Find the principal value of ltrbgt (a) `cosec^(-1) (-1)` (b) `cot^(-1) (-(1)/(sqrt3))`

Answer

(a) `-(pi)/(2)` (b) `(2pi)/(3)`

Explanation

(a) Let `theta` be the principal value of `cosec^(-1) (-1)`.
`theta in [-pi//2, pi//2] - {0} and cosec^(-1) (-1) = theta`
`:. theta = -(pi)/(2)` because `-(pi)/(2) in[-(pi)/(2), (pi)/(2)] - {0} and cosec (-(pi)/(2)) = -1`
Therefore, principal value of `cosec^(-1) (-1) " is " -(pi)/(2)`
(b) Let `theta` be the principal vlaue of `cot^(-1) (-(1)/(sqrt3))`
`:. theta in(0, pi) and cot theta = - (1)/(sqrt3)`
`:. theta = pi - (pi)/(3) = (2pi)/(3)`, because `(2pi)(3) in (0, pi) and cot(pi - (pi)/(3)) = - cot (pi)/(3) = -(1)/(sqrt3)`
Therefore, principal value of `cot^(-1) (-(1)/(sqrt3)) = (2 pi)/(3)`