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]
(..) Come on, Spider-Man. Come on, Spider-Man! Launch for intercept. Green light. Green light. Oh, yeah. Retroreflective panels engaged. Got a visual on the plane, but I'm feeling a little resistance. MASON: It's probably just drag on the new turbines. Look out for the cloaking cameras. (..)2017 Spider-Man: Homecoming