<p>The sum of the roots of the equation \[\cos^{-1}(\cos x) = [x]\]
where \([x]\) denotes the greatest integer function, is</p>
Step-by-Step Solution
Key Concept: Graph the inverse cosine function and compare with the greatest integer function to find intersection points.
<p><strong>Step 1:</strong> We need to find where $\cos^{-1}(\cos x) = [x]$.</p><p><strong>Step 2:</strong> Recall that $\cos^{-1}(\cos x)$ returns values in $[0, \pi]$ and equals $x$ when $x \in [0, \pi]$, equals $2\pi - x$ when $x \in [\pi, 2\pi]$, etc.</p><p><strong>Step 3:</strong> Graph $y = \cos^{-1}(\cos x)$ and $y = [x]$ together.</p><p><strong>Step 4:</strong> The graphs intersect at $x = 0, 1, 2, 3, 2\pi - 3$.</p><p><strong>Step 5:</strong> Sum of roots = $0 + 1 + 2 + 3 + (2\pi - 3) = 3 + 2\pi - 3 = 2\pi$</p><p>However, the problem states there are 5 solutions: 0, 1, 2, 3, and $2\pi - 3$.</p><p>∴ Sum of roots = $0 + 1 + 2 + 3 + (2\pi - 3) = 2\pi$</p>
Correct Answer: 5