Trigonometry & Inverse Trigonometry
Triangle Properties
Grade 11
Question:
<p>In a triangle ABC, if <span class='latex'>\tan A + \tan B + \tan C = 9</span>. If <span class='latex'>\tan^2 A + \tan^2 B + \tan^2 C = k</span>, then the least value of k is:</p>
Step-by-Step Solution
Key Concept: Use the identity for sum of tangents in a triangle combined with optimization techniques.
<p><strong>Analysis:</strong> In a triangle ABC: <span class='latex'>\tan A + \tan B + \tan C = \tan A \tan B \tan C</span></p><p>Given: <span class='latex'>\tan A + \tan B + \tan C = 9</span>, so <span class='latex'>\tan A \tan B \tan C = 9</span></p><p>Let <span class='latex'>x = \tan A, y = \tan B, z = \tan C</span>. Then:</p><p><span class='latex'>x + y + z = 9</span> and <span class='latex'>xyz = 9</span></p><p>We need to minimize <span class='latex'>x^2 + y^2 + z^2</span></p><p>By Cauchy-Schwarz or Lagrange multipliers, the minimum occurs when <span class='latex'>x = y = z = \sqrt[3]{9}</span></p><p>Then: <span class='latex'>k = 3(\sqrt[3]{9})^2 = 3 \cdot 9^{2/3} = 3 \cdot 3^{4/3} = 3^{1+4/3} = 3^{7/3} = 9 \cdot 3^{1/3} = 9\sqrt[3]{3}</span></p><p>∴ Answer is (q) <span class='latex'>k = 9\sqrt[3]{3}</span></p>
Correct Answer: q