What year did Captain Cook discover New South Wales?

Answer 1

The question proceeds from a false assumption. James Cook did not discover Australia; nor was he a Captain at the time he charted the eastern coast.

James Cook was still a Lieutenant in charge of the Endeavour when he charted the eastern coast of Australia and claimed it for Great Britain on 22 August 1770, calling it New South Wales. He charted the east coast between April and August of that year. For this reason, Cook is often wrongly credited with discovering Australia.

Answer 2

The simple answer is "April 1770".

However, there are two misconceptions in this question.
Firstly, James Cook was not a captain when he sailed up Australia's east coast. He was still a Lieutenant.
Secondly, he claimed the entire eastern seaboard as New South Wales, charting it, but theoretically not "discovering" it. Abel Tasman had already discovered the far north of what Cook called New South Wales as, at that time, the claimed territory extended beyond the tip of Cape York to Possession Island.

Answer 3

James Cook was not a captain, but a lieutenant when he claimed Australia for Great Britain on 22 August1770.