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]
(..) Copy all, Houston, and in work. Matt, immediate return to Explorer. Repeat, immediate return to Explorer. - Copy. Explorer, prep airlock. - Airlock engaged, ready to receive. - Houston, elaborate. Debris from the missile strike has caused a chain reaction, hitting other satellites (..)2013 Gravity
(..) make Houston anxious. No, Houston. Don't be anxious. Anxiety's not good for the heart. System is ready to reactivate. Hubble telescope engaged. Upgrade fully functional. That applause you hear is for you, Shariff. Congratulations. Kick back, take the rest of the day off. (..)2013 Gravity