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Question
Evaluate the following:
`cos^-1(1/2) + 2sin^-1(1/2)`
Solution
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)`.
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