Answer
$81.9^{\circ}$.
Work Step by Step
Apply Snell’s law.
$$n_{air}sin \theta_1=n_{glass}sin \theta_2$$
We are told that the angle of reflection is twice the angle of refraction.
We also know, from the law of reflection, that the angle of incidence equals the angle of reflection. Therefore, the angle of incidence is twice the angle of refraction.
$$\theta_1=2\theta_2$$
$$n_{air}sin2\theta_2=n_{glass}sin\theta_2$$
$$(1.00)sin2\theta_2=(1.51)sin\theta_2$$
Apply a “double-angle” trigonometric identity, $ sin2\theta_2= 2 sin\theta_2cos \theta_2$.
$$(1.00) 2 sin\theta_2cos \theta_2=(1.51)sin\theta_2$$
$$cos \theta_2=\frac{1.51}{2}$$
$$\theta_2=40.97^{\circ}$$
Now find the angle of incidence.
$$\theta_1=2\theta_2=81.9^{\circ}$$