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The first measurements of CP violation in the $B$ system will likely probe$\sin 2\alpha$, $\sin 2\beta$ and $\cos 2\gamma$. Assuming that the CP angles$\alpha$, $\beta$ and $\gamma$ are the interior angles of the unitaritytriangle, these measurements determine the angle set $(\alpha,\beta,\gamma)$except for a twofold discrete ambiguity. If one allows for the possibility ofnew physics, the presence of this discrete ambiguity can make its discoverydifficult: if only one of the two candidate solutions is consistent withconstraints from other measurements in the $B$ and $K$ systems, one is not surewhether new physics is present or not. We review the methods used to resolvethe discrete ambiguity and show that, even in the presence of new physics, theycan usually be used to uncover this new physics. There are some exceptions,which we describe in detail. We systematically scan the parameter space andpresent examples of values of $(\alpha,\beta,\gamma)$ and the new-physicsparameters which correspond to all possibilities. Finally, we show that if onerelaxes the assumption that the bag parameters $\BBd$ and $\BK$ are positive,one can no longer definitively establish the presence of new physics.Comment: 29 pages, LaTeX, 1 figures, presentation substantially reworked,  physics conclusions unchanged. This version will be published in Phys. Rev.

B-decayCPasymmetries, discrete ambiguities, and new physics