Question

Find the principal value of the following:

`sec^-1 (-sqrt2)`

Answer 1

Let `sec^-1 (-sqrt2)` = y

`-sqrt2` = sec y

sec y = `- sqrt2`

`1/(cos y) = - sqrt2`

Taking reciprocal cos y = `((-1)/sqrt2)` [where 0 ≤ y ≤ π]

cos y = `cos (pi - pi/4)  [cos pi/4 = 1/sqrt2 = cos (180^circ - theta) = - cos theta]`

`= cos ((4pi - pi)/4) = cos  (3pi)/4`

∴ The principal value of sec^-1 `(- sqrt2)` is `(3pi)/4`