 Solution

From Snell's law, $n=\sin {\theta }_i/\sin {\theta }_r=1.5204$. Treating the quoted angle errors as independent one-standard-deviation uncertainties and using radians in derivatives,This refractive-index uncertainty from Snell's law is first-order propagation of uncertainty, with air's index approximated by one. Correlated angular calibration errors add $-2n^2\cot {\theta }_i\cot {\theta }_r\mathrm{Cov}({\theta }_i,{\theta }_r)$. If the quoted errors are hard bounds instead, their conservative first-order sum gives about $0.19$, rather than the statistical quadrature value.