Question

Find the principal value of the following: sin^-1 `(1/sqrt(2))`

Answer 1

The principal value branch of sin^-1x is `[- pi/2, pi/2]`.

Let sin^-1`(1/sqrt(2)) = α, "where" (-pi)/(2) ≤ α ≤ pi/(2)`

∴ sin α = `(1)/(sqrt2) = sin  pi/(4)`

∴ α = `pi/(4)                  ...[∵ - pi/2 ≤ pi/4 ≤ pi/2 ]`

∴ the principal value of sin^-1`(1/sqrt(2)) "is" pi/(4)`.