Answer
$f(x,y,z)=x+y \sin z+k$ ; Conservative
Work Step by Step
The vector field $F$ is conservative when $curl F=0$
When $F=ai+bj+ck$, then we have $curl F=[c_y-b_z]i+[a_z-c_z]j+[b_x-a_y]k$
Now, $curl F=(\cos z-\cos z)i+(0-0)j+(0-0)k=0$
Thus, we have the vector field $F$ is conservative.
Consider $f(x,y,z)=x+g(y,z)\implies g'(y)=0$
and $g_y=\sin z$
Now, $g(y,z)=y \sin z+h(z) \implies f(x,y,z)=x+y \sin z+h(z)$
This implies that $h'(z)=0$
Thus, we have $f_z=y \cos z$
Hence, we get $f(x,y,z)=x+y \sin z+k$
