Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>If the sum of all values of \(\theta\), \(0 \le \theta \le 2\pi\) satisfying the equation<br/>\((8\cos^4 \theta - 3)(\cot \theta + \tan \theta - 2)(\cot \theta + \tan \theta + 2) = 12\)<br/>is \(k\pi\), then \(k\) is equal to:</p>
Step-by-Step Solution
Key Concept: Use the difference of squares formula and substitution to simplify the product, then solve systematically.
<p>Note that \((\cot \theta + \tan \theta - 2)(\cot \theta + \tan \theta + 2) = (\cot \theta + \tan \theta)^2 - 4\). Let \(u = \cot \theta + \tan \theta = \frac{2}{\sin 2\theta}\). Solve for the factors and find all solutions in \([0, 2\pi]\).</p>
Correct Answer: 8