Find the principal values of :
(1) `cos^(-1)((sqrt(3))/(2))`
(2) `sin^(-1)(-(1)/(sqrt(2)))`
(3) ` cot^(-1) (-(1)/(sqrt(3)))`

Explanation

(1) The principal value branch of `cos^(-1) x ` is `[0.pi]`.
Let `co^(-1)((sqrt(3))/(2))=alpha`, where `0le alpha le pi`
`therefore cos alpha =(sqrt(3))/(2)=cos.(pi)/(6)`
`therefore alpha=(pi)/(6)" ".............[because 0 le (pi)/(6)le pi]`
`therefore` the principal value of `cos^(-1)((sqrt(3))/(2)) is (pi)/(6)`.
(2) The principal of `sin^(-1) x is [-(pi)/(2),(pi)/(2)]`.
Let `sin^(-1)(-(1)/(sqrt(2)))=alpha`, where `-(pi)/(2) le alpha le (pi)/(2)`
`therefore sin alpha =-(1)/(sqrt(2))=-sin.(pi)/(4)`
`therefore sin alpha = sin (-(pi)/(4))`
`therefore alpha =-(pi)/(4)`
`therefore` the principal value of `sin^(-1)(-(1)/(sqrt(2)) is -(pi)/(4)`.
(3) The principal value branch of `cot^(-1)x is (0.pi)`.
Let `cot^(-1)(-(1)/(sqrt(3))=alpha` , where `0 lt alpha lt pi`
`therefore cot alpha=-(1)/(sqrt(3))=-cot.(pi)/(3)`.
`therefore cot alpha= cot (pi-(pi)/(3))`
`therefore cot alpha=cot (2pi)/(3)`
`therefore alpha=(2pi)/(3)" "..........[because 0 lt (2pi)/(3)lt pi]`
`therefore ` the principal value of `cot^(-1)(-(1)/(sqrt(3))) is (2pi)/(3)`.