Question

Evaluate the following:

`cos^-1(1/2) + 2sin^-1(1/2)`

Answer 1

Let `cos^-1(1/2)` = α, where 0 ≤ α ≤ π

∴ cos α = `1/2 = cos  (pi)/(3)`

∴ α = `pi/(3)                     ...[∵ 0 < pi/(3) < pi]`

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

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

∴ sin β = `(1)/(2) = sin  (pi)/(6)`

∴ β = `pi/(6)                  ...[∵ (-pi)/(2) ≤ pi/(6) ≤ pi/(2)]`

∴ `sin^-1(1/2) = pi/(6)`             ...(2)

`cos^-1(1/2) = pi/(3) and sin^-1(1/2) = pi/(6)`

∴ `cos^-1(1/2) + 2sin^-1(1/2)`

= `pi/(3) + 2(pi/6)`

= `pi/(3) + pi/(3)`

= `(2pi)/(3)`.