Question

Find the simplified form of `cos^-1 (3/5 cosx + 4/5 sin x)`, where x ∈ `[(-3pi)/4, pi/4]`

Answer 1

Given that `cos^-1 (3/5 cosx + 4/5 sin x)`

Put `3/5` = cos y

∴ `sqrt(1 - cos^2y)` = sin y

⇒ `sqrt(1 - 9/25)` = sin y

⇒ `4/5` = sin y

∴ `cos^-1 [3/5  cos x + 45 sin x]` = cos^–1[cos y cos x + sin y sin x]

= cos^–1 [cos (y – x)]

= y – x

= `tan^-1  4/3 - x`   ......`[tan y = siny/cosy = 4/3]`