Indignant means angry at something you think is unjust. Not merely cross — offended on principle.
The sense of injustice is what makes the word specific, so somebody indignant has been wronged, or believes they have. It takes at or about. There is a common ironic use, since describing somebody as indignant often suggests the writer thinks the offence was imaginary. The noun is indignation, and righteous indignation is the phrase for anger somebody is thoroughly enjoying. An indignity is a humiliation, which is the thing that causes it.
1adjectiveindignantAngry at treatment felt to be unfair or unjust.
“She was indignant at being asked to prove it.”
Indignant is Latin indignāns, from indignor, to consider a thing unworthy, so the anger is a judgement about what ought to have happened and not merely a temper.
Proto-Indo-European
*deḱ-
Latin
indignāns
English, today
indignant
Built from
Borrowed from Latin indignāns, present participle of indignor (“to consider as unworthy, be angry or displeased at”), from in- (“privative”) + dignor (“to deem worthy”), from dignus (“worthy”).
Etymology from Wiktionary, CC BY-SA