Question

A ray in air is incident on a glass plate whose index of refraction is 1.42. The angle of refraction is one half the angle of reflections. the angle of refraction is closest to:

Answer 1

Angle of incidence = theta1
Angle of refraction = theta2
Angle of reflection = theta3

Snell's law, "Law of refraction" states:
n1 * sin(theta1) = n2 * sin(theta2)

Angle of reflection is simply the angle of incidence:
theta1 = theta3

So,
Solve for theta2 in terms of theta1:
theta2 = arcsin(n1 * sin(theta1) / n2)

It is stated that theta2 = theta3 / 2
Restated:
theta3 = 2 * theta2

It is stated that n2 = 1.42
It is known that n1 = 1 (air)

Since we also know that theta3 = theta1

Now, we can write;

theta2 = arcsin(1 * sin(2 * theta2) / 1.42)
sin(theta2) = sin (2 * theta2) / 1.42
1.42 * sin(theta2) = sin(2 * theta2)

Using the double-angle forumla, rewrite as:
1.42 * sin(theta2) = 2 * sin(theta2) * cos(theta2)

Divide by sin(theta2)
1.42 = 2 * cos(theta2)

Solve for theta2
theta2 = arccos(1.42/2)
theta2 = arccos(0.71)
theta2 = 44.765 degrees

The angle of refraction is closest to 44.77 degrees

Answer 2