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`