Let `S = sqrt(2) - sin sqrt(3)` and `C = cossqrt(2) - cossqrt(3)` then which one of the following is correct ?

Options

A. `S gt 0` and `C gt 0`
B. `S gt 0` and `C lt 0`
C. `S lt 0` and `C gt 0`
D. `S lt 0` and `C lt 0`

Answer

To solve the problem, we need to analyze the expressions for \( S \) and \( C \) given as:

\[ S = \sqrt{2} - \sin(\sqrt{3}) \]
\[ C = \cos(\sqrt{2}) - \cos(\sqrt{3}) \]

### Step 1: Analyze \( S \)

1. **Evaluate \( \sqrt{2} \)**:
\[ \sqrt{2} \approx 1.414 \]

2. **Evaluate the range of \( \sin(\sqrt{3}) \)**:
- The value of \( \sqrt{3} \) is approximately \( 1.732 \).
- The sine function oscillates between -1 and 1 for all real numbers. Thus:
\[ -1 \leq \sin(\sqrt{3}) \leq 1 \]

3. **Determine the minimum value of \( S \)**:
- Since \( \sin(\sqrt{3}) \) can be at most 1, we have:
\[ S = \sqrt{2} - \sin(\sqrt{3}) \geq \sqrt{2} - 1 \approx 1.414 - 1 = 0.414 \]
- Therefore, \( S > 0 \).

### Step 2: Analyze \( C \)

1. **Evaluate \( \cos(\sqrt{2}) \) and \( \cos(\sqrt{3}) \)**:
- We know \( \sqrt{2} \approx 1.414 \) and \( \sqrt{3} \approx 1.732 \).
- The cosine function decreases in the interval \( [0, \pi] \). Thus:
\[ \cos(\sqrt{2}) > \cos(\sqrt{3}) \]

2. **Determine the signs of \( C \)**:
- Since both \( \sqrt{2} \) and \( \sqrt{3} \) are in the range \( [0, \pi] \), we can evaluate:
\[ C = \cos(\sqrt{2}) - \cos(\sqrt{3}) \]
- Since \( \cos(\sqrt{2}) \) is positive and \( \cos(\sqrt{3}) \) is also positive but less than \( \cos(\sqrt{2}) \), we conclude:
\[ C > 0 \]

### Conclusion

From our analysis, we find that both \( S \) and \( C \) are greater than zero:

\[ S > 0 \quad \text{and} \quad C > 0 \]

Thus, the correct option is:

\[ \text{Option 1: } S > 0 \text{ and } C > 0 \]