v · w = |v| |w| cosθ
v × w = |v| |w| sinθ n
n is the unit vector perpendicular to both v and w. You can find n from the cross product given, and divide to get tanθ = sinθ / cosθ.
Divide the 2nd equation by the 1st, to get sinθ / cosθ on the right.
v × w / v · w = (|v| |w| sinθ n) / (|v| |w| cosθ)
〈2, 5, 4〉 / 2 = (sinθ / cosθ) n
〈1, 5/2, 2〉 = tanθ n
Since n is a unit vector, tanθ equals the magnitude of the vector on the left.