Overlapping diffraction orders Steps:

• Apply the grating equation d sin θ = m λ to both wavelengths for coincidence at same θ.
• Verify order pairing: 3 × 436 nm = 1308 nm and 2 × 654 nm = 1308 nm, so third order of 436 nm overlaps second order of 654 nm.
• Determine spacing: grating has 5.00 × 10⁴ lines/m, so d = 1 / 5.00 × 10⁴ = 2.00 × 10^{-5} m = 20,000 nm.
• Compute sin θ = 1308 / 20,000 = 0.0654; θ = arcsin(0.0654) ≈ 4.0°. Why C is correct:
• 4.0° satisfies d sin θ = 1308 nm from the grating equation for overlapping orders. Why the others are wrong:
• A: 1.9° implies sin θ ≈ 0.033, requiring d ≈ 40 μm (twice actual spacing).
• B: 2.5° matches first-order separation for d = 5 μm (ignores given line density).
• D: 7.8° implies sin θ ≈ 0.136, requiring d ≈ 10 μm (half actual spacing).

Final answer: C