Answer
$1$.
Work Step by Step
To find the Jacobian, we calculate the determinant of the 2x2 matrix $\frac{\partial(x, y)}{\partial(r, t)}$ as follows $$
\operatorname{Jac}(G)=\frac{\partial(x, y)}{\partial(r, t)}=\left|\begin{array}{ll}
{\frac{\partial x}{\partial r}} & {\frac{\partial x}{\partial t}} \\
{\frac{\partial y}{\partial r}} & {\frac{\partial y}{\partial t}}
\end{array}\right|=\left|\begin{array}{ll}
{ \sin t} & {r\cos t} \\
{1} & {\sin t}
\end{array}\right| =\sin^2 t-r\cos t.
$$
Now, at the point $(r,t)= (1,\pi)$, we have $
\operatorname{Jac}(G)=\sin^2 \pi-\cos \pi=1.$
