Find the principal values of the following :
` "cosec"^(-1) (-sqrt(2))`

Answer

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

### Step 1: Understand the definition of cosecant inverse
The cosecant inverse function, \( \csc^{-1}(x) \), is defined for \( x \leq -1 \) or \( x \geq 1 \). Therefore, \( -\sqrt{2} \) is within the domain of the function.

### Step 2: Relate cosecant to sine
We know that:
\[ \csc(\theta) = \frac{1}{\sin(\theta)} \]
Thus, if \( y = \csc^{-1}(-\sqrt{2}) \), then:
\[ \csc(y) = -\sqrt{2} \]
This implies:
\[ \sin(y) = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2} \]

### Step 3: Determine the angle for sine
The sine function equals \(-\frac{\sqrt{2}}{2}\) at specific angles. The angles where this occurs in the unit circle are:
\[ y = -\frac{\pi}{4} + 2k\pi \quad \text{and} \quad y = -\frac{3\pi}{4} + 2k\pi \quad (k \in \mathbb{Z}) \]

### Step 4: Identify the principal value
The principal value of \( \csc^{-1}(x) \) is defined to be in the range \( [-\frac{\pi}{2}, -0) \cup (0, \frac{\pi}{2}] \). Among the angles we found, the one that falls within this range is:
\[ y = -\frac{\pi}{4} \]

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

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