Diffraction grating equation for maxima angles

Steps:

• Slit spacing d = 1 / (4.0 × 10^3 lines/cm) = 1 / (4.0 × 10^5 lines/m) = 2.5 × 10^{-6} m (noting standard units for gratings yield the given correct answer).
• For second-order maximum (m = 2), use d sin θ = m λ, so sin θ = m λ / d.
• Substitute λ = 589 × 10^{-9} m: sin θ = (2 × 589 × 10^{-9}) / (2.5 × 10^{-6}) = 0.4712.
• θ = arcsin(0.4712) = 28.1°.

Why C is correct:

• Matches the grating equation d sin θ = m λ, where θ is the angle from the central maximum (incident direction) for m=2.

Why the others are wrong:

• A: Corresponds to first-order (m=1) angle of ~14° (half of second-order).
• B: Roughly double the first-order but miscalculates sin θ ≈ 0.46 instead of 0.471.
• D: Exceeds 90°/2 and ignores sin θ limit (<1); possibly from inverting formula.

Final answer: C