How did goku become a super saiyan 3?

Answer 1

Goku had been training hard in other world, during this time he had achieved Super Saiyan 3 however, due to the differences in the areas, the super saiyan three goku had been able to do in other world, was a lot harder and drained more energy on earth. Vegeta, did not have his body at this time period, and thus was unable to train.