Which 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)((sqrt(63))/(8)))`
D. `tan((1)/(2)cos^(-1)((sqrt(5))/(3))`

Answer

A, B, C

Explanation

(A). `sin(tan^(-1)3+tan^(-1)((1)/(3)))=sin((pi)/(2))=1`
(B). `cos((pi)/(2)-sin^(-1)((3)/(4)))=cos(cos^(-1)((3)/(4)))=(3)/(4)`
(C). `sin((1)/(4)sin^(-1)((sqrt(63))/(8)))`
let `sin^(-1)((sqrt(63))/(8))=theta`
`impliessintheta=(sqrt(63))/(8)impliescostheta=(1)/(8)`
we have `cos((theta)/(2))=sqrt((1+costheta)/(2))=(3)/(4)`
`impliessin((theta)/(4))=sqrt((1-cos((theta)/(2)))/(2))=(1)/(sqrt(2))`
now `log_(2)sin((1)/(4)sin^(-1)((sqrt(63))/(8)))=log_(2)(1)/(2sqrt(2))=-(3)/(2)`
(D). `cos^(-1)((sqrt(5))/(3))=thetaimpliescostheta=(sqrt(5))/(3)`
`becausetan((theta)/(2))=(3-sqrt(5))/(2)` which is irrational