Find the simplified form

Question:

Find the simplified form of cos-1 [3/5 cos x + 4/5 sin x], x ∈ [-3π/4, π/4].

Solution:

We have,

cos-1 [3/5 cos x + 4/5 sin x], x ∈ [-3π/4, π/4]

Now, let cos α = 3/5

So, sin α = 4/5 and tan α = 4/3

cos-1 [3/5 cos x + 4/5 sin x]

⇒ cos-1 [3/5 cos x + 4/5 sin x] = cos-1 [cos α cos x + sin α sin x]

= cos-1 [cos (α – x)]

= α – x

= tan-1 4/3 – x

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