Find the principal values of the following
(i) `sin^(-1) (sin 1)` (ii) `sin^(-1) (sin 2)`
(iii) `sin^(-1) (sin 10)` (iv) `sin^(-1) (sin 20)`
(v) `sin^(-1) (sin 100)` (vi) `sin^(-1) (sin. (29 pi)/(5))`

Explanation

(i) Since `1 in [-pi//2, pi//2], sin^(-1) (sin 1) = 1`
(ii) Since `2 in [-pi//2, pi//2], sin^(-1) (sin 2) != 2`
`:. Sin^(-1) (sin 2) = sin^(-1) (sin (pi) -2))`
`= pi - 2 " as " (pi - 2) in [-pi//2, pi//2]`
(iii) `sin^(-1) (sin 10) = sin^(-1) (sin (3 pi - 10))`
`= 3 pi - 10 " as " (3 pi - 10) in [-pi//2, pi//2]`
(iv) `sin^(-1) (sin (-20)) = sin^(-1) (sin (6 pi - 20))`
(v) `sin^(-1) (sin 100) = sin^(-1) (sin (100 - 32 pi))`
`= 100 - 32 pi " as " (100 - 32 pi) in [-pi//2, pi//2]`
(vi) `sin^(-1) (sin.(29 pi)/(5)) = sin^(-1) (sin(6 pi - (pi)/(5)))`
`= sin^(-1) (sin(-(pi)/(5)))`
`= -(pi)/(5)`