<p><strong>142.</strong> If \(x=\sin^{-1}(\sin 10)\) and \(y=\cos^{-1}(\cos 10)\), then \(y-x\) is equal to:</p>
Step-by-Step Solution
Key Concept: The inverse trigonometric functions return values within their principal ranges: sin⁻¹ returns [-π/2, π/2] and cos⁻¹ returns [0, π]. Since 10 radians lies outside these ranges, we must reduce it using periodicity and symmetry properties before applying the inverse functions.
<p><strong>Step 1: Find x = sin⁻¹(sin 10)</strong></p><p>Since 10 radians and 3π ≈ 9.42, we have 10 ≈ 3π + 0.58, which lies between 3π and 7π/2.</p><p>Note: π ≈ 3.14, so 2π ≈ 6.28, 3π ≈ 9.42, and 7π/2 ≈ 11.0</p><p>For angles in (3π, 7π/2): sin(10) = sin(3π + (10 - 3π)) = -sin(10 - 3π)</p><p>Since 10 - 3π ≈ 0.58 ∈ (0, π/2), we have sin(10) = -sin(10 - 3π)</p><p>Therefore: x = sin⁻¹(-sin(10 - 3π)) = -(10 - 3π) = 3π - 10</p><p><strong>Step 2: Find y = cos⁻¹(cos 10)</strong></p><p>For angles in (3π, 7π/2): cos(10) = cos(2π·2 - (10 - 4π)) = cos(10 - 4π)</p><p>Since 10 - 4π ≈ 10 - 12.56 = -2.56, we compute: 10 = 3π + (10 - 3π), and cos(10) = -cos(10 - 3π)</p><p>More directly: cos(10) = cos(2π - (2π - 10)) = cos(2π - 10) since cos is even in the principal range</p><p>Actually, 10 - 3π ≈ 0.58, so: cos(10) = cos(4π - 10) where 4π - 10 ≈ 2.57 ∈ (0, π)</p><p>Therefore: y = cos⁻¹(cos(4π - 10)) = 4π - 10</p><p><strong>Step 3: Calculate y - x</strong></p><p>y - x = (4π - 10) - (3π - 10) = 4π - 10 - 3π + 10 = π</p><p>∴ Answer: A (y - x = π)</p>
Correct Answer: A