Question

Find the value of the following:

`cos^(-1) (1/2) + 2 sin^(-1) (1/2)`

Answer 1

We know that the range of the principal value branch of cos^–1 and sin^–1 are [0, π] and `[-pi/2, pi/2]` respectively.

Let `cos^(-1) (1/2) = x`

⇒ `1/2 = cos x`

Then, `1/2 = cos (pi/3)`, where `pi/3 ∈ [0, pi]`

Let `sin^(-1) (1/2) = y`

⇒ `1/2 = sin y`

Then, `1/2 = sin (pi/6)`, where `pi/6 ∈ [-pi/2, pi/2]`

∴ `cos^(-1) (1/2) + 2 sin^(-1) (1/2)`

= `pi/3 + 2 * pi/6`

= `pi/3 + pi/3`

= `(2pi)/3`