Find the principal values of the following :
`sin^(-1)(1/(sqrt (2)))`

Answer

To find the principal value of \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \), we can follow these steps:

### Step 1: Set the equation
Let \( x = \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \).

### Step 2: Apply the sine function
To eliminate the inverse sine, we apply the sine function to both sides:
\[ \sin(x) = \frac{1}{\sqrt{2}} \]

### Step 3: Identify the angle
We need to find the angle \( x \) such that \( \sin(x) = \frac{1}{\sqrt{2}} \). We know from trigonometric values that:
\[ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

### Step 4: Determine the principal value
Since the range of the inverse sine function \( \sin^{-1}(y) \) is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), the principal value of \( x \) is:
\[ x = \frac{\pi}{4} \]

### Conclusion
Thus, the principal value of \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \) is:
\[ \frac{\pi}{4} \]

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