If `0 le x, y le pi "and sin"x + "sin" y = 2, "then"x + y = `

Options

A. `pi`
B. `(pi)/(2)`
C. `3pi`
D. none of these

Explanation

To solve the problem, we need to find the values of \(x\) and \(y\) given that:

\[ \sin x + \sin y = 2 \]

with the constraints \(0 \leq x, y \leq \pi\).

### Step-by-Step Solution:

1. **Understanding the Sine Function**:
The sine function, \(\sin\), has a maximum value of 1. Therefore, for the equation \(\sin x + \sin y = 2\) to hold true, both \(\sin x\) and \(\sin y\) must equal 1.

**Hint**: Recall that the sine function reaches its maximum value of 1 at specific angles.

2. **Setting Up the Equations**:
We can set up the equations:
\[ \sin x = 1 \quad \text{and} \quad \sin y = 1 \]

3. **Finding Values of \(x\) and \(y\)**:
The sine function equals 1 at:
\[ x = \frac{\pi}{2} \quad \text{and} \quad y = \frac{\pi}{2} \]

**Hint**: Remember that \(\sin \theta = 1\) occurs at \(\theta = \frac{\pi}{2}\) within the interval \([0, \pi]\).

4. **Calculating \(x + y\)**:
Now, we can find \(x + y\):
\[ x + y = \frac{\pi}{2} + \frac{\pi}{2} = \pi \]

5. **Conclusion**:
Thus, the final answer is:
\[ x + y = \pi \]

### Final Answer:
\[ \boxed{\pi} \]