I recall a few of the steps made assumptions on our ability to calculate. I think, for example, they narrowed down the set of all vector spaces to just those spaces that were differentiable. I may be mis-remembering the precise detail but it was something along those lines, and this was just one of a few instances of this kind of "throw away cases that we are unable to calculate" along the path. In some cases the narrowing was justified but in a few the instructor admitted that the entire reason we were excluding possible sets of solutions was because they would otherwise make the next steps impossible.
I think you're confusing what an exact solution is supposed to be with your own approximation of the exact solution. In your own example, cos(123) would represent a closed form solution to the problem. That solution doesn't cease to be exact if you decide to express it as a finite power series.
It's not that closed form answers are required by the insistence of anyone, I just thought it's just of purely mathematical interest of what kinds of problems there are, like problems placed in P vs in NP.