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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.

Lemma sin_inj x y : -(PI/2)  <= x <= PI/2 -> -(PI/2) <= y <= PI/2 -> sin x = sin y -> x = y.