Answer
$F$ is a conservative vector field with potential function $$f(x,y)=xsiny+cosz+C$$
Work Step by Step
As we are given that $F(x,y,z)=sinyi+xcosyj-sinzk$
$F(x,y)=Pi+Qj+Rk$ will be conservative when
$$R_y=Q_z,P_y=Q_x$$, and $$P_z=R_x$$
Hence, $F$ is a conservative vector field with potential function $$f(x,y)=xsiny+cosz+C$$
