Let's call the bottom left vertex of the cube C.
AB is the diagonal distance between the opposing vertices of the cube A and B. This length is the hypotenuse of the right triangle whose legs are one edge of the cube (the line segment AC) and the diagonal on the bottom face of the cube (the line segment BC).
BC is itself a hypotenuse of a right triangle whose legs are both adjoining edges of the cube.
By the Pythagorean theorem, BC will have length that satisfies
[tex]a^2+a^2=(BC)^2\implies BC=\sqrt{2a^2}=a\sqrt2[/tex]
And again by the Pythagorean theorem, AB will have length that satisfies
[tex](AC)^2+(BC)^2=(AB)^2\iff a^2+(\sqrt{2a^2})^2=(AB)^2\implies AB=\sqrt{3a^2}=a\sqrt3[/tex]
The angle [tex]\theta[/tex] can be written in terms of any of the six trigonometric ratios. One would be
[tex]\tan\theta=\dfrac{AC}{BC}=\dfrac a{a\sqrt2}=\dfrac1{\sqrt2}[/tex]
From this ratio we can determine that [tex]\theta=\dfrac\pi4[/tex].
