Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>If the sum of all solutions of the equation \(3\cot^2 \theta + 10\cot \theta + 3 = 0\) in \([0, 2\pi]\) is \(k\pi\) where \(k \in \mathbb{I}\), then find the value of \(k\).</p>
Step-by-Step Solution
Key Concept: Solve the quadratic equation for cotangent, then find all angles in the given interval and sum them.
<p>Solve the quadratic in \(\cot \theta\): Let \(u = \cot \theta\), then \(3u^2 + 10u + 3 = 0\). Using the quadratic formula or factoring: \((3u + 1)(u + 3) = 0\), giving \(u = -\frac{1}{3}\) or \(u = -3\). For each value, find all \(\theta \in [0, 2\pi]\) and sum them to get \(k\pi\).</p>
Correct Answer: 5