Trigonometry & Inverse Trigonometry
Trigonometric identities
Grade 11
Question:
<p>In triangle ABC, if <code>cos(a) = 1/2 = 1/4</code> from triangle OED, and <code>θ = π - 2a</code>, find the area of triangle ABC where <code>BD = 2cot(θ/2) = 2cot(π/2 - a) = 2tan(a) = 2√15</code> and <code>AC = 3</code>.</p>
Step-by-Step Solution
Key Concept: Use the angle relationship and trigonometric identities to find the base and height, then calculate the area.
<p><strong>Step 1:</strong> From triangle OED: <code>cos(a) = 1/4</code></p><p><strong>Step 2:</strong> Given <code>θ = π - 2a</code></p><p><strong>Step 3:</strong> Calculate <code>BD = 2cot(θ/2) = 2cot(π/2 - a) = 2tan(a)</code></p><p><strong>Step 4:</strong> From <code>cos(a) = 1/4</code>, we get <code>sin(a) = √(1 - 1/16) = √15/4</code>, so <code>tan(a) = √15</code></p><p><strong>Step 5:</strong> Thus <code>BD = 2√15</code></p><p><strong>Step 6:</strong> Area of triangle ABC = <code>(1/2) × AC × BD = (1/2) × 3 × 2√15 = 3√15</code></p><p><strong>Step 7:</strong> <code>A² = (3√15)² / 15 = 135/15 = 9</code></p>
Correct Answer: 9