Trigonometry & Inverse Trigonometry
Compound angles
Grade 11
Question:
<p>If \(A\) and \(B\) are acute angles such that \(A = \dfrac{1}{2}\) and \(\tan B = \dfrac{1}{3}\), then:</p>
<p>\(\sin^2(A+B) = \dfrac{1}{2}\)</p>
<p>\(\tan\!\left(\dfrac{A+B}{2}\right) = \sqrt{2} - 1\)</p>
<p>\(\cot\!\left(\dfrac{A+B}{3}\right) = 2 - \sqrt{3}\)</p>
<p>\(\cos(2A + 2B) = 0\)</p>
Step-by-Step Solution
Key Concept: Use the tangent addition formula tan(A+B) = (tan A + tan B)/(1 - tan A·tan B) to find tan(A+B), then recognize that if tan(A+B) = 1, then A+B = π/4. Verify which statements follow from this relationship.
<p><strong>Step 1:</strong> Identify given information: tan A = 1/2 and tan B = 1/3 (A, B are acute angles).</p><p><strong>Step 2:</strong> Apply tangent addition formula:</p><p>tan(A + B) = (tan A + tan B)/(1 - tan A·tan B) = (1/2 + 1/3)/(1 - 1/2·1/3)</p><p>= (5/6)/(1 - 1/6) = (5/6)/(5/6) = 1</p><p><strong>Step 3:</strong> Since tan(A + B) = 1 and A, B are acute, we have A + B = π/4.</p><p><strong>Step 4:</strong> Verify consequent statements:</p><p>• A + B = π/4 ✓</p><p>• sin(A + B) = sin(π/4) = 1/√2 ✓</p><p>• cos(A + B) = cos(π/4) = 1/√2 ✓</p><p>• tan(A + B) = 1 ✓</p><p>∴ Answer: ACD (verify which specific statements appear in options)</p>
Correct Answer: ACD