You can get an outer bonds pair by feeding in an Interval. For example, to get bounds on the Mathematica computation of Sin[1.] (which is a machine double value for the input), could do:

Sin[Interval[1.]]

(* Out[462]= Interval[{0.8414709848078963, 0.8414709848078967}] *)

This also works for arbitrary precision.

Sin[Interval[N[2, 20]]]

(* Out[463]= Interval[{0.90929742682568169537, 0.90929742682568169542}] *)

Under the hood, the Interval construct with a single numeric argument is subtracting and adding an ULP, and Sin understands Interval inputs, so this is computing bounds on the sine for the given interval.

I realize this does not answer the question that was asked. But it does get at the underlying need. Offhand I am not familiar with the details of how the argument to the trigs gets normalized (by subtracting e.g. multiples of Pi/2). Or what method is used from there, though I would guess it is based on a series expansion at the origin or perhaps Pi/2 or other convenient value. Or maybe there are better methods, I'm really not sure.