27COS x 81sin x33 COS x 34 sin x⇒ 3(3 cos x + 4 sin x)Now,3 cos x + 4 sin x can be WRITTEN as:\(\sqrt {{{\left( 3 \right)}^2} + {{\left( 4 \right)}^2}} \sin \left( {x + {{\TAN }^{ - 1}}\frac{3}{4}} \right)\) \(= 5\sin \left( {x + {{\tan }^{ - 1}}\frac{3}{4}} \right)\) ---(1)[∵ a sin x + b cos x = R sin (x + θ)Where \(R = \sqrt {{a^2} + {b^2}} ,\;\;θ = \left. {{{\tan }^{ - 1}}\frac{b}{a}} \right]\)∴ The minimum VALUE of (1) is (-5)∴ The minimum value of3(3 cos x + 4 sin x) ⇒ 3-5\(= \frac{1}{{{3^5}}} = \frac{1}{{243}}\)