The value of a + b such that the inequality a ≤ 5 cos ⁡ θ + 3 c o s θ + π 3 + 3 ≤ b holds true for all the real values of θ is (equality holds on both sides atleast once for real values of θ )

Step-by-step solution

The given expression is 5 cos ⁡ θ + 3 cos ⁡ θ cos ⁡ 60 o - sin ⁡ θ sin ⁡ 60 o + 3 = 5 cos ⁡ θ + 3 cos ⁡ θ 1 2 - 3 sin ⁡ θ 3 2 + 3 = cos ⁡ θ 5 + 3 2 + sin ⁡ θ - 3 3 2 + 3 = 13 2 cos ⁡ θ - 3 3 2 sin ⁡ θ + 3 13 2 cos ⁡ θ - 3 3 2 sin ⁡ θ ∈ - k , k where k = 13 2 2 + - 3 3 2 2 = 169 4 + 27 4 = 14 2 = 7 ⇒ 13 2 cos ⁡ θ - 3 3 2 sin ⁡ θ ∈ - 7,