Question

Find the principal value of the following:

cosec^-1 (- 2)

Answer 1

Let cosec^-1 (- 2) = x ⇒ cosec x = - 2

⇒ cosec x = - cosec \(\frac{\pi}{6}\)

⇒ cosec x = cosec \((-\frac{\pi}{6})\)

[∵ cosec (- θ) = - cosec θ]

We know that the range of the principal value of

cosec^-1 x is \([\frac{-\pi}{2},​​\frac{\pi}{2}]\) - [0] and cosec \((-\frac{\pi}{6})\) = -2.

Thus, the principal value of cosec^-1 (- 2) is \(-\frac{\pi}{6}\).