<p>Find the maximum value of \(3\cos\theta + 5\sin\left(\theta - \dfrac{\pi}{6}\right)\).</p>
Step-by-Step Solution
Key Concept: Expand sin(θ - π/6) using the sine difference formula, then combine all cosine and sine terms into a single sinusoidal expression of the form R·sin(θ + φ), whose maximum value is R.
<p><strong>Step 1:</strong> Expand sin(θ - π/6) using the sine difference formula:</p><p>sin(θ - π/6) = sin θ cos(π/6) - cos θ sin(π/6) = (√3/2)sin θ - (1/2)cos θ</p><p><strong>Step 2:</strong> Substitute into the original expression:</p><p>f(θ) = 3cos θ + 5[(√3/2)sin θ - (1/2)cos θ]</p><p>f(θ) = 3cos θ + (5√3/2)sin θ - (5/2)cos θ</p><p><strong>Step 3:</strong> Combine like terms:</p><p>f(θ) = (3 - 5/2)cos θ + (5√3/2)sin θ = (1/2)cos θ + (5√3/2)sin θ</p><p><strong>Step 4:</strong> Express as R·sin(θ + φ) form. For a·cos θ + b·sin θ, the maximum is √(a² + b²):</p><p>R = √[(1/2)² + (5√3/2)²] = √[1/4 + 75/4] = √(76/4) = √19</p><p>∴ Maximum value = √19 ≈ <strong>4.3589</strong></p>
Correct Answer: 4.3589