A ray of light is incident grazingly on face $AB$ to a right angled prism as shown. It emerges out of face $AC$, as shown, $e=$angle of emergence. Refractive indices of different media have been shown in the diagram. Choose the correct alternatives

By Snell's law,

                    $\frac{{\mu }_2}{{\mu }_1}=\frac{\sin {90}^{\circ }}{\sin r_1}$

or              $\sin r_1=\left(\frac{{\mu }_1}{{\mu }_2}\right)$

Similarly,     $\frac{\sin e}{\sin r_2}={\mu }_2$

or                  $\sin r_2=\frac{\sin e}{{\mu }_2}$

or      $\sin \left({90}^{\circ }-r_1\right)=\frac{\sin e}{{\mu }_2}$

or                 $\cos r_1=\frac{\sin e}{{\mu }_2}$

or        $\sqrt{1-{\sin }^2{r}_1}=\frac{\sin e}{{\mu }_2}$

or      $\sqrt{1-{\left(\frac{{\mu }_1}{{\mu }_2}\right)}^2}=\frac{\sin e}{{\mu }_2}$

$\therefore {\mu }_1^2+{\sin }^{2}e={\mu }_2^2$

If, $\mathrm{e}={90}^0 \mathrm{and} {\mathrm{\mu }}_2=\sqrt{3}$, then ${\mathrm{\mu }}_1=\sqrt{2}$

Now, using,  $\sin r_1=\left(\frac{{\mu }_1}{{\mu }_2}\right)=\frac{\sqrt{2}}{\sqrt{3}}$

$r_1={\sin }^{-1}\left(\sqrt{\frac{2}{3}}\right)$$$