Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>If <span style='font-family: Arial'>cos(α + β) = 4/5, sin(α - β) = 5/13</span> and α, β lie between 0 and π/4, then <span style='font-family: Arial'>tan 2α</span> is equal to</p>
<p>(a) 56/33</p>
<p>(b) 16/63</p>
<p>(c) 33/56</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Express 2α as the sum of (α+β) and (α-β), then use angle addition formulas to find sin 2α and cos 2α.
<p><strong>Step 1:</strong> From <span style='font-family: Arial'>cos(α + β) = 4/5</span>, we get <span style='font-family: Arial'>sin(α + β) = 3/5</span></p><p><strong>Step 2:</strong> From <span style='font-family: Arial'>sin(α - β) = 5/13</span>, we get <span style='font-family: Arial'>cos(α - β) = 12/13</span></p><p><strong>Step 3:</strong> Use <span style='font-family: Arial'>sin 2α = sin[(α + β) + (α - β)] = sin(α + β)cos(α - β) + cos(α + β)sin(α - β)</span></p><p><strong>Step 4:</strong> <span style='font-family: Arial'>sin 2α = (3/5)(12/13) + (4/5)(5/13) = 36/65 + 20/65 = 56/65</span></p><p><strong>Step 5:</strong> <span style='font-family: Arial'>cos 2α = cos[(α + β) + (α - β)] = cos(α + β)cos(α - β) - sin(α + β)sin(α - β)</span></p><p><strong>Step 6:</strong> <span style='font-family: Arial'>cos 2α = (4/5)(12/13) - (3/5)(5/13) = 48/65 - 15/65 = 33/65</span></p><p><strong>Step 7:</strong> <span style='font-family: Arial'>tan 2α = sin 2α / cos 2α = (56/65) / (33/65) = 56/33</span></p><p>∴ Answer is A.</p>
Correct Answer: A