Question:

<p><strong>Step 1:</strong> Convert the interval bounds to standard form.</p><p>199π/2 = 99.5π and 100π = 100π</p><p>This represents angles from 99.5π to 100π, spanning 0.5π radians.</p><p><strong>Step 2:</strong> Determine the quadrant. Since 99π = 49(2π) + π, we have 99.5π in the 2nd quadrant (where sin θ > 0, cos θ < 0) and 100π = 50(2π) is on the positive x-axis.</p><p><strong>Step 3:</strong> Analyze α = sin θ|sin θ| and β = cos θ|cos θ|.</p><p>For θ ∈ [199π/2, 100π]: sin θ ≥ 0 throughout, so α = sin²θ ≥ 0</p><p>For θ ∈ [199π/2, 100π]: cos θ ≤ 0 throughout (moving from 2nd quadrant toward negative x-axis, then to origin), so β = -cos²θ ≤ 0</p><p><strong>Step 4:</strong> Establish the relationship.</p><p>Since α = sin²θ ≥ 0 and β = -cos²θ ≤ 0, we have α + β = sin²θ - cos²θ = -cos(2θ)</p><p>The key relations are: α ≥ 0, β ≤ 0, and α + β ≤ 1</p><p>∴ Answer: B</p>