Asymptotic expansion of $\sin\left(\pi + \exp(-1/\epsilon)\right)$

Question

I need to find the two term asymptotic expansion of $\sin\left(\pi + \exp(-1/\epsilon)\right)$ as $\epsilon$ tends to zero, but the exponential term is confusing me...

Answer 1

Using Maple I am obtaining

$$-{{\rm e}^{-{\epsilon}^{-1}}}+{\frac {1}{6}}\,{{\rm e}^{-3\,{\epsilon}^{-1}}}-{ \frac {1}{120}}\,{{\rm e}^{-5\,{\epsilon}^{-1}}}+{\frac {1}{5040}}\,{ {\rm e}^{-7\,{\epsilon}^{-1}}}-{\frac {1}{362880}}\,{{\rm e}^{-9\,{ \epsilon}^{-1}}}+O \left( {{\rm e}^{-11\,{\epsilon}^{-1}}} \right) $$