Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p><strong>171.</strong> The least positive value of \(x\) satisfying the equation \(\dfrac{\sin x}{\cos 3x} + \dfrac{\sin 3x}{\cos 9x} + \dfrac{\sin 9x}{\cos 27x} = 0\) is:</p>
<p>\(\dfrac{\pi}{26}\)</p>
<p>\(\dfrac{\pi}{27}\)</p>
<p>\(\dfrac{\pi}{9}\)</p>
<p>\(\dfrac{\pi}{3}\)</p>
Step-by-Step Solution
Key Concept: Recognize that each term has the form sin(θ)/cos(3θ), which can be rewritten using the identity sin(θ) = sin(3θ - 2θ) to create a telescoping series structure.
<p><strong>Step 1:</strong> Rewrite using the identity sin(θ) = sin(3θ - 2θ). Notice that:</p><p>sin(θ)/cos(3θ) = [sin(3θ - 2θ)]/cos(3θ) = [sin(3θ)cos(2θ) - cos(3θ)sin(2θ)]/cos(3θ)</p><p>= sin(3θ)·cot(3θ)·cos(2θ) - sin(2θ)</p><p><strong>Step 2:</strong> Alternatively, use the key observation: sin(θ)/cos(3θ) = [sin(3θ) - sin(3θ - θ)]/cos(3θ)·(1/2sin(θ)) suggests telescoping.</p><p><strong>Step 3:</strong> More directly: multiply through and recognize that</p><p>sin(x)/cos(3x) + sin(3x)/cos(9x) + sin(9x)/cos(27x) telescopes to give: tan(27x) - tan(x) = 0</p><p><strong>Step 4:</strong> This requires tan(27x) = tan(x), which means 27x - x = nπ, so 26x = nπ</p><p>Therefore x = nπ/26</p><p><strong>Step 5:</strong> The least positive value occurs at n = 1: x = π/26</p><p>∴ Answer: B</p>
Correct Answer: B