Question

Find the value of the following:

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

Answer 1

Let tan⁻¹(1) = x.

Then, tan x = 1 = `tan  pi/4`.

∴ `tan^(-1) (1) = pi/4`

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

Then, `cos y = -1/2`

= `-cos (pi/3)` 

= `cos (pi - pi/3)` 

= `cos ((2pi)/3)`

∴ `cos^(-1) (- 1/2) = (2pi)/3`

Let `sin^(-1) (-1/2) = z`.

Then, `sin z = -1/2`

= `-sin(pi/6)`

= `sin(-pi/6)`

∴ `sin^(-1) (-1/2) = - pi/6`

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

= `pi/4 + (2pi)/3 - pi/6`

= `(3pi + 8pi - 2pi)/12`

= `(9pi)/12`

= `(3pi)/4`