Sequences & Series
Geometric Progression
Grade 11
Question:
<p>The sum of all values of \(\theta\) in interval \([0, 4\pi]\) for which \(\sin\theta, \cos\theta, \tan\theta\) taken in that order constitute a geometric progression is</p>
<p>(A) ...</p>
<p>(B) \(24\pi\)</p>
<p>(C) \(16\pi\)</p>
<p>(D) \(8\pi\)</p>
Step-by-Step Solution
Key Concept: Use the G.P. condition \(\cos^2\theta = \sin\theta \cdot \tan\theta\) and convert to a single trigonometric equation.
<p><strong>Step 1:</strong> For G.P., \(\cos^2\theta = \sin\theta \cdot \tan\theta = \sin\theta \cdot \frac{\sin\theta}{\cos\theta}\).</p><p><strong>Step 2:</strong> Simplify: \(\cos^2\theta = \frac{\sin^2\theta}{\cos\theta}\), so \(\cos^3\theta = \sin^2\theta\).</p><p><strong>Step 3:</strong> Use \(\sin^2\theta = 1 - \cos^2\theta\) and solve \(\cos^3\theta = 1 - \cos^2\theta\).</p><p><strong>Step 4:</strong> Find all solutions in \([0, 4\pi]\) and sum them to get \(8\pi\).</p>
Correct Answer: D