(*****************************************************************)
(** * Using series definitions of cos and sin                    *)
(*****************************************************************)
Definition sin_ub (a:R) : R := sin_approx a 4.
Definition cos_lb (a:R) : R := cos_approx a 3.
Definition cos_ub (a:R) : R := cos_approx a 4.

Lemma cos_ge_0 : forall x:R, - (PI / 2) <= x -> x <= PI / 2 -> 0 <= cos x.