  A proposal on determination of the effective electro-weak mixing angle ${\rm
sin^2\theta_{eff}^f\;(f=c,b)}$ at a super Z-factory (an $e^+e^-$ collider
designed with the possible highest luminosity and running around the center
mass energy $\sqrt s=m_Z$) is proposed. It is to determine how the effective
mixing angles depend on the flavors, especially the flavors $c$, $b$ (with the
determined effective ones, the electro-weak mixing angles ${\rm
sin^2\theta_W^f\;(f=c,b)}$ of Standard Model can be derived out precisely). The
key point of the proposal is that at such a super Z-factory, via measuring the
forward-backward ($A_{FB}$), left-right ($A_{LR}$) and combined left-right
forward-backward ($A^{FB}_{LR}$) asymmetries of the produced doubly
heavy-flavored hadrons (such as $B_c$, $B_c^*$ and the baryons $H_{QQ'q}$:
$\Xi_{cc}$, $\Xi_{bc}$ and $\Xi_{bb}$ etc) in the relevant production processes
$e^+e^-\to B_c+\cdots$ and $e^+e^-\to H_{QQ'q}+\cdots$ one may complement the
final determination of the mixing angle ${\rm sin^2\theta_W^f\;(f=c,b)}$ of
Standard Model. The advantage of the proposed way to determine the effective
electro-weak mixing angles is that the doubly heavy flavor(s) and the out-going
direction of the produced doubly-heavy hadron can be experimental determined
precisely, i.e. there are no such errors caused by missing identification of
the heavy flavor(s) and by determining the so-called thrust axis of the
produced jets, which can not be avoided in the early determination at LEP-I and
SLC.
