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

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

1.	Find the principal value of ltrbgt (a) `cosec^(-1) (-1)` (b) `cot^(-1) (-(1)/(sqrt3))`
Answer» Correct Answer - (a) `-(pi)/(2)` (b) `(2pi)/(3)` (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)`

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