  Vector bosons, such as Dark Photon $A'$ or $Z'$, can couple to muons and be
produced in the bremsstrahlung reaction $\mu^- + N \rightarrow \mu^- + N +
A'(Z')$. Their possible subsequent invisible decay can be detected in fixed
target experiments through missing energy/momentum signature. In such
experiments, not only is the energy transfer to $A'(Z')$ important, but also
the recoil muon angle $\psi_{\mu'}$. In this paper, we derive the total and the
double differential cross-sections involved in this process using the phase
space Weisz\"acker-Williams and improved Weisz\"acker-Williams approximations,
as well as using exact-tree-level calculations. As an example, we compare the
derived cross-sections and resulting signal yields in the NA64$\mu$ experiment
that uses a 160 GeV muon beam at the CERN Super Proton Synchrotron accelerator.
We also discuss its impact on the NA64$\mu$ expected sensitivity to explore the
$(g-2)_\mu$ anomaly favoured region with a $Z'$ boson considering $10^{12}$
muons accumulated on target.
