The difference between an angle of 45.1234° and another of 45.1235° is just 0.36 arcseconds, or 20 centimeters in position, a negligible quantity.
However, calculating positions and routes is done with trigonometric functions. Around 0° or 180° the differences of one degree are very, very small.
Therefore, astronomers use every digit they have and round only the final result (e.g. 15 significant digits in Excel).
The problem exists even for mariners:
Sine(90°) = 1
Sine(89.5°) = 0.999961923…, rounded to 4 decimal points it returns 1 too.
For calculations with trigonometric functions, 4 decimal points for every single step seem to be insufficient.