The maximum value of 3 cos ⁡ θ + 5 sin ⁡ θ - π 6 for any real value of θ is :

Step-by-step solution

Given f θ = 3 cos ⁡ θ + 5 sin ⁡ θ - π 6 ⇒ f θ = 3 cos ⁡ θ + 5 sin ⁡ θ · 3 2 - cos ⁡ θ · 1 2 ⇒ f θ = 5 3 2 sin ⁡ θ + 1 2 cos ⁡ θ Now using the concept a cos x + b sin x + c   ∈   c - a 2 + b 2 ,   c + a 2 + b 2 , we can write Maximum value of f θ is 5 3 2 2 + 1 2 2 = 75 4 + 1 4 = 19