Trigonometry & Inverse Trigonometry
Inverse trigonometric equations
Grade None
Question:
<p>If \(\cos^{-1}\!\left(\dfrac{2}{3x}\right) + \cos^{-1}\!\left(\dfrac{3}{4x}\right) = \dfrac{\pi}{2}\) \(\left(x > \dfrac{3}{4}\right)\), then \(x\) is equal to:</p>
<p>(a) \(\dfrac{\sqrt{146}}{12}\)</p>
<p>(b) \(\dfrac{\sqrt{145}}{11}\)</p>
<p>(c) \(\dfrac{\sqrt{145}}{10}\)</p>
<p>(d) \(\dfrac{\sqrt{145}}{12}\)</p>
Step-by-Step Solution
Key Concept: Use the complementary angle property: if cos⁻¹(a) + cos⁻¹(b) = π/2, then cos⁻¹(a) = sin⁻¹(b), which means a = √(1-b²). This converts the equation into an algebraic relation.
<p><strong>Step 1:</strong> Use the complementary angle property: cos⁻¹(a) + cos⁻¹(b) = π/2 implies cos⁻¹(a) = sin⁻¹(b).</p><p>Therefore: cos⁻¹(2/3x) = sin⁻¹(3/4x)</p><p><strong>Step 2:</strong> If cos⁻¹(2/3x) = sin⁻¹(3/4x), then 2/3x = √(1 - (3/4x)²)</p><p><strong>Step 3:</strong> Square both sides: (2/3x)² = 1 - (3/4x)²</p><p>4/9x² = 1 - 9/16x²</p><p><strong>Step 4:</strong> Combine fractions: 4/9x² + 9/16x² = 1</p><p>(64 + 81)/(144x²) = 1</p><p>145/(144x²) = 1</p><p>x² = 145/144</p><p><strong>Step 5:</strong> x = √(145/144) = √145/12 (taking positive root since x > 3/4)</p><p>∴ Answer: D</p>
Correct Answer: D