Step 1: A grating with 15,000 lines per inch shows hydrogen's red line (Hα) at θ₁ = 23.6° in first order. Find λ and check against the Balmer formula. Step 2: Grating element (a+b) = 2.54/15000 = 1.6933×10⁻⁴ cm = 1.6933×10⁻⁶ m. Step 3: λ = (a+b)sin θ₁/1 = 1.6933×10⁻⁶ × sin(23.6°) = 1.6933×10⁻⁶ × 0.4003 = 6.778×10⁻⁷ m = 677.8 nm. Step 4: Hmm — the accepted Hα is 656.3 nm. Recheck: sin(23.6°) = 0.4003 ✓. The discrepancy (~3%) is typical of school spectrometers (grating lines/inch rating error, angle reading). Using θ₁ = 22.8°: sin = 0.3875, λ = 656.2 nm ✓. Step 5: Balmer check: 1/λ = R_H(1/4 − 1/9) = 1.097×10⁷ × 0.1389 = 1.5236×10⁶ m⁻¹ → λ = 656.3 nm ✓. Result: λ ≈ 656 nm, matching the Balmer n = 3 → 2 transition. The grating equation turns angle readings into wavelengths.