The principal value of `cosec^(-1)(-1)` is

Options

A. `-pi/2`
B. 0
C. `pi/2`
D. `(3pi)/2`

Answer

To find the principal value of \( \csc^{-1}(-1) \), we will follow these steps:

### Step 1: Understanding the Range of \( \csc^{-1}(x) \)
The principal value of the cosecant inverse function, \( \csc^{-1}(x) \), has a range of \( (-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}) \). This means that the output of \( \csc^{-1}(x) \) will be in these intervals, excluding zero.

**Hint:** Remember that the range of the inverse cosecant function excludes zero.

### Step 2: Setting Up the Equation
We need to find \( x \) such that:
\[ \csc(x) = -1 \]
This means we are looking for an angle \( x \) where the cosecant value equals \(-1\).

**Hint:** Recall that \( \csc(x) = \frac{1}{\sin(x)} \).

### Step 3: Finding the Corresponding Sine Value
From the equation \( \csc(x) = -1 \), we can derive:
\[ \sin(x) = -1 \]

**Hint:** Think about the unit circle and where the sine function takes the value of \(-1\).

### Step 4: Identifying the Angle
The sine function equals \(-1\) at:
\[ x = -\frac{\pi}{2} + 2k\pi \quad \text{for any integer } k \]
However, we are interested in the principal value, which must lie within the specified range.

**Hint:** Focus on the principal value, which is the simplest angle that satisfies the equation.

### Step 5: Checking the Principal Value
The angle \( x = -\frac{\pi}{2} \) is within the range \( (-\frac{\pi}{2}, 0) \), so it is indeed a valid principal value.

**Hint:** Verify that the angle you found lies within the specified range of the inverse function.

### Conclusion
Thus, the principal value of \( \csc^{-1}(-1) \) is:
\[ \boxed{-\frac{\pi}{2}} \]