Question

Find the principal value of \( \text{cosec}^{-1}(-\sqrt{2}) \).

Hint:

To find the principal value of inverse cosecant, it's often easier to convert the problem into its sine equivalent, i.e., \( \text{cosec}^{-1}(x) = \sin^{-1}(\frac{1}{x}) \). Then, find the value within the range of \( \sin^{-1}(x) \), which is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).

Answer 1

Step 1: Understanding the Concept:
The principal value of an inverse trigonometric function is the value that lies within its defined principal value branch.
For \( \text{cosec}^{-1}(x) \), the principal value branch is \( [-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\} \).
Step 2: Detailed Explanation:
Let \( y = \text{cosec}^{-1}(-\sqrt{2}) \).
By the definition of an inverse function, this implies:
\[ \text{cosec}(y) = -\sqrt{2} \] We know that \( \text{cosec}(y) = \frac{1}{\sin(y)} \).
So, \( \sin(y) = -\frac{1}{\sqrt{2}} \).
We know that \( \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} \).
Since \( \sin(-x) = -\sin(x) \), we can write:
\[ \sin(-\frac{\pi}{4}) = -\sin(\frac{\pi}{4}) = -\frac{1}{\sqrt{2}} \] So, \( y = -\frac{\pi}{4} \).
Step 3: Final Answer:
We must check if this value lies in the principal value range of \( \text{cosec}^{-1}(x) \), which is \( [-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\} \).
Since \( -\frac{\pi}{2} \leq -\frac{\pi}{4} \leq \frac{\pi}{2} \) and \( -\frac{\pi}{4} \neq 0 \), the value is valid.
Therefore, the principal value of \( \text{cosec}^{-1}(-\sqrt{2}) \) is \( -\frac{\pi}{4} \).