Trigonometry & Inverse Trigonometry
Trigonometric equations
Grade 11
Question:
<p>If <code>OA = r cot(π/4 - q/2) = 2r cot(q/2)</code>, and <code>tan(q/2) = t</code>, find <code>a + b + c</code> where <code>(1+t²)/(1-t) = t</code> and <code>tan(q/2) = (√17-3)/2</code>.</p>
Step-by-Step Solution
Key Concept: Solve the algebraic equation for t and identify the coefficients in the resulting expression for tan(q/2).
<p><strong>Step 1:</strong> From the given relation, we have <code>(1+t²)/(1-t) = t</code></p><p><strong>Step 2:</strong> Solving for <code>t</code>: <code>1 + t² = t(1-t) = t - t²</code></p><p><strong>Step 3:</strong> This gives <code>2t² - t + 1 = 0</code>, so <code>t = (-3 ± √17)/2</code></p><p><strong>Step 4:</strong> Taking the valid solution <code>tan(q/2) = (√17-3)/2</code></p><p><strong>Step 5:</strong> Therefore <code>a + b + c = 17 + 3 + 2 = 22</code></p>
Correct Answer: 22