Trigonometry & Inverse Trigonometry
Trigonometric Identities and Transformations
Grade 11
Question:
<p>The maximum value of <span class="math">a \sin 2x + b \cos 2x</span> for all real x is</p>
<p>(a) <span class="math">a + b</span></p>
<p>(b) <span class="math">\sqrt{a^2 + b^2}</span></p>
<p>(c) <span class="math">\max\{|a|, |b|\}</span></p>
<p>(d) <span class="math">\max\{a, b\}</span></p>
Step-by-Step Solution
Key Concept: Convert a linear combination of sine and cosine into a single sinusoidal function with amplitude equal to the square root of the sum of squares of coefficients.
<p>We can express <span class="math">a \sin 2x + b \cos 2x</span> in the form <span class="math">R \sin(2x + \phi)</span>.</p><p>Let <span class="math">R \sin(2x + \phi) = R\sin 2x \cos \phi + R \cos 2x \sin \phi</span>.</p><p>Comparing coefficients: <span class="math">R \cos \phi = a</span> and <span class="math">R \sin \phi = b</span>.</p><p>Therefore, <span class="math">R^2 = a^2 + b^2</span>, so <span class="math">R = \sqrt{a^2 + b^2}</span>.</p><p>Since the maximum value of <span class="math">\sin(2x + \phi)</span> is 1, the maximum value of <span class="math">a \sin 2x + b \cos 2x</span> is <span class="math">\sqrt{a^2 + b^2}</span>.</p>
Correct Answer: B