Keeping a cool head should definitely be a goal. Becoming angry is almost never well-received by either your opponent or the audience. That's not to say you should be cold and distant, either. People are not often persuaded purely by facts or even by self-interest. They are most often persuaded by those they trust, and in order to establish that trust, you need to be calm, honest, and personable. Once you have that, then you can show that the facts are on your side and have it actually influence opinion.
This can work even on the Internet. You can get true believers to admit that they might be wrong, but it's a lot of work, and a prerequisite is that people have stopped shouting and sniping at each other. I've at least temporarily convinced a couple of Young-Earthers here that the Young-Earth model is not compatible with the scientific findings of the last few centuries. Calling a person stupid or uninformed is not the way to do it. Instead, you have to show them that what they were taught about science is wrong. I would often agree that evolution and other theories, according to what they were taught, could not work, but that those theories do not actually work the way they were taught. I would then show them how they actually work, and the evidence predicted by them. Eventually they came to realize that they might have had it wrong all along not because they were stupid, but they were unfortunate enough to be educated by people who didn't know or didn't care what they were talking about, a state of affairs that is depressingly common in many US states.
