(*****************************************************************)
(** * 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 sin_eq_O_2PI_1 (x:R) :
0 <= x -> x <= 2 * PI ->
x = 0 \/ x = PI \/ x = 2 * PI -> sin x = 0.