sigh vs let out

sigh is a noun but let out is not a noun.

Word	Noun	Adjective	Verb	Adverb
sigh	Yes	No	Yes	No
let out	No	No	Yes	No

As verbs, let out is a hypernym of sigh; that is, let out is a word with a broader meaning than sigh:

Other hypernyms of sigh include emit, let loose, utter.

sigh (noun)	let out (noun)
an utterance made by exhaling audibly
a sound like a person sighing

sigh (verb)	let out (verb)
heave or utter a sigh; breathe deeply and heavily	make (clothes) larger
utter with a sigh	make known to the public information that was previously known only to a few people or that was meant to be kept a secret
	express audibly; utter sounds (not necessarily words)
	bring out of a specific state