engage vs sign up

engage and sign up both are verbs.

Word	Noun	Adjective	Verb	Adverb
engage	No	No	Yes	No
sign up	No	No	Yes	No

As verbs, sign up is a hyponym of engage; that is, sign up is a word with a more specific, narrower meaning than engage:

Other hyponyms of engage include featherbed, fill, engage, ship, contract, sign, sign on, rat, farm out, job, subcontract.

engage (verb)	sign up (verb)
get caught	join a club, an activity, etc. with the intention to join or participate
consume all of one's attention or time	engage by written agreement
give to in marriage
keep engaged
engage for service under a term of contract
hire for work or assistance
ask to represent; of legal counsel
engage or hire for work
carry out or participate in an activity; be involved in
carry on (wars, battles, or campaigns)