Use 'indignant' when reacting to situations that seem unjust, not just for any annoyance or irritation. They were indignant when the honest employee was fired without reason.
Pro Tip 2/3
Strong Emotion
'Indignant' implies a deep emotional response - more than simply being upset or displeased. He felt deeply indignant after his work was plagiarized.
Pro Tip 3/3
Moral Aspect
Indignation often involves a moral or ethical perspective on why something is wrong. She was indignant that the new policy penalized the less fortunate.