Systems of Differential Equations · Superchallenge problem

Let $\Phi(t)$ be the $2\times2$ matrix solution of $\Phi'=A(t)\Phi$, where $A(t)=\begin{pmatrix}\sin t&t\\e^{t}&\cos t\end{pmatrix}$, with $\Phi(0)=\begin{pmatrix}1&2\\3&4\end{pmatrix}$. Find $\det\Phi(\pi)$.