`sin^(-1)(sin3)+sin^(-1)(sin4)+sin^(-1)(sin5)` is equal to

Options

A. `-1`
B. `-2`
C. 12
D. `2-pi`

Explanation

To solve the expression \( \sin^{-1}(\sin 3) + \sin^{-1}(\sin 4) + \sin^{-1}(\sin 5) \), we will use the properties of the inverse sine function and the periodicity of the sine function.

### Step 1: Analyze \( \sin^{-1}(\sin x) \)

The function \( \sin^{-1}(\sin x) \) has specific cases based on the value of \( x \):

1. If \( x \) is in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \), then \( \sin^{-1}(\sin x) = x \).
2. If \( x \) is in the range \( [\frac{\pi}{2}, \frac{3\pi}{2}] \), then \( \sin^{-1}(\sin x) = \pi - x \).
3. If \( x \) is in the range \( [\frac{3\pi}{2}, \frac{5\pi}{2}] \), then \( \sin^{-1}(\sin x) = x - 2\pi \).
4. If \( x \) is in the range \( [\frac{5\pi}{2}, \frac{7\pi}{2}] \), then \( \sin^{-1}(\sin x) = 3\pi - x \).

### Step 2: Calculate \( \sin^{-1}(\sin 3) \)

Since \( 3 \) is in the range \( [\frac{\pi}{2}, \frac{3\pi}{2}] \) (approximately \( 1.57 \) to \( 4.71 \)), we use the second case:

\[ \sin^{-1}(\sin 3) = \pi - 3 \]

### Step 3: Calculate \( \sin^{-1}(\sin 4) \)

Since \( 4 \) is also in the range \( [\frac{\pi}{2}, \frac{3\pi}{2}] \), we again use the second case:

\[ \sin^{-1}(\sin 4) = \pi - 4 \]

### Step 4: Calculate \( \sin^{-1}(\sin 5) \)

Since \( 5 \) is in the range \( [\frac{3\pi}{2}, \frac{5\pi}{2}] \), we use the third case:

\[ \sin^{-1}(\sin 5) = 5 - 2\pi \]

### Step 5: Combine the results

Now we can combine all three results:

\[ \sin^{-1}(\sin 3) + \sin^{-1}(\sin 4) + \sin^{-1}(\sin 5) = (\pi - 3) + (\pi - 4) + (5 - 2\pi) \]

### Step 6: Simplify the expression

Combining the terms:

\[ = \pi - 3 + \pi - 4 + 5 - 2\pi \]
\[ = (2\pi - 2\pi) + (-3 - 4 + 5) \]
\[ = 0 - 2 = -2 \]

### Final Answer

Thus, the final answer is:

\[ \sin^{-1}(\sin 3) + \sin^{-1}(\sin 4) + \sin^{-1}(\sin 5) = -2 \]