What is the prinicple value of `cosec^(-1)(-sqrt2)` ?

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. `-(pi)/(4)`
D. 0

Answer

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

### Step 1: Define the angle
Let \( \theta = \csc^{-1}(-\sqrt{2}) \). By the definition of the cosecant function, we have:
\[ \csc(\theta) = -\sqrt{2} \]

### Step 2: Relate cosecant to sine
Recall that cosecant is the reciprocal of sine:
\[ \csc(\theta) = \frac{1}{\sin(\theta)} \]
Thus, we can write:
\[ \sin(\theta) = \frac{1}{\csc(\theta)} = \frac{1}{-\sqrt{2}} = -\frac{1}{\sqrt{2}} \]

### Step 3: Determine the angle
The sine value \( -\frac{1}{\sqrt{2}} \) corresponds to angles in the third and fourth quadrants. The reference angle where \( \sin(\theta) = \frac{1}{\sqrt{2}} \) is \( \frac{\pi}{4} \). Therefore, the angles where \( \sin(\theta) = -\frac{1}{\sqrt{2}} \) are:
\[ \theta = -\frac{\pi}{4} \quad \text{(in the fourth quadrant)} \]
and
\[ \theta = \frac{5\pi}{4} \quad \text{(in the third quadrant)} \]

### Step 4: Check the range of the inverse cosecant function
The principal value of \( \csc^{-1}(x) \) is defined to be in the range \( (-\frac{\pi}{2}, -\frac{\pi}{4}] \cup [\frac{\pi}{4}, \frac{\pi}{2}) \). Since \( -\frac{\pi}{4} \) lies within this range, we can conclude that:
\[ \theta = -\frac{\pi}{4} \]

### Step 5: Conclusion
Thus, the principal value of \( \csc^{-1}(-\sqrt{2}) \) is:
\[ \boxed{-\frac{\pi}{4}} \]