Question

Find the value of `sin(2cos^-1  sqrt(5)/3)`.

Answer 1

`sin(2cos^-1  sqrt(5)/3)`

Let `cos^-1  sqrt(5)/3` = α,

∴ cos α = `sqrt(5)/3`

From figure, sin α = `2/3`

Consider, sin 2 α = 2 sin α cos α

= `2*2/3*sqrt(5)/3`

= `(4sqrt(5))/9`

∴ 2α = `sin^-1((4sqrt(5))/9)`

i.e. `2cos^-1  sqrt(5)/3 = sin^-1  (4sqrt(5))/9`

∴ Given expression becomes

`sin(2cos^-1  sqrt(5)/3)`

= `sin[sin^-1  (4sqrt(5))/9]`

= `(4sqrt(5))/9`