How to find the angle between a diagonal of a cube and one of its faces?
I am having trouble picturing how this looks.
The diagonal means a spatial diagonal, a line segment connecting two vertices that do not lie on the same face.
The angle in question is formed by this line and any face of the cube. By symmetry, it does not matter what face is used: all these angles are equal.
If by "diagonal" you mean a diameter (not contained in a face), then look at the right triangle formed by the diagonal, the projection of the diagonal onto the face, and the edge of the cube connecting them. This triangle has sides $c\sqrt{3}$, $c\sqrt{2}$, and $c$, respectively (here $c$ is the length of a side of the cube). The angle in question is between the first two sides, so it is $\sin^{-1}\sqrt{\frac{1}{3}}$, which is the same as $\cos^{-1}\sqrt{\frac{2}{3}}$ and $\tan^{-1}\sqrt{\frac{1}{2}}$.