Question

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

Hint:

To find the principal value of inverse trigonometric functions, consider the principal range of the function. For \( \csc^{-1}(x) \), the range is \( [0, \pi] \).

Answer 1

Step 1: Recall that \( {cosec}^{-1}(x) \) is the inverse of the cosec function, which gives the angle whose cosec is \( x \). In this case, we need to find the angle whose cosec is \( -\sqrt{2} \). 

Step 2: \( {cosec} \theta = -\sqrt{2} \), so \( \sin \theta = -\frac{1}{\sqrt{2}} \). The angle \( \theta \) whose sine is \( -\frac{1}{\sqrt{2}} \) is \( \theta = \frac{3\pi}{4} \). 

Step 3: Therefore, the principal value of \( {cosec}^{-1} \left( -\sqrt{2} \right) \) is \( \frac{3\pi}{4} \).