We create two vectors: one from the last joint position to the endpoint of the last bone (the effector) and another from the same last joint to the target we are trying to reach. Now we can compute the angle between these two vectors by computing the dot product of their unit vectors. Notice that we have the angle between two vectors, but we need to make sure we rotate in the appropriate direction to make the shortest path possible. Again, we can use our AI knowledge by performing a cross product between the two vectors to obtain a third, perpendicular vector. Then, we use the sign of the Z coordinate of the newly computed vector. This sign determines the direction at which we must rotate, and the arc-cosine of the dot product will give you the angle.