Actually that's kind of how people work....
Anger causes us to be more optimistic. Risks seem smaller when angry and your actions more likely to succeed. Anger feels good at times and can be very useful.
It's fine to be angry. Completely normal, balanced and healthy. I can assure you that being angry, with moderation, will stop you from being so upset and it will clearly let the other person know they upset you.
And if it upsets you it's not dumb or unjustified. If something upsets you then it's important, it wouldn't have upset you if it wasn't. And if that's someone else's fault you've got a right to be angry at them. Don't go berserk, that's something else. But anger? A great way to deal with it.
Anger is completely normal, healthy and balanced behaviour when done in moderation. There's actually studies who show that people who get angry every now and then have a reduced chance of getting heart disease, compared to more fearful people.
Seriously, get angry sometimes. Curse someone in traffic, tell your friends about this freaking ass-hole you know etc. It'll make you feel much better and give yourself room to deal with it.
As for people being angry at you, it's just emotion. They're upset and they're dealing with it. Listen to it, take the blame you deserve and dismiss the blame you don't. Talk with them and solve the problem.
Anger isn't some vile abhorrent human emotion that's the cause of all misery. Anger is a completely natural, healthy and balanced emotion for when you're upset because of someone (or thing) else. And, as with almost everything, should be moderated.