`sin^(-1)(sin 3) + sin^(-1) ( sin 4) + sin^(-1) ( sin 5) ` when simplified reduces to

Options

A. an irrational number
B. a rational number
C. an even prime
D. a negative integer

Answer

To simplify 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: Understanding the Range of \( \sin^{-1} \)
The function \( \sin^{-1}(x) \) (or arcsin) returns values in the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). Therefore, we need to express the angles \( 3 \), \( 4 \), and \( 5 \) in a way that fits within this range.

### Step 2: Simplifying \( \sin^{-1}(\sin 3) \)
Since \( 3 \) radians is greater than \( \frac{\pi}{2} \) (approximately \( 1.57 \)), we can use the property:
\[ \sin x = \sin(\pi - x) \]
Thus,
\[ \sin^{-1}(\sin 3) = \pi - 3 \]

### Step 3: Simplifying \( \sin^{-1}(\sin 4) \)
Similarly, for \( 4 \) radians, which is also greater than \( \frac{\pi}{2} \):
\[ \sin^{-1}(\sin 4) = \pi - 4 \]

### Step 4: Simplifying \( \sin^{-1}(\sin 5) \)
For \( 5 \) radians, which is again greater than \( \frac{\pi}{2} \):
\[ \sin^{-1}(\sin 5) = \pi - 5 \]

### Step 5: Adding the Results
Now we can add these simplified expressions together:
\[ \sin^{-1}(\sin 3) + \sin^{-1}(\sin 4) + \sin^{-1}(\sin 5) = (\pi - 3) + (\pi - 4) + (\pi - 5) \]
This simplifies to:
\[ 3\pi - (3 + 4 + 5) = 3\pi - 12 \]

### Final Result
Thus, the simplified expression is:
\[ 3\pi - 12 \]