Question

Find the principal value of the following:

`cos^(-1) (-1/sqrt(2))`

Answer 1

Let `cos^(-1) (-1/sqrt(2)) = y`

Then, `cos y = -1/sqrt(2)`

= `-cos (pi/4)`

= `cos(pi - pi/4)`

= `cos((3pi)/4)`

We know that the range of the principal value branch of cos⁻¹ is [0, π] and `cos ((3pi)/4) = -1/sqrt(2)`.

Where `(3pi)/4 ∈ [0, pi]`

Therefore, the principal value of `cos^(-1) (-1/sqrt(2))` is `(3pi)/4`.