STAINED STEEL SINK

There was a small stain on my sink and I sprayed a cleaning product on the sink and forgot about it. Later I  saw dark brown stains all over the sink. I tried to clean it off, but the stains couldn't be removed. 

I looked for answers from Professor Internet and was told to try using baking soda mixed with liquid dish soap to make a good paste to gently rub on stains. Then rinse the stainless steel surface thoroughly, and towel dry.
If the stains still remain then the next alternative is to use vinegar. If that still doesn't work, I was to get a
stainless steel cleaner and polisher.

I will skip the baking soda and instead start with the vinegar and see if it works.  If not I'll go for the steel cleaner.