We discuss a simple approach to measure the weak mixing angles φs and φ d of the Bs and B d systems in the presence of New Physics. We present a new expression that allows one to measure directly φ NP d,s if New Physics contributes significantly to the mixing only. We apply the method to specific penguin-mediated B → P P , B → P V and B → V V modes. We provide a very stringent and simple bound on the direct CP asymmetries of all these modes, the violation of which is a signal of New Physics in decay. Within the same theoretical framework, an updated prediction for the branching ratio of Bs → K 0 * K 0 * is presented, which can be compared with a recent LHCb analysis.In our quest for New Physics (NP) in flavour processes, the weak angles φ M involved in the mixing of B d (φ d ) and B s (φ s ) mesons have been studied with a great attention recently, since they have shown interesting discrepancies with respect to the Standard Model [1,2]. On the one hand, there is a tension in the SM fit between the measurement of φ d and the branching ratio of B → τ ν (see also Refs. [3,4] where the discrepancies with respect to other inputs are discussed, e.g., inclusive or exclusive semileptonic decays, or mixing-dependent observables). On the other hand, the most recent angular analyses of B s → J/Ψφ by CDF, DØ and LHCb [5][6][7] seem to reduce substantially the space for New Physics in φ s . Finally, there still remains an important discrepancy with the Standard Model through the dimuon asymmetry measured at DØ [8,9].In this context, it is useful to devise alternative methods to extract the mixing angle of both B d (φ d ) and B s (φ s ) systems from experiment. In this paper we will take the point of view of assuming that New Physics contributes significantly only to the mixing phases whereas the formalism of the CKM matrix can still be used to analyse decay amplitudes. Under this assumption we present a clean method to extract the NP contribution to the weak mixing phase φ d,s from selected B d,s → P P, V P and V V decays.Some time ago [10], we proposed a test on the value of sin φ s in the SM by measuring certain longitudinal branching ratios and CP asymmetries of B d,s mesons decaying into vectors, for modes mediated by penguin diagrams -we focused on the potential of the B s →K 0 * K 0 * decay at that time. The method has advantages both from the theoretical and experimental points of view. On the theoretical side, it reduces the required theoretical input mainly to a single hadronic quantity ∆, defined as the difference between "tree" and "penguin" contributions. More precisely, for aB Q meson decaying through a b → q penguin-mediated process, the decompositionwith the CKM factors λUq , can be used to define ∆ as the difference of the hadronic matrix elements that are multiplied by λ (q) u and λ (q) c respectively, i.e.,In the case of penguin-mediated decays, the evaluation of this quantity using QCD factorisation (QCDF) is expected to be particularly robust, as it was built specifically to cancel the infrared divergence...