Solution for PDE

Solution for PDE

I have the following PDE: $$u_t+(1+\epsilon \sin x)u_x=0$$ where $\epsilon
\to 0$ and I want to find the explicit form of the solution including
terms up to $O(\epsilon)$ and as initial condition I have $x(0)=s$. Now, I
manage it to write down $dx/(1+\epsilon \sin x) = dt$ and then using
$1/(1+\epsilon \sin x) = 1 - \epsilon\sin x + O(\epsilon^2)$ I find out:
$$ x-s+\epsilon\bigl(\cos(x)-\cos(s)\bigr)=t \tag{$*$}$$
The problem is that my solution is $u=f(s)$ and therefore I need to
express $s$ as a function of $x$ and $t$ from $(*)$.
Any suggestions?