Can the complement of an unrecognizable language be a recognizable language?

Question

I know that complement of a language that is recursively enumerable, but not recursive, is definitely not recursively enumerable (or unrecognizable). So my question is what can be said about the complement of an unrecognizable language?

Answer 1

Sure: the Halting language (sometimes called $K$) is recursively enumerable whereas its complement ($\overline{K}$) is famously not.

Generally speaking, the existence of such languages follows from the implication you state -- the complement of enumerable yet undecidable languages can not be enumerable -- by the existence of precisely enumerable yet undecidable languages.

That said, there are also non-enumerable languages whose complement isn't either.