Ok. Just idea. I don't know where you are with math so ask if something is unclear.

You know where robot's hand is and the current angle of it in comparing with something (x,y,z,alpha). And you know where robot should go (x1,y1,z1,alpha1).

What you need to do is just to assume that robot is rotated in the right direction and the hand is in (x1,y1,z1,alpha1). Then you can calculate position of its other end. And then is easy to determine if remaining two segments can or cannot make the requested position. In the most of cases you'll have only one solution.

I suggest you to do all calculations in the plane where all segments are. You need only the angles between the segments so you won't need to turn back to 3d.