Trigonometry & Inverse Trigonometry
Range of Trigonometric Functions
Grade 11
Question:
<p>If <span class="math">A = \cos(\cos x) + \sin(\cos x)</span>, then the least and greatest value of A are</p>
<p>(a) 0, 2</p>
<p>(b) –1, 1</p>
<p>(c) <span class="math">-\sqrt{2}, \sqrt{2}</span></p>
<p>(d) <span class="math">0, \sqrt{2}</span></p>
Step-by-Step Solution
Key Concept: Convert the sum of sine and cosine into a single sinusoidal function using the amplitude formula, then find the range based on the restricted domain.
<p>Let <span class="math">t = \cos x</span>, where <span class="math">t \in [-1, 1]</span>. Then <span class="math">A = \cos t + \sin t = \sqrt{2}\sin(t + \pi/4)</span>.</p><p>Since <span class="math">t \in [-1, 1]</span>, we have <span class="math">t + \pi/4 \in [-1 + \pi/4, 1 + \pi/4] \approx [0.785, 1.785]</span>.</p><p>The sine function on this interval ranges from <span class="math">\sin(-1 + \pi/4) = \sin(0.785) \approx 0.707</span> to maximum value 1 at <span class="math">\pi/2 \approx 1.571</span>.</p><p>Therefore, <span class="math">A \in [0, \sqrt{2}]</span>, giving minimum 0 and maximum <span class="math">\sqrt{2}</span>.</p>
Correct Answer: D