<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 in restricted ranges: sin⁻¹ returns values in [-π/2, π/2] and cos⁻¹ returns values in [0, π]. Since 10 radians lies outside these ranges, we must use the periodicity and symmetry properties to reduce the argument to the correct range.
<p><strong>Step 1:</strong> Find x = sin⁻¹(sin 10).</p><p>Since 10 radians > π/2, first reduce 10 to the range [-π/2, π/2].</p><p>Note that 10 = 3π + (10 - 3π) where 3π ≈ 9.42, so 10 - 3π ≈ 0.58</p><p>Also, 10 = 4π - (4π - 10) where 4π ≈ 12.57, so 4π - 10 ≈ 2.57</p><p>Actually, 3π < 10 < 4π. We have sin(10) = sin(10 - 3π) since sin has period 2π, but 10 - 3π ≈ 0.58 ∈ (0, π/2).</p><p>Therefore: <strong>x = 10 - 3π</strong></p><p><strong>Step 2:</strong> Find y = cos⁻¹(cos 10).</p><p>Since 10 radians is outside [0, π], we need to reduce it. Since 3π < 10 < 4π, we use: cos(10) = cos(4π - 10).</p><p>Now 4π - 10 ≈ 12.57 - 10 = 2.57 ∈ (0, π).</p><p>Therefore: <strong>y = 4π - 10</strong></p><p><strong>Step 3:</strong> Calculate y - x.</p><p>y - x = (4π - 10) - (10 - 3π) = 4π - 10 - 10 + 3π = 7π - 20</p><p>∴ Answer: A (which should be 7π - 20)</p>
Correct Answer: A