If `A=sqrt(sin2-sin sqrt(3)), B=sqrt(cos2-cos sqrt(3))`, then which of the following statement is true ?

Options

A. A and B both are real numbers and `A gt B`
B. A and B both are real numbers and `A lt B`
C. Exactly one of A and B is not real number
D. Both A and B are not real numbers

Answer

To solve the problem, we need to analyze the expressions for \( A \) and \( B \) given by:

\[ A = \sqrt{\sin(2) - \sin(\sqrt{3})} \]
\[ B = \sqrt{\cos(2) - \cos(\sqrt{3})} \]

We will determine whether \( A \) and \( B \) are real numbers by checking the conditions under which the expressions inside the square roots are non-negative.

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

We start with \( A \):

\[ A = \sqrt{\sin(2) - \sin(\sqrt{3})} \]

For \( A \) to be real, the expression inside the square root must be non-negative:

\[ \sin(2) - \sin(\sqrt{3}) \geq 0 \]

This implies:

\[ \sin(2) \geq \sin(\sqrt{3}) \]

### Step 2: Evaluate \( \sin(2) \) and \( \sin(\sqrt{3}) \)

Now we need to find the values of \( \sin(2) \) and \( \sin(\sqrt{3}) \):

- The value of \( \sin(2) \) can be approximated using a calculator:
\[ \sin(2) \approx 0.909 \]

- The value of \( \sqrt{3} \) is approximately \( 1.732 \):
\[ \sin(\sqrt{3}) = \sin(1.732) \approx 0.987 \]

### Step 3: Compare \( \sin(2) \) and \( \sin(\sqrt{3}) \)

Now we compare:

\[ 0.909 < 0.987 \]

Thus,

\[ \sin(2) < \sin(\sqrt{3}) \]

This means:

\[ \sin(2) - \sin(\sqrt{3}) < 0 \]

Therefore, \( A \) is not a real number.

### Step 4: Analyze \( B \)

Next, we analyze \( B \):

\[ B = \sqrt{\cos(2) - \cos(\sqrt{3})} \]

For \( B \) to be real, we need:

\[ \cos(2) - \cos(\sqrt{3}) \geq 0 \]

This implies:

\[ \cos(2) \geq \cos(\sqrt{3}) \]

### Step 5: Evaluate \( \cos(2) \) and \( \cos(\sqrt{3}) \)

Now we find the values of \( \cos(2) \) and \( \cos(\sqrt{3}) \):

- The value of \( \cos(2) \) can also be approximated:
\[ \cos(2) \approx -0.416 \]

- For \( \cos(\sqrt{3}) \):
\[ \cos(\sqrt{3}) = \cos(1.732) \approx -0.187 \]

### Step 6: Compare \( \cos(2) \) and \( \cos(\sqrt{3}) \)

Now we compare:

\[ -0.416 < -0.187 \]

Thus,

\[ \cos(2) < \cos(\sqrt{3}) \]

This means:

\[ \cos(2) - \cos(\sqrt{3}) < 0 \]

Therefore, \( B \) is also not a real number.

### Conclusion

Since both \( A \) and \( B \) are not real numbers, the correct option is:

**Option 4: Both \( A \) and \( B \) are not real numbers.**