we have no idea how to precisely find a new location of the end of an arm in space to be able to add a new point to a line strip.
You do this by performing the matrix math and transformations yourself.
(from comment)
To do this we are suppose to multiply the matrices and get some information out of glGetFloatv
Please don't do this. Especially not if you're supposed to build a pretransformed line strip geometry yourself. OpenGL is not a matrix math library and there's absolutely no benefit to use OpenGL's fixed function pipeline matrix functions. But it has a lot of drawbacks. Better use a real matrix math library.
Your robot arm technically consists of a number of connected segments where each segment is transformed by the composition of transformations of the segments upward in the transformation hierachy.
M_i = M_{i-1} · (R_i · T_i)
where R_i and T_i are the respective rotation and translation of each segment. So for each segment you need the individual transform matrix to retrieve the point of the line segment.
Since you'll place each segment's origin at the tip of the previous segment you'd transform the homogenous point (0,0,0,1) with the segment's transformation matrix, which has the nice property of being just the 4th column of the transformation matrix.
This leaves you with the task of creating the transformation matrix chain. Doing this with OpenGL is tedious. Use a real math library for this. If your tutor insists on you using the OpenGL fixed function pipeline please ask him to show you the reference for the functions in the specicifications of a current OpenGL version (OpenGL-3 and later); he won't find them because all the matrix math functions have been removed entirely from modern OpenGL.
For math libraries I can recommend GLM, Eigen (with the OpenGL extra module) and linmath.h (self advertisement). With each of these libraries building transformation hierachies is simple, because you can create copies of each intermediary matrix without much effort.