A ray of light enters from a rarer to a denser medium. The angle of incidence is Then the reflected and refracted rays are mutually perpendicular to each other. The critical angle for the pair of media is

From law of reflection, \(\angle \mathrm{i}=\angle \mathrm{r}\) and
\(
\frac{\sin \mathrm{r}^{\prime}}{\sin \mathrm{i}}=\frac{\mu_{\mathrm{d}}}{\mu_{\mathrm{r}}}
\)
From the figure



\(
r^{\prime}=\left(90^{\circ}-i\right)
\)
From Eq. (ii) \(\frac{\sin \left(90^{\circ}-i\right)}{\sin i}=\frac{\mu_{\mathrm{d}}}{\mu_{\mathrm{r}}}\) or \(\quad \frac{\cos \mathrm{i}}{\sin \mathrm{i}}=\frac{\mu_{\mathrm{d}}}{\mu_{\mathrm{r}}} \Rightarrow \cot \mathrm{i}=\frac{\mu_{\mathrm{d}}}{\mu_{\mathrm{r}}}\)
But \(\frac{\mu_{\mathrm{d}}}{\mu_{\mathrm{r}}}=\sin \mathrm{C}\) (where is critical angle) \(\therefore \quad \cot \mathrm{i}=\sin \mathrm{C} \Rightarrow \mathrm{C}=\sin ^{-1}(\cot \mathrm{i})\)