Which one of the following is not a rational number ?

Options

A. `sin (tan^(-1) 3 + tan^(-1).(1)/(3))`
B. `cos ((pi)/(2) - sin^(-1).(3)/(4))`
C. `"log"_(2) (sin ((1)/(4) sin^(-1).(sqrt63)/(8)))`
D. `tan ((1)/(2) cos^(-1).(sqrt5)/(3))`

Answer

A, B, C

Explanation

(1) `sin (tan^(-1) 3 + tan^(-1).(1)/(3)) = sin.(pi)/(2) =1`
(2) `cos((pi)/(2) - sin^(-1).(3)/(4)) = cos (cos^(-1).(3)/(4)) = (3)/(4)`
(3) `sin((1)/(4) sin^(-1). (sqrt63)/(8))`
Let `sin^(-1).(sqrt63)/(8) = theta`. Then,
`sin theta = (sqrt63)/(8) rArr cos theta = (1)/(8)`
We have `cos.(theta)/(2) = sqrt((1 + cos theta)/(2)) = (3)/(4)`
`rArr sin.(theta)/(4) = sqrt((1 - cos.(theta)/(2))/(2)) = (1)/(2sqrt2)`
Now, `"log"_(2) sin ((1)/(4) sin^(-1).(sqrt63)/(8)) = "log"_(2) (1)/(2sqrt2) = -(3)/(2)`
(4) `cos^(-1).(sqrt5)/(3) = theta " or " cos theta = (sqrt5)/(3)`
`:. tan.(theta)/(2) = sqrt(1 - cos theta)/(1 + cos theta)) = (3-sqrt5)/(2)`, which is irrational