Trigonometry & Inverse Trigonometry
Half-angle substitution
Grade 11
Question:
<p>Find the value of <strong>a + b + c</strong> where <strong>OA</strong> = <strong>r</strong> cot(π/4 − q/2) = 2r cot(q/2), and tan(q/2) = (−3 ± √17)/2.</p>
Step-by-Step Solution
Key Concept: Convert the cotangent expression using the tangent half-angle substitution and solve the resulting quadratic equation.
<p><strong>Step 1:</strong> Given OA = r cot(π/4 − q/2) = 2r cot(q/2)</p><p><strong>Step 2:</strong> Let tan(q/2) = t, then (1 + t²)/(1 − t) = t</p><p><strong>Step 3:</strong> Solving: t = (−3 ± √17)/2</p><p><strong>Step 4:</strong> Therefore tan(q/2) = (√17 − 3)/2</p><p><strong>Step 5:</strong> Thus a + b + c = 17 + 3 + 2 = 22</p>
Correct Answer: 22