Hi there,
I hope this might help:
The first thing to do is to split the triangle into two right angled triangles by drawing a line perpendicular from A to a (thus giving the height of the triangle). Now you have two right angled triangles and you can exploit this using some cool trig stuff. First thing is you should note that sinc = opposite over hypotenuse, which will be the height over b. You can calculate the height now and use your area value to solve for a (using area=1/2 base x height). I hope that helps, if you need any more details just let me know.
You should know that you can calculate the area of a triangle using $\displaystyle \displaystyle \textrm{Area} = \frac{1}{2}ab\sin{C}$. You know the area, $\displaystyle \displaystyle b$ and $\displaystyle \displaystyle \sin{C}$, use this information to solve for $\displaystyle \displaystyle a$.