Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade 12
Question:
<p>The given equation \(3\tan^{-1}(2-\sqrt{3}) - \tan^{-1}\left(\dfrac{1}{x}\right) = \tan^{-1}\left(\dfrac{1}{2}\right)\) is solved. Find the value of \(x\).</p>
Step-by-Step Solution
Key Concept: Recognize that 2-√3 = tan(15°) by using the tangent subtraction formula, then apply the addition formula for inverse tangent functions strategically to simplify 3tan⁻¹(2-√3).
<p><strong>Step 1:</strong> Recognize that <strong>2 - √3 = tan(15°) = tan(π/12)</strong></p><p>Verify: tan(45° - 30°) = (tan 45° - tan 30°)/(1 + tan 45° tan 30°) = (1 - 1/√3)/(1 + 1/√3) = (√3 - 1)/(√3 + 1) = 2 - √3 ✓</p><p><strong>Step 2:</strong> Therefore: 3tan⁻¹(2 - √3) = 3 · (π/12) = <strong>π/4</strong></p><p><strong>Step 3:</strong> Substitute into the original equation:</p><p>π/4 - tan⁻¹(1/x) = tan⁻¹(1/2)</p><p><strong>Step 4:</strong> Rearrange:</p><p>tan⁻¹(1/x) = π/4 - tan⁻¹(1/2)</p><p><strong>Step 5:</strong> Apply the tangent subtraction formula: tan(A - B) = (tan A - tan B)/(1 + tan A tan B)</p><p>tan[π/4 - tan⁻¹(1/2)] = (1 - 1/2)/(1 + 1·(1/2)) = (1/2)/(3/2) = <strong>1/3</strong></p><p><strong>Step 6:</strong> Therefore:</p><p>tan⁻¹(1/x) = tan⁻¹(1/3)</p><p>∴ <strong>x = 3</strong></p>
Correct Answer: 3