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(** * Using series definitions of cos and sin                    *)
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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.

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Lemma sin_eq_0_1 : forall x:R, (exists k : Z, x = IZR k * PI) -> sin x = 0.