Question

Prove the following: 

`sin^-1(1/sqrt(2)) -3sin^-1(sqrt(3)/2) = -(3π)/(4)`

Answer 1

Let `sin^-1(1/sqrt(2)) = α, "where" - pi/(2) ≤ α ≤ pi/(2)`

∴ sin α = `(1)/sqrt(2) = sin  pi/(4)`

∴ α = `pi/(4)                       ...[∵ - pi/(2) ≤ pi/(4) ≤ pi/(2)]`

∴ `sin^-1(1/sqrt(2)) = pi/(4)`           ...(1)

Let `sin^-1(sqrt(3)/2) = β, "where" - pi/(2) ≤ β ≤ pi/(2)`

∴ sin β = `sqrt(3)/(2) = sin  pi/(3)`

∴ β = `pi/(3)                     ...[∵ - pi/(2) ≤ pi/(3) ≤ pi/(2)]`

∴ `sin^-1(sqrt(3)/2) = pi/(3)`          ...(2)

L.H.S. = `sin^-1(1/sqrt(2)) - 3sin^-1(sqrt(3)/2)`

= `pi/(4) - 3(pi/3)`      ...[By (1) and (2)]

= `pi/(4) - pi`

= `-(3pi)/(4)`

= R.H.S.