Find the value of `sin^(-1) (sin. (4pi)/5)`

Answer

To find the value of \( \sin^{-1}(\sin(4\pi/5)) \), we can follow these steps:

### Step 1: Understand the function
The function \( \sin^{-1}(x) \) (or arcsin) is defined for \( x \) in the range of \([-1, 1]\) and gives an output in the range of \([- \frac{\pi}{2}, \frac{\pi}{2}]\).

### Step 2: Find the sine value
First, we need to find \( \sin(4\pi/5) \). We know that:
\[ \sin(4\pi/5) = \sin(\pi - 4\pi/5) = \sin(\pi/5) \]
This is because \( \sin(\pi - x) = \sin(x) \).

### Step 3: Use the property of inverse sine
Now, we can write:
\[ \sin^{-1}(\sin(4\pi/5)) = \sin^{-1}(\sin(\pi/5)) \]

### Step 4: Determine the output range
Since \( \pi/5 \) is in the range of \( [0, \frac{\pi}{2}] \), it is also in the range of \( \sin^{-1} \). Therefore:
\[ \sin^{-1}(\sin(\pi/5)) = \pi/5 \]

### Conclusion
Thus, the value of \( \sin^{-1}(\sin(4\pi/5)) \) is:
\[ \boxed{\frac{\pi}{5}} \]