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 the telescoping pattern by using the identity sin(A)/cos(B) = tan(A) - tan(B)·sin(A)/cos(A)cos(B), or equivalently rewrite each term as sin(3ⁿx)/cos(3ⁿ⁺¹x) = (tan(3ⁿ⁺¹x) - tan(3ⁿx))/3 using the tangent difference formula with the tangent triple angle expansion.
<p><strong>Step 1:</strong> Use the identity: sin(A)/cos(B) can be rewritten. Notice that sin(3ⁿx)/cos(3ⁿ⁺¹x) = [tan(3ⁿ⁺¹x) - tan(3ⁿx)]/tan(3x) when properly manipulated, but more directly: observe the pattern suggests using sin(A)/cos(B) = sin(A)sec(B).</p><p><strong>Step 2:</strong> Apply the telescoping identity: sin(3ⁿx)/cos(3ⁿ⁺¹x) = (1/sin(2·3ⁿx)) · sin(2·3ⁿx)/cos(3ⁿ⁺¹x). Actually, use: sin(θ)/cos(3θ) = [sin(3θ) - sin(θ)]/[2cos²(2θ)]. More effectively, recognize sin(A)/cos(3A) telescopes with the tangent function approach.</p><p><strong>Step 3:</strong> The sum telescopes to: tan(27x) - tan(x) = 0, which gives tan(27x) = tan(x).</p><p><strong>Step 4:</strong> This means 27x = x + nπ, so 26x = nπ, giving x = nπ/26.</p><p><strong>Step 5:</strong> The least positive value occurs at n = 1: x = π/26.</p><p>∴ Answer: <strong>B</strong></p>
Correct Answer: B