i) Show that `sin^(-1){sin((3pi)/(4))} ne (3pi)/4` and find its value.
ii) Show that `cos^(-1){cos(-pi/3)} ne -pi/3` and find its value.
iii) Show that `tan^(-1){tan((5pi)/6)}ne (5pi)/6` and find its value.

Explanation

i) We know that the principal-value branch of `sin^(-1)` is `[-pi/2,pi/2]`.
`therefore sin^(-1){sin(3pi)/4} ne (3pi)/4`.
Now, `sin^(-1){sin(3pi)/4)=sin^(-1){sin(pi-pi/4)}=sin^(-1){sinpi/4}[therefore sin(pi-pi/4)=sinpi/4]`
`=pi/4 {therefore pi/4 in [-pi/2, pi/2]}`.
`therefore sin^(-1){sin (3pi)/4}=pi/4`.
ii) We know that the principal-value branch of `cos^(-1){cos(-pi/3)} ne -pi/3`.
Now, `cos^(-1){cos(-pi/3)}=cos^(-1){cospi/3} [therefore cos(-theta)=costheta]=pi/3 [therefore pi/3 in [0,pi]]`.
`therefore cos^(-1){cos(-pi/3)}=pi/3`.
iii) We know that the principal-value branch of `tan^(-1)` is `(-pi/2,pi/2)`.
`therefore tan^(-1){tan(5pi)/6} ne (5pi)/6`.
Now, `tan^(-1)(tan(5pi)/6}=tan^(-1){tan(pi-pi/6)}=tan^(-1){tan(-pi/6)} [therefore tan(pi-theta)=tan(-theta)]`.
`=-pi/6 [therefore -pi/6 in (-pi/2, pi/2)]`
`therefore tan^(-1){tan(5pi)/6}=-pi/6`.