1. Refraction across a 30^o prism

We want to find the angular displacement θ3 of the ray after having passed through the prism.

We have
θ1 = 90^o - 60^o = 30^o

According Snell's law, we have:
1 sin θ1 = 1.5 sin θ2

Therefore
sin θ2 = (1/1.5) sin 30^o = 0.333
θ2 = 19.50^o

(90 - 19.50) + 30 = 180 - (90 - θ3)
(90 - 19.50) + 30 = 180 - (90 - θ3)
θ3 = 10.53

According Snell's law, we have:

1.5 sin θ3 = 1 sin θ4
1.5 sin(10.53) = sin θ4
sin θ4 = 0.27

θ4 = 15.90^o

2. Brewster Angle

The Brewster angle is the angle of incidence that corresponds to the reflected ray perpendicular to the refracted ray.

We want to find the relationship between θ1 and the two refraction indexex n1 and n2.

We have

θ1 + θ2 = 90^o

According Snell's law, we have:
n1 sin θ1 = n2 sin θ2

Therefore
n1 sin θ1 = n2 sin θ2 = n2 sin (90 - θ1)
= n2 cos θ1

Hence; tan θ1 = n2/n1

tan θ1 = n2/n1