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

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

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

Answer»

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