<p>Find the value of \(\cos\left(\cos^{-1}\frac{\pi}{3} + \sin^{-1}\sin\frac{2\pi}{3}\right)\).</p>
Step-by-Step Solution
Key Concept: We must carefully evaluate the inverse trigonometric functions using their principal ranges: cos⁻¹ has range [0, π], and sin⁻¹ has range [-π/2, π/2]. We cannot directly simplify cos⁻¹(π/3) since π/3 ≈ 1.047 > 1, which is outside the domain of cosine inverse.
<p><strong>Step 1: Evaluate sin(2π/3)</strong></p><p>We have sin(2π/3) = sin(π - π/3) = sin(π/3) = √3/2</p><p><strong>Step 2: Evaluate sin⁻¹(sin(2π/3))</strong></p><p>Since sin⁻¹ has range [-π/2, π/2], and 2π/3 is NOT in this range, we use:</p><p>sin⁻¹(sin(2π/3)) = sin⁻¹(√3/2) = π/3</p><p>This is because sin(π/3) = √3/2 and π/3 ∈ [-π/2, π/2].</p><p><strong>Step 3: Recognize the issue with cos⁻¹(π/3)</strong></p><p>The expression cos⁻¹(π/3) is problematic because π/3 ≈ 1.047, which exceeds 1 and is outside the domain [-1, 1] of the inverse cosine function. This suggests the problem statement may contain a typo. If the intended expression was cos⁻¹(√3/2), we proceed as follows:</p><p><strong>Step 4: Assuming the expression is cos⁻¹(√3/2) + sin⁻¹(sin(2π/3))</strong></p><p>cos⁻¹(√3/2) = π/6 (since cos(π/6) = √3/2 and π/6 ∈ [0, π])</p><p>sin⁻¹(sin(2π/3)) = π/3</p><p><strong>Step 5: Add the angles</strong></p><p>cos⁻¹(√3/2) + sin⁻¹(sin(2π/3)) = π/6 + π/3 = π/6 + 2π/6 = 3π/6 = π/2</p><p><strong>Step 6: Evaluate the outer cosine</strong></p><p>cos(π/2) = 0</p><p><strong>∴ Answer: 0</strong></p>
Correct Answer: 0