  Using a $3.19~\mathrm{fb}^{-1}$ data sample collected at an $e^+e^-$
center-of-mass energy of $E_{\rm cm}=4.178$ GeV with the BESIII detector, we
measure the branching fraction of the leptonic decay $D_s^+\to\mu^+\nu_\mu$ to
be $\mathcal{B}_{D_s^+\to\mu^+\nu_\mu}=(5.49\pm0.16_{\rm stat.}\pm0.15_{\rm
syst.})\times10^{-3}$. Combining our branching fraction with the masses of the
$D_s^+$ and $\mu^+$ and the lifetime of the $D_s^+$, we determine
$f_{D_s^+}|V_{cs}|=246.2\pm3.6_{\rm stat.}\pm3.5_{\rm syst.}~\mathrm{MeV}$.
Using the $c\to s$ quark mixing matrix element $|V_{cs}|$ determined from a
global standard model fit, we evaluate the $D_s^+$ decay constant
$f_{D_s^+}=252.9\pm3.7_{\rm stat.}\pm3.6_{\rm syst.}$\,MeV. Alternatively,
using the value of $f_{D_s^+}$ calculated by lattice quantum chromodynamics, we
find $|V_{cs}| = 0.985\pm0.014_{\rm stat.}\pm0.014_{\rm syst.}$. These values
of $\mathcal{B}_{D_s^+\to\mu^+\nu_\mu}$, $f_{D_s^+}|V_{cs}|$, $f_{D_s^+}$ and
$|V_{cs}|$ are each the most precise results to date.
