Find the principal values of the following :
If `sin^(-1) x = y ` then :

Options

A. `0 lt y lt pi`
B. `-pi/2 le y le pi/2 `
C. `0 lt y lt pi `
D. ` -pi/2 lt y lt pi/2 `

Answer

To find the principal values of the function \( \sin^{-1} x = y \), we will follow these steps:

### Step 1: Understand the Function
The function \( \sin^{-1} x \) is the inverse of the sine function. It takes a value \( x \) and returns an angle \( y \) such that \( \sin y = x \).

### Step 2: Identify the Domain
The domain of \( \sin^{-1} x \) is restricted to the interval:
\[ -1 \leq x \leq 1 \]
This means that \( x \) can take any value between -1 and 1, inclusive.

### Step 3: Identify the Range
The range of \( \sin^{-1} x \) is the set of possible values for \( y \). For the inverse sine function, the range is:
\[ -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \]
This means that \( y \) can take any value from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), inclusive.

### Step 4: Conclusion
Thus, the principal value of \( \sin^{-1} x \) is given by:
\[ y \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]

### Final Answer
The principal values of \( \sin^{-1} x \) are:
\[ y \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]

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