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]
(..) - How were you to know the bank was being robbed? - That is correct. What is correct? I did not know... the bank was being robbed because I was engaged in my sworn duty... as a police officer. You didn't even arrest the old beggar. There was some question as to whether the (..)1975 The Return of the Pink Panther