Question

Find the principal value of the following:

`"cosec"^(-1)(-sqrt2)`

Answer 1

Let `"cosec"^(-1)(-sqrt2)` = y

Then, cosec y = `-sqrt2 = - "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) = -sqrt2`.

Therefore, the principal value of `"cosec"^(-1) (-sqrt2)` is `-pi/4`.