Trigonometry & Inverse Trigonometry
Maximum and minimum values
Grade 11
Question:
<p>Maximum value of \(\cos x (\sin x + \cos x)\) is equal to:</p>
<p>(a) \(\sqrt{2}\)</p>
<p>(b) \(2\sqrt{2}\)</p>
<p>(c) \(\frac{\sqrt{2}+1}{2}\)</p>
<p>(d) \(\sqrt{2}+1\)</p>
Step-by-Step Solution
Key Concept: To find the maximum of a trigonometric expression, convert it to a single trigonometric function using substitution and calculus, or rewrite it in the form R·sin(x+α) or R·cos(x+α) to identify the amplitude.
<p><strong>Step 1: Expand the expression</strong></p><p>Let f(x) = cos x(sin x + cos x) = sin x cos x + cos²x</p><p><strong>Step 2: Rewrite using double angle formulas</strong></p><p>f(x) = (1/2)sin(2x) + cos²x</p><p>Since cos²x = (1 + cos(2x))/2, we have:</p><p>f(x) = (1/2)sin(2x) + (1 + cos(2x))/2 = (1/2)sin(2x) + (1/2)cos(2x) + 1/2</p><p><strong>Step 3: Combine sine and cosine terms</strong></p><p>f(x) = (1/2)[sin(2x) + cos(2x)] + 1/2</p><p>Using the identity a·sin(θ) + b·cos(θ) = √(a² + b²)·sin(θ + φ):</p><p>sin(2x) + cos(2x) = √(1² + 1²)·sin(2x + π/4) = √2·sin(2x + π/4)</p><p><strong>Step 4: Find the maximum</strong></p><p>f(x) = (1/2)·√2·sin(2x + π/4) + 1/2</p><p>The maximum value of sin(2x + π/4) is 1, so:</p><p>f(x)_max = (1/2)·√2·(1) + 1/2 = √2/2 + 1/2 = (√2 + 1)/2</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C