Find the principal value of each of the following:
i) `cos^(-1)(-1/sqrt(2))`,
ii) `cos^(-1)(-1/2)`,
iii) `cot^(-1)(-1/sqrt(3))`

Explanation

i) We know that the range of the principal value of `cos^(-1)` is `[0,pi]`.
Let `cos^(-1)(-1/sqrt(2))=theta`. Then,
`costheta=-1/sqrt(2)=-cospi/4=cos(pi-pi/4)=cos(3pi)/4`.
`therefore theta=(3pi)/4 in [0, pi]`.
Hence, the principal value of `cos^(-1)(-1/sqrt(2))` is `(3pi)/4`.
ii) We know that the range of the principal value of `cos^(-1)` is `[0,pi]`.
Let `cos^(-1)(-1/2)=theta`. Then,
`costheta=-1/2=-cospi/3=cos(pi-pi/3)=cos""(2pi)/3`.
iii) We know that the range of the principal value of `cot^(-1)` is `(0,pi)`.
Let `cot^(-1)(-1/sqrt(3))=theta`. Then,
`cottheta=-1/sqrt(3)=-cotpi/3=cot(pi-pi/3)=cot(2pi)/3`.
`therefore theta=(2pi)/3 in (0,pi)`.
Hence, the principal value of `cot^(-1)(-1/sqrt(3))` is `(2pi)/3.`