What you are asking for is a conjugacy of the dynamical systems $g(x)=4x(1-x)$ and $h(x)=\sin(\pi x)$. Since $g$ and $h$ are both full unimodal maps of $[0,1]$, there will exist such a conjugacy, but it is very unlikely that you will be able to write it down explicitly. Also, it will certainly be continuous, but likely not absolutely continuous.
You are asking that $f(g(x))=h(f(x))$. You are requiring that $f(0)=0$ and $f(1)=1$. Substituting $x=\frac 12$, you obtain $1=f(1)=h(f(\frac 12))$. Since $h^{-1}(1)=\{\frac 12\}$, it follows that $f(\frac 12)=\frac 12$. You can continue in a similar way to obtain the value of $f$ at further preimages of $\frac 12$. This defines $f$ on a dense set.