Use lemon juice or white vinegar and a sprinkling of coarse salt with a soft brush.

That is what I use to clean my copper pots. It was a Martha Stewart tip from years ago. It takes some elbow grease, but works. I tried Brasso once, and it made the copper lose its luster temporarily.

If you want to keep the dark patina on the sink, we recommend just washing the sink after heavy use with a soft cloth, warm water, and a dab of soft dish soap. The patina is a thin layer of oxidization on the copper which can be removed with acid or abrasives, but the finish will heal itself with time and regular use.