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

Answer

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

### Step 1: Understand the function
The function we are dealing with is the inverse sine function, denoted as \( \sin^{-1}(x) \). The range of the inverse sine function is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

### Step 2: Use the property of inverse sine
For any negative value \( -x \), we can use the property of the inverse sine function:
\[ \sin^{-1}(-x) = -\sin^{-1}(x) \]
In our case, we have:
\[ \sin^{-1}(-\frac{1}{2}) = -\sin^{-1}(\frac{1}{2}) \]

### Step 3: Find \( \sin^{-1}(\frac{1}{2}) \)
Now, we need to find \( \sin^{-1}(\frac{1}{2}) \). We know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Thus, we have:
\[ \sin^{-1}(\frac{1}{2}) = \frac{\pi}{6} \]

### Step 4: Substitute back into the equation
Now, substituting back into our equation from Step 2:
\[ \sin^{-1}(-\frac{1}{2}) = -\sin^{-1}(\frac{1}{2}) = -\frac{\pi}{6} \]

### Final Answer
Therefore, the principal value of \( \sin^{-1}(-\frac{1}{2}) \) is:
\[ -\frac{\pi}{6} \]

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