A ray of light entering from air into a denser medium of refractive index $\frac{4}{3},$as shown in figure. The light ray suffers total internal reflection at the  adjacent surface as shown. The maximum value of angle $\mathrm{\theta }$ should be equal to :

At maximum angle θ ray at point B goes in gazing emergence, at all less values of θ, TIR occurs. At point B

$\begin{array}{l}\frac{4}{3}\times \sin {\theta }^{\prime \prime }=1\times \sin {90}^{\text{〔ำ }}\\ {\theta }^{\prime \prime }={\sin }^{-1}\left(\frac{3}{4}\right)\\ {\theta }^{\mathrm{\prime }}=\left(\frac{\pi }{2}-{\theta }^{\prime \prime }\right)\end{array}$

At point A

$\begin{array}{l}1\times \sin \theta =\frac{4}{3}\times \sin {\theta }^{\mathrm{\prime }}\\ \sin \theta =\frac{4}{3}\times \sin \left(\frac{\pi }{2}-{\theta }^{\prime \prime }\right)\\ \sin \theta =\frac{4}{3}\cos \left[{\cos }^{-1}\frac{\sqrt{7}}{4}\right]\end{array}$

$\begin{array}{r}\sin \theta =\frac{4}{3}\times \frac{\sqrt{7}}{4}\\ \theta ={\sin }^{-1}\left(\frac{\sqrt{7}}{3}\right)\end{array}$