The principal value of `"cosec"^(-1)(-sqrt(2))` is

Options

A. `-pi/4`
B. `(3pi)/4`
C. `(5pi)/4`
D. none of these

Explanation

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

### Step 1: Set up the equation
Let \( x = \csc^{-1}(-\sqrt{2}) \). This implies that:
\[ \csc x = -\sqrt{2} \]

### Step 2: Use the definition of cosecant
Recall that cosecant is the reciprocal of sine. Therefore, we can write:
\[ \csc x = \frac{1}{\sin x} \]
Thus, we have:
\[ \frac{1}{\sin x} = -\sqrt{2} \]

### Step 3: Solve for sine
Taking the reciprocal gives us:
\[ \sin x = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2} \]

### Step 4: Identify the angle
The sine function is negative in the third and fourth quadrants. The reference angle for \( \sin x = \frac{\sqrt{2}}{2} \) is \( \frac{\pi}{4} \). Therefore, the angles where \( \sin x = -\frac{\sqrt{2}}{2} \) are:
\[ x = -\frac{\pi}{4} \quad \text{(fourth quadrant)} \]
and
\[ x = \frac{5\pi}{4} \quad \text{(third quadrant)} \]

### Step 5: Determine the principal value
However, since we are looking for the principal value of \( \csc^{-1}(-\sqrt{2}) \), we need to ensure that \( x \) lies within the range of \( \csc^{-1} \), which is \( (-\frac{\pi}{2}, 0) \). The angle \( -\frac{\pi}{4} \) is within this range.

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