Trigonometry & Inverse Trigonometry
Inverse Trigonometric Equations
Grade 12
Question:
<p>Given <br>\(\cos^{-1}\left(\dfrac{2}{3x}\right) + \cos^{-1}\left(\dfrac{3}{4x}\right) = \dfrac{\pi}{2}\)<br>Find the value of \(x\).</p>
<p>\(x = \pm \dfrac{\sqrt{145}}{12}\)</p>
<p>\(x = \pm \dfrac{\sqrt{145}}{10}\)</p>
<p>\(x = \pm \dfrac{\sqrt{135}}{12}\)</p>
<p>\(x = \pm \dfrac{\sqrt{145}}{6}\)</p>
Step-by-Step Solution
Key Concept: When cos⁻¹(a) + cos⁻¹(b) = π/2, we have the complementary angle relationship: cos⁻¹(a) = sin⁻¹(b), which means a = √(1-b²). Use this identity to eliminate inverse trigonometric functions.
<p><strong>Step 1:</strong> Use the complementary angle property. If cos⁻¹(a) + cos⁻¹(b) = π/2, then cos⁻¹(a) = π/2 - cos⁻¹(b), which means cos⁻¹(a) = sin⁻¹(b).</p><p><strong>Step 2:</strong> Therefore: a = sin(sin⁻¹(b)) = b, or equivalently, a = √(1 - b²) in the complementary form. Here we use: (2/3x)² + (3/4x)² = 1</p><p><strong>Step 3:</strong> Substitute a = 2/(3x) and b = 3/(4x):</p><p>4/(9x²) + 9/(16x²) = 1</p><p><strong>Step 4:</strong> Find common denominator (144x²):</p><p>64/(144x²) + 81/(144x²) = 1</p><p>145/(144x²) = 1</p><p><strong>Step 5:</strong> Solve for x²:</p><p>145 = 144x²</p><p>x² = 145/144</p><p>x = √(145)/12 (taking positive root since arguments must be positive)</p><p><strong>Verification:</strong> Check that both arguments stay in domain [0,1]: 2/(3x) ≈ 0.656 ✓ and 3/(4x) ≈ 0.755 ✓</p><p>∴ Answer: x = √145/12</p>
Correct Answer: A