Question:

<p><strong>Step 1:</strong> Simplify the interval bounds. Convert 199π/2 = 99.5π = 99π + π/2. The interval [199π/2, 100π] has length 100π - 199π/2 = π/2, spanning a quarter period.</p><p><strong>Step 2:</strong> Determine position: 199π/2 = 99π + π/2, and since 99 is odd, we're in the second quadrant (at angle π/2 from an odd multiple of π). The interval [199π/2, 100π] covers from the end of the second quadrant through the third and into the fourth quadrant, or equivalently, the interval [π/2, π] in standard position (modulo 2π).</p><p><strong>Step 3:</strong> Analyze signs: In the interval [π/2, π], sin θ ≥ 0 and cos θ ≤ 0, so α = sin θ·|sin θ| = sin²θ ≥ 0 and β = cos θ·|cos θ| = -cos²θ ≤ 0.</p><p><strong>Step 4:</strong> Find ranges: Since sin²θ ∈ [0,1] and cos²θ ∈ [0,1] over this interval, we have α ∈ [0,1] and β ∈ [-1,0].</p><p><strong>Step 5:</strong> Verify the relationship: α + β = sin²θ - cos²θ = -cos(2θ) ∈ [-1,1], and α² + β² = sin⁴θ + cos⁴θ.</p><p>∴ Answer: B</p>