We study ideal cotorsion pairs associated to almost exact structures in extension closed subcategories of triangulated categories. This approach allows us extend the recent approximation theory developed by Fu, Herzog et al. for categories above mentioned context. In last part paper we apply order projective classes (in particular localization or smashing subcategories) compactly generated

Recall that for a triangulated category T , a Bousfield localization is an exact functor L : T → T which is coaugmented (there is a natural transformation Id → L; sometimes L is referred to as a pointed endofunctor) and idempotent (there is a natural isomorphism Lη = ηL : L → LL). The kernel ker(L) is the collection of objects X such that LX = 0. If T is closed under coproducts, it’s a localizi...