Hi, good question,
Because its a rotation on the Bloch sphere and not a rotation in the plane. Qubit states are not vectors in the plane. Your final matrix (cos phi sin phi etc ... ) is the rotation matrix of angle phi in the plane. On the Bloch sphere there is this factor 1/2 which comes from the way we parametrize states. Therefore the matrix with alpha/2 is indeed a rotation of angle alpha on the Bloch sphere for quantum qubit states.
I hope this clarifies
Nicolas Macris
