2010 AMC 12B Problem 13

In $\triangle ABC$ , $\cos(2A-B)+\sin(A+B)=2$ and $AB=4$ . What is $BC$ ?

$\textbf{(A)}\ \sqrt{2} \qquad \textbf{(B)}\ \sqrt{3} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2\sqrt{2} \qquad \textbf{(E)}\ 2\sqrt{3}$

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.