Find the principal value of the following:
(i) `cosec^(-1)(2)` (ii) `tan^(-1) (-sqrt3)` (iii) `cos^(-1) (-(1)/(sqrt2))`

Explanation

(i) Let `cosec^(-1) (2) = y`. Then,
`cosec y = 2 = cosec ((pi)/(6))`
We know that the range of the principal values branch of the function `cosec^(-1) " is " [-(pi)/(2), (pi)/(2)] - {0} and cosec (pi)/(6) = 2`
Therefore, the principal value of `cosec^(-1) (2) " is " (pi)/(6)`
(ii) Let `tan^(-1) (-sqrt3) = y`. Then,
`tan y = -sqrt3 = - tan. (pi)/(3) = tan.(-(pi)/(3))`
We know that the range of the principal values branch of the function `tan^(-1) " is " (-(pi)/(2), (pi)/(2)) and tan (-(pi)/(3)) = -sqrt3` ltrbgt Therefore, the principal value of `tan^(-1)(-sqrt3) " is " -(pi)/(3)`
(iii) Let `cos^(-1) (-(1)/(sqrt2)) = y`. Then, `cos y = -(1)/(sqrt2) = - cos. ((pi)/(4)) = cos (pi - (pi)/(4)) = cos. ((3pi)/(4))`
We know that the range of the principal values branch of the function `cos^(-1) " is " [0, pi] and cos. ((3pi)/(4)) = -(1)/(sqrt2)`
Therefore, the principal value of `cos^(-1) (-(1)/(sqrt2)) " is " (3pi)/(4)`