The principal value of `sin^(-1)(1/2)` is

Options

A. `pi/6`
B. `(5pi)/6`
C. `-pi/6`
D. Both (1) & (2)

Answer

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

### Step 1: Understand the definition of the inverse sine function
The function \( \sin^{-1}(x) \) or arcsine gives us the angle \( y \) such that \( \sin(y) = x \). The principal value of \( \sin^{-1}(x) \) is defined in the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

**Hint:** Remember that the range of \( \sin^{-1}(x) \) is restricted to \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

### Step 2: Set up the equation
We need to find \( y \) such that:
\[ \sin(y) = \frac{1}{2} \]

**Hint:** Think about the angles whose sine values you know.

### Step 3: Identify the angle
From the unit circle or trigonometric values, we know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

**Hint:** Recall common angles and their sine values.

### Step 4: Check if the angle is in the principal range
The angle \( \frac{\pi}{6} \) is approximately \( 0.5236 \) radians, which lies within the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

**Hint:** Always verify if the angle you found is within the defined range for the inverse function.

### Step 5: Conclusion
Thus, the principal value of \( \sin^{-1}(1/2) \) is:
\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]

**Final Answer:** \( \frac{\pi}{6} \)