Let Σ be a bilinear control system on R2 whose matrices generate the Lie algebra sl(2) of the Lie group Sl(2) : the group of order two real matrices with determinant 1. In this work we focus on the extremals of a quadratic cost optimal problem for the angle system PΣ defined by the projection of Σ onto the real projective line P1. It has been proved in [2] that through the Cartan-Killing form the cotangent bundle of P1 can be identified with a cone C in sl(2). Via the Pontryagin Maximum Principle, we explicitly show the extremals by using the mentioned identification and the special form of the trajectories associated with the lifting of vector fields on PΣ. We analyze both: the controllable case and when the system bf P Σ give rise to control sets. Some examples are shown.