Question

Find the value of the given expression.

`sin^(-1) (sin  (2pi)/3)`

Answer 1

We know that sin^–1 (sin x) = x if `x ∈ [-pi/2, pi/2]`, which is the principal value branch of sin^–1 x.

Here, `(2pi)/3 ∉ [(-pi)/2, pi/2]`

Now, `sin^(-1) (sin  (2pi)/3)` can be written as: 

`sin^(-1) (sin  (2pi)/3) `

= `sin^(-1) [sin (pi - (2pi)/3)] `

= `sin^(-1) (sin  pi/3)` where `pi/3 ∈ [(-pi)/2, pi/2]`

∴ `sin^(-1) (sin  (2pi)/2)`

= `sin^(-1) (sin  pi/3) `

= `pi/3`