Central Idea: The ray will get transmitted through face $AC$ if $i_{AC}<i_C$

Consider the ray diagram is shown below.

A ray of light incident on face $AB$ at an angle $\theta$

$r_1=$ Angle of refraction on face $AB$

$r_2=$ Angle of incidence at face $AC$

For transmission of light through face $AC$

$i_{AC}<i_C$

or $A-r_1<i_C$

or $\sin \left(A-r_1\right)<\sin i_C$

or $\sin \left(A-r_1\right)<\frac{1}{\mu }$

$A-r_1<{\sin }^{-1}\left(\frac{1}{\mu }\right)$

or $\sin r_1>\sin \left[A-{\sin }^{-1}\left(\frac{1}{\mu }\right)\right]$

Now, applying Snell's law at the face $AB$

$1\times \sin \theta =\mu \sin r_1$

or $\sin r_1=\frac{\sin \theta }{\mu }$

$\Rightarrow \frac{\sin \theta }{\mu }>\sin \left[A-{\sin }^{-1}\left(\frac{1}{\mu }\right)\right]$

or $\theta >{\sin }^{-1}\left[\mu \sin \left\{A-{\sin }^{-1}\left(\frac{1}{\mu }\right)\right\}\right]$