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