Question

Find the value of `cos^-1 (1/2) + tan^-1 (1/sqrt(3))`

Answer 1

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

∴ cos x = `1/2`

= `cos  pi/3` 

The principal value branch of cos⁻¹ is [0, π] and `0 ≤ pi/3 ≤ pi`

∴ x = `pi/3`

∴ `cos^-1  1/2 = pi/3`

Let `tan^-1 (1/sqrt(3))` = y

∴ tan y = `1/sqrt(3`

= tan `pi/6`

The principal value branch of tan⁻¹ is `((-pi)/2, pi/2)` and `- pi/2, < pi/6 < pi/2`

∴ y = `pi/6`

∴ `tan^-1 (1/sqrt(3)) = pi/6`

∴ `cos^-1 (1/2) + tan^-1 (1/sqrt(3)) = pi/3 + pi/6`

= `(3pi)/6`

= `pi/2`