STATEMENT-1: `sin 2 > sin 3` STATEMENT-2: If `x,y in (pi/2, pi), x lt y,` then `sin x gt siny`

Answer

To determine the validity of the two statements provided, we will analyze each statement step by step.

### Step 1: Analyze Statement 1
**Statement 1:** `sin 2 > sin 3`

1. **Convert radians to degrees:**
- \(2\) radians is approximately \(114.59^\circ\)
- \(3\) radians is approximately \(171.89^\circ\)

2. **Evaluate the sine values:**
- We know that the sine function increases in the interval \( (0, \frac{\pi}{2}) \) and decreases in the interval \( (\frac{\pi}{2}, \pi) \).
- Since \(2\) radians is in the interval \( ( \frac{\pi}{2}, \pi) \) and \(3\) radians is also in the interval \( ( \frac{\pi}{2}, \pi) \), we can analyze their sine values based on their positions within this interval.

3. **Graphical interpretation:**
- The sine function reaches its maximum at \( \frac{\pi}{2} \) and decreases towards \( \pi \).
- Since \(2 < 3\), it follows that \( \sin(2) > \sin(3) \).

Thus, **Statement 1 is true.**

### Step 2: Analyze Statement 2
**Statement 2:** If \(x, y \in \left(\frac{\pi}{2}, \pi\right)\), \(x < y\), then \(\sin x > \sin y\).

1. **Understanding the intervals:**
- The interval \( \left(\frac{\pi}{2}, \pi\right) \) corresponds to angles between \(90^\circ\) and \(180^\circ\).
- In this interval, the sine function is decreasing.

2. **Choose specific values:**
- Let \(x = \frac{2\pi}{3}\) (which is approximately \(120^\circ\)) and \(y = \frac{5\pi}{6}\) (which is approximately \(150^\circ\)).
- Here, \(x < y\).

3. **Evaluate the sine values:**
- \(\sin\left(\frac{2\pi}{3}\right) = \sin(120^\circ) = \frac{\sqrt{3}}{2}\)
- \(\sin\left(\frac{5\pi}{6}\right) = \sin(150^\circ) = \frac{1}{2}\)

4. **Comparison:**
- Since \(\frac{\sqrt{3}}{2} > \frac{1}{2}\), it follows that \(\sin x > \sin y\).

Thus, **Statement 2 is also true.**

### Conclusion
Both statements are true:
- **Statement 1:** \( \sin 2 > \sin 3 \) is true.
- **Statement 2:** If \( x < y \) in \( \left(\frac{\pi}{2}, \pi\right) \), then \( \sin x > \sin y \) is true.