Find the value of the following :
`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 Range of \( \sin^{-1}(x) \)
The function \( \sin^{-1}(x) \) (also known as arcsin) has a domain of \( x \) in the interval \([-1, 1]\) and a range of \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

### Step 2: Check the Value of \( \sin(4\pi/5) \)
Since \( 4\pi/5 \) is greater than \( \frac{\pi}{2} \) and less than \( \pi \), we can find its sine value using the identity:
\[ \sin(\pi - \theta) = \sin(\theta) \]
Thus, we can express \( \sin(4\pi/5) \) as:
\[ \sin(4\pi/5) = \sin(\pi - 4\pi/5) = \sin(\pi/5) \]

### Step 3: Substitute Back into the Inverse Function
Now we can substitute back into the inverse sine function:
\[ \sin^{-1}(\sin(4\pi/5)) = \sin^{-1}(\sin(\pi/5)) \]

### Step 4: Apply the Range of \( \sin^{-1}(x) \)
Since \( \pi/5 \) lies within the range of \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we have:
\[ \sin^{-1}(\sin(\pi/5)) = \pi/5 \]

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