ensure correct solutions.

Learn how to solve the equation \( \sin \theta = \frac{m}{2} \) for \( m \) with detailed steps and calculations. This problem involves trigonometric concepts and inequalities, focusing on finding the range of \( m \) within a given interval of \( \theta \). Suitable for high school students studying trigonometry.

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