Trigonometry & Inverse Trigonometry
Properties of Triangle
Grade 11
Question:
<p>96. In the △ABC, A > B. If the measures of A and B satisfy the equation \(3\sin x - 4\sin^3 x - k = 0\), \(0 < k < 1\) then the measure of C is</p>
<p>(a) \(\dfrac{\pi}{3}\)</p>
<p>(b) \(\dfrac{\pi}{2}\)</p>
<p>(c) \(\dfrac{2\pi}{3}\)</p>
<p>(d) \(\dfrac{5\pi}{6}\)</p>
Step-by-Step Solution
Key Concept: The equation 3sin x - 4sin³x = k is the triple angle formula sin(3x) = k. Since A > B and both satisfy this equation with the same k value, we use the property that sin(3A) = sin(3B) to find the relationship between A and B.
<p><strong>Step 1:</strong> Recognize that 3sin x - 4sin³x = sin(3x) (triple angle formula). So the equation becomes sin(3x) = k, where 0 < x < π/3.</p><p><strong>Step 2:</strong> Both A and B satisfy sin(3A) = sin(3B) = k with A > B and 0 < A, B < π/3, so 0 < 3A, 3B < π.</p><p><strong>Step 3:</strong> For sin(3A) = sin(3B) in the range (0, π), we have two cases: 3A = 3B (impossible since A ≠ B) or 3A + 3B = π (supplementary angles).</p><p><strong>Step 4:</strong> From 3A + 3B = π, we get A + B = π/3.</p><p><strong>Step 5:</strong> Since A + B = π/3 and A > B > 0, we have A > π/6 and B < π/6, with A + B + C = π giving C = 2π/3.</p><p>∴ Answer: C</p>
Correct Answer: C