Refractive-index uncertainty from Snell's law (${\sigma }_n^2/n^2={\cot }^{2}i{\sigma }_i^2+{\cot }^{2}r{\sigma }_r^2-2\cot i\cot r\mathrm{Cov}(i,r)$)

For air approximated by index one, $n=\sin i/\sin r$, so $dn/n=\cot idi-\cot rdr$. Taking the variance gives the displayed first-order error formula, with angles and their errors in radians. Independence removes the covariance term; hard error bounds require a different combination rule. Common angular-zero errors can produce covariance and should not automatically be added in quadrature.