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

Answer

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

### Step 1: Identify the angle
The angle \( \frac{5\pi}{4} \) is greater than \( \pi \) (which is \( \frac{4\pi}{4} \)). The sine function has a periodicity of \( 2\pi \), and we need to express \( \frac{5\pi}{4} \) in a form that falls within the range of the \( \sin^{-1} \) function.

### Step 2: Rewrite the angle
We can rewrite \( \frac{5\pi}{4} \) as:
\[ \frac{5\pi}{4} = \pi + \frac{\pi}{4} \]
This shows that \( \frac{5\pi}{4} \) is in the third quadrant.

### Step 3: Use the sine identity
Using the identity \( \sin(\pi + \theta) = -\sin(\theta) \), we can find:
\[ \sin\left(\frac{5\pi}{4}\right) = \sin\left(\pi + \frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) \]
Since \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \), we have:
\[ \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \]

### Step 4: Substitute into the inverse sine function
Now we substitute this result into the inverse sine function:
\[ \sin^{-1}(\sin(\frac{5\pi}{4})) = \sin^{-1}\left(-\frac{\sqrt{2}}{2}\right) \]

### Step 5: Determine the angle for the inverse sine
The value of \( \sin^{-1}(-\frac{\sqrt{2}}{2}) \) corresponds to an angle in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). The angle that satisfies this is:
\[ -\frac{\pi}{4} \]

### Step 6: Final result
Thus, we conclude that:
\[ \sin^{-1}(\sin(\frac{5\pi}{4})) = -\frac{\pi}{4} \]

### Summary
The final answer is:
\[ -\frac{\pi}{4} \]