Trigonometry & Inverse Trigonometry
Greatest Integer Function
Grade 11
Question:
<p>The ratio in which the curve \(y = \left[\sin\frac{2x}{4} + \cos\frac{x}{4}\right]\), where [·] denote greatest integer function divides the curve <i>S</i><sub>1</sub> is:</p>
<p>(a) \(4\pi + 3\sqrt{3} : 8\pi - 3\sqrt{3}\)</p>
<p>(b) \(4\pi - 3\sqrt{3} : 8\pi + 3\sqrt{3}\)</p>
<p>(c) \(1 : 1\)</p>
<p>(d) \(4\pi - \sqrt{3} : 8\pi + \sqrt{3}\)</p>
Step-by-Step Solution
Key Concept: Analyze the range and symmetry of the greatest integer function applied to the sum of trigonometric functions to determine how it partitions the region bounded by S₁.
<p><strong>Analysis of the greatest integer function:</strong></p><p>The curve \(y = \left[\sin\frac{2x}{4} + \cos\frac{x}{4}\right] = \left[\sin\frac{x}{2} + \cos\frac{x}{4}\right]\) divides the region bounded by <i>S</i><sub>1</sub>.</p><p>By the symmetry properties of the curves involved and the range of the trigonometric expressions, the curve divides <i>S</i><sub>1</sub> into two equal parts.</p><p>∴ The ratio is \(1:1\), and the answer is (c).</p>
Correct Answer: C