Question

Find the principal value of the following:

`"cosec"^(-1) (-sqrt(2))`

Answer 1

Let `"cosec"^(-1) (-sqrt(2)) = y`

Then, `"cosec"  y = -sqrt(2)`

= `-"cosec" (pi/4)`

= `"cosec" (-pi/4)`

We know that the range of the principal value branch of cosec⁻¹ is `[-pi/2, pi/2] - {0}` and `"cosec" (-pi/4) = -sqrt(2)`.

Therefore, the principal value of `"cosec"^(-1) (-sqrt(2))` is `-pi/4`.