If you want to calm someone down, sympathize with them whilst describing what's upsetting them in descending orders of magnitude.
I understand why you're angry
you're right to be frustrated
This would annoy me too.
As they accept the acknowledgements they want they should also accept the declining emphasis on emotion and become calmer
Giving students in my elementary class the illusion of choice. If you ask "do you want to start your work?" Or "isn't it time we got something done?" I modify it to-
"Would you like to do your assignment with a pencil or blue pen?" They are too consumed with this choice that they have forgotten that they didn't want to do it in the first place.
Also, every recess-
"Did you push jimmy?" "No."
"Ok umm two people saw you do it, and reported it to me. Are you sure you didn't?" "In sure."
Then change it to "can you tell me why you pushed him?"
"Well it's because..."
Gets'em every time
If you work with clients, prevent them from getting defensive and angry by not using the word "you".
It's not "you didn't send the attachment", it's "the attachment didn't send". Don't blame them, blame the thing you're talking about.
This is really useful in sales. I use to sell car warranty's and a big thing was using "we" instead of "you" to get on the customer's page.
"You need to protect your vehicle so you don't end up paying large repair bills out of pocket," sounds like you're telling the person what to do and selling on fear.
"We should keep warranties on our vehicles so we don't end up getting stuck paying high repair bills when they inevitably break down, you know what I mean?" Gets them to agree getting them into yes mode and sounds relatable and truthful rather than like you're selling them.
I learned this from my primary school principal. When you're angry, raise your voice. When you're really angry, lower it.