Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>If \(\tan 2x \cdot \tan x = 1\), then \(x\) is</p>
<p>(a) \(\frac{\pi}{3}\)</p>
<p>(b) \((6n \pm 1)\frac{\pi}{6}\)</p>
<p>(c) \((4n \pm 1)\frac{\pi}{6}\)</p>
<p>(d) \((2n \pm 1)\frac{\pi}{6}\)</p>
Step-by-Step Solution
Key Concept: Use the double angle formula tan 2x = 2tan x/(1 - tan² x) and convert the equation into a condition on tan x. The condition tan 2x · tan x = 1 means 2x and x are complementary angles (their tangents multiply to 1).
<p><strong>Step 1:</strong> Start with tan 2x · tan x = 1, which can be rewritten as tan 2x = 1/tan x = cot x.</p><p><strong>Step 2:</strong> Since tan 2x = cot x, we have tan 2x = tan(π/2 - x).</p><p><strong>Step 3:</strong> The general solution for tan A = tan B is A = B + nπ, where n ∈ ℤ. Therefore: 2x = π/2 - x + nπ</p><p><strong>Step 4:</strong> Solve for x: 2x + x = π/2 + nπ → 3x = π/2 + nπ → x = π/6 + nπ/3</p><p><strong>Step 5:</strong> Rewrite nπ/3 as (n/3)π. Since n is any integer, we can write n/3 in the form (6k ± 1)/6 for appropriate integers k. This gives: x = π/6 + nπ/3 = (6n ± 1)π/6</p><p><strong>Step 6:</strong> Verification: When x = π/6: tan(π/3) · tan(π/6) = √3 · (1/√3) = 1 ✓</p><p><strong>∴ Answer:</strong> B</p>
Correct Answer: B