Answer
$cos ^{-1}(\frac{1}{ \sqrt 3} )\approx 55 ^ \circ$
Work Step by Step
Consider a cube with side length $1$ in which one diagonal goes from $(0,0,0) $ to $(1,1,1)$. An edge goes from $(0,0,0) $ to $(0,0,1)$ .
$ \theta = cos ^{-1}(\frac{\lt 1,1,1 \gt \cdot \lt 0,0,1 \gt}{\sqrt {1^2+1^2+1^2}\sqrt {0^2+0^2+0^2}})$
$\theta = cos ^{-1}(\frac{1}{ \sqrt 3} )\approx 55 ^ \circ$
