This isn't why. You can always use a convenient Taylor series to approximate cosine anything to arbitrary precision. For the three-body problem, small changes to initial conditions diverge into incalculably chaotic behavior.

> We consider “cos 123” an exact solution even though to numerically calculate it requires a power series approximation.

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.