I frequently use it as a sarcastic retort, but, honestly; there will be subtle hints.
How people express anger is unique to them. They may use profanity. They may change their sentence structure. They may pick their words more deliberately and deliver them with more force. They may raise their voice. They may lower it. They may smile, no really, this can be a tell. Their idiomatic use of the language may shift slightly. They may give very little outward sign, but their behavior towards the person who pissed them off will change. They may use more pejorative terms for the subject of their ire. Or, they can yell, scream, and throw a tantrum, that is a legitimate option. Obviously, they won’t be doing all these things at once, and this isn’t an exhaustive list.
It really depends on who your characters are.
That said, the tells for a character should be mostly consistent to them. A character who gets quiet, methodical, and downright malicious, when they’re angry probably isn’t going to then get into a screaming match, unless it’s to manipulate someone else.
Tells that alter how a character’s dialog flows can be effective, but very difficult to actually pull off. It’s something worth experimenting with. It’s a useful tool to have as a writer, because it will allow you more flexibility with how you structure dialog for your characters. Just, be ready to tweak it around a bit until you get the hang of it.
Also, as a personal nitpick, unless someone’s actually being charged with a crime, or introducing an airborne piece of literary canon to the scene, “throwing the book” just doesn’t work. It’s a decent, but sadly abused idiom. Give the poor thing a rest.
As for what you’re looking to say in an argument? Remember your characters’ goals.
Every argument starts with two characters who want different things from a scene. Even if one of them just wants the other to go away and leave them alone.
At that point, your characters’ jobs are to advocate for their position, or undermine the other’s. As the scene starts they have a lot of options for how they can approach that goal, but as they choose their approach, they’ll whittle down their options. You can think of this as the characters building inertia in their approach. Someone who’s aggressively derisive of the other character’s position will have a much harder time walking it back, then someone who takes a more neutral approach. Alternately, you can have a character who’s building up their opponent’s position, only to start ripping it apart once their opponent’s emotional guard is down.
I realize the irony in answering your fears about your scenes being too mechanical with a strictly mechanical approach to character interaction. The point is to remember that the scene needs to go places. If it starts with two characters in a room talking about what to do next, they need to state their case, respond to their opponents’ objections or alternatives, and the actual point of the argument needs to move naturally.
For the dialog itself, you will probably benefit from showing your characters anger, rather than explicitly telling the audience that one of your characters is slightly irked now. Focus on how their demeanor changes, not what those changes mean.