Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>If \(0 < x < \pi\) and \(\cos x + \sin x = \dfrac{1}{2}\), then \(\tan x\) is</p>
<p>\(\dfrac{(1-\sqrt{7})}{4}\)</p>
<p>\(\dfrac{(4-\sqrt{7})}{3}\)</p>
<p>\(-\dfrac{(4+\sqrt{7})}{3}\)</p>
<p>\(\dfrac{(1+\sqrt{7})}{4}\)</p>
Step-by-Step Solution
Key Concept: Use the tangent addition formula tan(A + B) = (tan A + tan B)/(1 - tan A tan B) and recognize that when the denominator equals zero, the sum equals π/2 (or an odd multiple thereof).
<p><strong>Step 1:</strong> Let α = tan⁻¹(a) and β = tan⁻¹(b), so tan α = a and tan β = b, where 0 < α, β < π/2.</p><p><strong>Step 2:</strong> Apply the tangent addition formula: tan(α + β) = (a + b)/(1 - ab).</p><p><strong>Step 3:</strong> Since ab > 1 and a, b > 0, the denominator (1 - ab) < 0, making tan(α + β) undefined.</p><p><strong>Step 4:</strong> When tan(α + β) is undefined and α + β ∈ (0, π), we have α + β = π/2.</p><p><strong>Step 5:</strong> Therefore: tan⁻¹(a) + tan⁻¹(b) = π/2.</p><p>∴ Answer: C</p>
Correct Answer: C