He won't be able to attend because he's 'otherwise engaged'. [=he is busy doing something else]
('US') He is 'engaged' in research. = ('Brit') He is 'engaged' on research.
She 'engaged' him 'in' conversation. [=she started a conversation with him]
(..) young man. What the fuck? Goddamn, shit's changed in a year. What are we doin' here? Okay, sir, we're engaged in procuring your tattoo. Welcome to the Identity Processing Program of America. Please insert your forearm into the forearm receptacle. (..)2006 Idiocracy