Find the principal value of each of the following:
i) `sin^(-1)(1/sqrt(2))`
ii) `cos^(-1)(sqrt(3))/2`
iii) `tan^(-1)(sqrt(3))`
iv) `"cosec"^(-1)(2)`

Explanation

i) We know that the range of the principal-value branch of `sin^(-1)` is `[-pi/2, pi/2]`.
Let `sin^(-1)(1/sqrt(2))=theta`. Then,
`sintheta=1/sqrt(2)=sinpi/4 rArr theta=pi/4 in [-pi/2 pi/2]`
Hence, the principal value of `sin^(-1)(1/sqrt(2))` is `pi/4`.
ii) We know that the range of the principal-value branch of `cos^(-1)` is `[0,pi]`
Let `cos^(-1)(sqrt(3)/2)=theta`. Then,
`costheta=sqrt(3)/2=cospi/6 rArr theta=pi/6 in [0,pi]`.
Hence, the principal value of `cos^(-)(sqrt(3)/2)` is `pi/6`.
iii) We know that the range of the principal-value branch is `tan^(-1)` is `(-pi/2, pi/2)`.
Let `tan^(-1)(sqrt(3))=theta`. Then,
`tantheta=sqrt(3)=tanpi/3 rArr theta=pi/3 in (-pi/2, pi/2)`.
Hence, the principal value of `"cosec"^(-1)(2)` is `pi/6`.