Center on mass trigonometric trick

hey there im doing a lot of center of mass problem most of them with collisions , and most of the solutions I see, have a big part of "trigonometric trciks", like the way u watch the vectors in cm frame and the way in the lab frame. does anyone met this kind of solutions and have some tricks or advices that wil be great.
btw im in classical physics (course)
thx

 

well it pretty hard to explain but ill just say that the proble,m is of two balls striking each other and in the first graph is the cm graph the other one is the lab frame.
I hope it will help/ and there are a lot of problems like that