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 evaluate the expressions for \( S \) and \( C \) and determine their signs.

### Step 1: Evaluate \( S \)
Given:
\[ S = \sqrt{2} - \sin(\sqrt{3}) \]

First, we calculate the approximate values:
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt{3} \approx 1.732 \)

Next, we need to find \( \sin(\sqrt{3}) \). Since \( \sqrt{3} \) is approximately \( 1.732 \), we know that:
\[ \sin(1.732) \text{ is less than } 1 \text{ (as } \sin(x) \text{ is maximum at } 1 \text{ for } x \leq \frac{\pi}{2} \text{)} \]

Thus, we can conclude:
\[ S = \sqrt{2} - \sin(\sqrt{3}) > \sqrt{2} - 1 > 0 \]

### Step 2: Evaluate \( C \)
Given:
\[ C = \cos(\sqrt{2}) - \cos(\sqrt{3}) \]

Now, we calculate \( \cos(\sqrt{2}) \) and \( \cos(\sqrt{3}) \):
- \( \cos(\sqrt{2}) \) is the cosine of approximately \( 1.414 \)
- \( \cos(\sqrt{3}) \) is the cosine of approximately \( 1.732 \)

Both angles are in the range where cosine is positive but decreasing. Specifically:
- \( \cos(1.414) \) is positive but less than 1.
- \( \cos(1.732) \) is also positive but less than \( \cos(1.414) \).

Since \( 1.732 > 1.414 \), we have:
\[ C = \cos(\sqrt{2}) - \cos(\sqrt{3}) > 0 \text{ (because } \cos(\sqrt{2}) > \cos(\sqrt{3}) \text{)} \]

### Conclusion
Both \( S \) and \( C \) are positive:
\[ S > 0 \quad \text{and} \quad C > 0 \]

Thus, the correct answer is that both \( S \) and \( C \) are greater than zero.