For the total internal reflection of a substance, we can express the relationship between the critical angle, [tex] \theta_{c} [/tex], and the indices of refraction as
[tex] n_{1} sin(\theta_{c}) = n_{2} [/tex] (1)
where [tex]n_{1}[/tex] and [tex]n_{2}[/tex] are the indices of refraction. And under polarization, we can express Snell's Law as
[tex]n_{1} sin(\theta_{p}) = n_{2}sin(90-\theta_{p}) [/tex]
Rearranging this we have
[tex]n_{1} sin(\theta_{p}) = n_{2}cos(\theta_{p})[/tex]
[tex] \frac{sin(\theta_{p}}{cos(\theta_{p})} = \frac{n_{2}}{n_{1}} [/tex]
[tex] tan(\theta_{p}) = \frac{n_{2}}{n_{1}} [/tex] (2)
Substituting equations (1) into (2), we have
[tex] tan(\theta_{p}) = \frac{n_{1} sin(\theta_{c})}{n_{1}} [/tex]
[tex] tan(\theta_{p}) = sin(\theta_{c}) [/tex]
Substituting the value of the critical angle, we have
[tex] tan(\theta_{p}) = sin(42) [/tex]
[tex] \theta_{p} = tan^{-1}(sin42) = 33.79 [/tex]
Answer: 33.79°
