How to plot periodic functions that are defined over a specific region

Let's say I've the following function (that is also periodic over the rest of the region I did not define):

$$\begin{cases} 4\sin\left(2x\right)\space\space\space\space\space\space\space\space\space\text{when}\space0\le x<\pi\\ \\ 4\exp\left(-2x\right)\space\space\space\space\space\text{when}\space\pi\le x<2\pi \end{cases}\tag1$$

So, after $2\pi$ the sine part of the function start again. How can I plot that function in Mathematica 10.0?

Piecewise[{{Sin[x], Mod[x, 2 Pi] < \[Pi]}}, -Abs@Cos[x]]

Plot[Piecewise[{{Sin[x], Mod[x, 2 Pi] < \[Pi]}}, -Abs@Cos[x]], {x, -8, 8}]