Trigonometry & Inverse Trigonometry
Composite Trigonometric Functions
Grade 11
Question:
<p>If <i>P</i> = cos(cos <i>x</i>) + sin(cos <i>x</i>), then the least and greatest value of <i>P</i> respectively are:</p>
<p>(A) −1 and 1</p>
<p>(B) 0 and 2</p>
<p>(C) −√2 and √2</p>
<p>(D) 0 and √2</p>
Step-by-Step Solution
Key Concept: Express \(\cos u + \sin u\) as \(\sqrt{2}\sin(u + π/4)\) and find the range on the restricted domain.
<p>Let <i>u</i> = cos <i>x</i>. Then <i>u</i> ∈ [−1, 1].</p><p><i>P</i> = cos <i>u</i> + sin <i>u</i> = √2 sin(<i>u</i> + π/4).</p><p>For <i>u</i> ∈ [−1, 1], we have <i>u</i> + π/4 ∈ [−1 + π/4, 1 + π/4] ≈ [0.785, 1.785].</p><p>The minimum of √2 sin(<i>u</i> + π/4) on this interval occurs at <i>u</i> = −1: <i>P</i> = √2 sin(π/4 − 1) ≈ 0.</p><p>The maximum occurs at <i>u</i> = 1: <i>P</i> = √2 sin(π/4 + 1) ≈ √2.</p><p>∴ Answer is (D).</p>
Correct Answer: D