Trigonometry & Inverse Trigonometry
Greatest Integer Function with Trigonometric Equations
Grade 11
Question:
<p>The number of solutions of the equation
\[[y + [y]] = 2\cos x\]
where
\[y = \frac{1}{3}[\sin x + [\sin x + [\sin x]]]\]
and \([\cdot]\) denotes the greatest integer function, is</p>
Step-by-Step Solution
Key Concept: Use the greatest integer function property and analyze when the floor of sine equals cosine by comparing their graphs.
<p><strong>Step 1:</strong> Start with the equation $$[y + [y]] = 2\cos x$$</p><p><strong>Step 2:</strong> Using the property $$[x + h] = [x] + h$$ for $x \in \mathbb{I}$ and $0 \leq h < 1$, we get: $$[y] + [y] = 2\cos x$$</p><p><strong>Step 3:</strong> Simplify: $$2[y] = 2\cos x$$ which gives $$[y] = \cos x$$ ... (i)</p><p><strong>Step 4:</strong> Compute $y$: $$y = \frac{1}{3}([\sin x])^3 = [\sin x]$$ ... (ii)</p><p><strong>Step 5:</strong> From (i) and (ii): $$[\sin x] = \cos x$$</p><p><strong>Step 6:</strong> Analyzing the graphs of $y = [\sin x]$ and $y = \cos x$ shows they do not intersect.</p><p>∴ The number of solutions is <strong>0</strong>.</p>
Correct Answer: 0