Question

What is the value of `tan^-1(1) cos^-1(- 1/2) + sin^-1(- 1/2)`

Options

• `(3pi)/4`

• `(2pi)/4`

• `4pi`

• `3pi`

Answer 1

`(3pi)/4`

Explanation:

Let `tan^-1(1)` = x

⇒ `tan x = 1 = tan  pi/4`

∴ `tan^-1(1) = pi/4`

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

⇒ `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

⇒ `sin z = - 1/2 - 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`