Here is a screenshot of a problem from my textbook
My book says
$$S₁= 2t (\tan(\theta_{2})\cos(\theta_{1}))$$
I know the distance for $\overline{AC}$ is
$$2t (\tan(\theta_{2})) $$
but where did $\cos(\theta_1) $ come from?
How is $S_{1}=$ distance between $\overline{AC}$ and $(\cos(\theta_{1}))$?
Please see the diagram.
From $\triangle ADB$, we can calculate that $DB = t\tan\theta_2$. Now $AC = 2DB = 2t\tan\theta_2$.
Denote point $X$ such that $\triangle AXC$ is a right triangle, with the right angle being at $X$. $\angle ACX = \theta_1$, by the theorem of corresponding angles. Since $AC$ is the hypotenuse of this triangle, and $CX$ is the adjacent side to the angle $ACX$, $s_1 = AC\cos\theta_1 = 2t\tan\theta_2\cos\theta_1$, as required.
Just to add on, the comment on the question bringing up unstated assumptions is a very fair one. But this problem seems to be one on the classical theory of refraction and partial/total internal reflection in the topic of optics (in physics), so the ray diagram and its inherent assumptions are fairly obvious. Nevertheless, it would be good for the asker to confirm this.