(*****************************************************************)
(** * 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_eq_0_0 (x:R) :
cos x = 0 ->  exists k : Z, x = IZR k * PI + PI / 2.