= Refractive-index uncertainty from Snell's law
{title2=$\sigma_n^2/n^2=\cot^2 i\,\sigma_i^2+\cot^2 r\,\sigma_r^2-2\cot i\cot r\,\operatorname{Cov}(i,r)$}
For air approximated by index one, $n=\sin i/\sin r$, so $dn/n=\cot i\,di-\cot r\,dr$. 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.
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