Trigonometry & Inverse Trigonometry
Triangle Solution
Grade 11
Question:
<p>In an acute angled triangle <i>ABC</i>, given that <i>a</i> = 6, <i>b</i> = 3 and \(\cos(A - B) = -\frac{1}{4}\), find angle <i>C</i>.</p>
<p>(a) \(\frac{3\pi}{4}\)</p>
<p>(b) \(\frac{\pi}{2}\)</p>
<p>(c) \(\frac{\pi}{4}\)</p>
<p>(d) \(\frac{3\pi}{4}\)</p>
Step-by-Step Solution
Key Concept: Use the tangent half-angle formula for the difference of angles combined with the cosine condition to find C.
<p><strong>Step 1:</strong> Use the identity $\tan\frac{A-B}{2} = \frac{a-b}{a+b}\cot\frac{C}{2}$</p><p><strong>Step 2:</strong> With <i>a</i> = 6, <i>b</i> = 3: $\tan\frac{A-B}{2} = \frac{3}{9}\cot\frac{C}{2} = \frac{1}{3}\cot\frac{C}{2}$</p><p><strong>Step 3:</strong> From $\cos(A-B) = -\frac{1}{4}$ and using the relation with the tangent expression, we can solve to get $C = \frac{3\pi}{4}$</p><p>∴ Answer is <i>A</i>.</p>
Correct Answer: A