<p>The value of \(\sin\left(\cos^{-1}\dfrac{1}{2} + \sin^{-1}\dfrac{\sqrt{3}}{2}\right)\) is ______.</p>
Step-by-Step Solution
Key Concept: Convert inverse trigonometric values to standard angles, then add them to find the argument, and finally apply sine to get the result. Recognize that cos⁻¹(1/2) = π/3 and sin⁻¹(√3/2) = π/3.
<p><strong>Step 1:</strong> Identify the inverse trigonometric values.</p><p>Let α = cos⁻¹(1/2), then cos(α) = 1/2 and α ∈ [0, π]</p><p>This gives α = π/3</p><p><strong>Step 2:</strong> Identify the second inverse value.</p><p>Let β = sin⁻¹(√3/2), then sin(β) = √3/2 and β ∈ [-π/2, π/2]</p><p>This gives β = π/3</p><p><strong>Step 3:</strong> Add the angles.</p><p>cos⁻¹(1/2) + sin⁻¹(√3/2) = π/3 + π/3 = 2π/3</p><p><strong>Step 4:</strong> Apply sine to the sum.</p><p>sin(2π/3) = sin(π - π/3) = sin(π/3) = √3/2</p><p>∴ Answer: √3/2</p><p><em>Note: If the expected answer is 1, the question may have different values. With given values, the answer is <strong>√3/2</strong>. Please verify the problem statement as standard evaluation yields this result.</em></p>
Correct Answer: 1