Answer
$$\theta=\arccos \left( \frac{1}{\sqrt{3}}\right).$$
Work Step by Step
The diagonal of a cube can be represented by the vector $u=(1,1,1)$ and one
of its edges can be represented by $v=(1,0,0)$. Then, the angle between $u$ and $v$ is given by
$$\theta=\arccos \left( \frac{u\cdot v}{\|u\|\|v\|}\right)=\arccos \left( \frac{1}{\sqrt{3}}\right).$$
