Where does $\cos(pi/2)$ and $\sin(pi/2)$ come from in quaternion rotation? Can you provide a simple unit quaternion rotation example?

Question

I have seen many methods of rotation such as $p' = qpq^{-1}$ and $q = \cos (pi/2),v \sin (pi/2)$ , but I became slightly confused by how a unit quaternion is performed around a vector line.