て form - what does てやる mean when it is not used for giving?

Okay, according to the (monolingual) dictionary that I use, てやる is used to indicate "doing something with strong will or determination" (my translation) e.g. 飛び降りてやる! But this definition seems a little too vague for me; is there anything that it actually translates to in English?

I've seen some sources claim that it still means "do" but I'm unsure, because I have come across constructs like やってやる！