  In $\Lambda_b^0 \to \Lambda_c^+ (\to \Lambda^0 \pi^+) \tau^- \bar{\nu}_\tau $
decay, the solid angle of the final-state particle $\tau^-$ cannot be
determined precisely since the decay products of the $\tau^-$ include an
undetected $\nu_\tau$. Therefore, the angular distribution of this decay cannot
be measured. In this work, we construct a {\it measurable} angular distribution
by considering the subsequent decay $\tau^- \to \pi^- \nu_\tau$. The full
cascade decay is $\Lambda_b^0 \to \Lambda_c^+ (\to \Lambda^0 \pi^+)\tau^- (\to
\pi^- \nu_\tau)\bar{\nu}_\tau$. The three-momenta of the final-state particles
$\Lambda^0$, $\pi^+$, and $\pi^-$ can be measured. Considering all Lorentz
structures of the new physics (NP) effective operators and an unpolarized
initial $\Lambda_b$ state, the five-fold differential angular distribution can
be expressed in terms of ten angular observables ${\cal K}_i (q^2, E_\pi)$. By
integrating over some of the five kinematic parameters, we define a number of
observables, such as the $\Lambda_c$ spin polarization $P_{\Lambda_c}(q^2)$ and
the forward-backward asymmetry of $\pi^-$ meson $A_{FB}(q^2)$, both of which
can be represented by the angular observables $\widehat{{\cal K}}_i (q^2)$. We
provide numerical results for the entire set of the angular observables
$\widehat{{\cal K}}_i (q^2)$ and $\widehat{{\cal K}}_i$ both within the
Standard Model and in some NP scenarios, which are a variety of best-fit
solutions in seven different NP hypotheses. We find that the NP which can
resolve the anomalies in $\bar{B} \to D^{(*)} \tau^- \bar{\nu}_\tau$ decays has
obvious effects on the angular observables $\widehat{{\cal K}}_i (q^2)$, except
$\widehat{{\cal K}}_{1ss} (q^2)$ and $\widehat{{\cal K}}_{1cc} (q^2)$.
