Instead of trying to talk to the person while he/she is angry, it is best to let him/her vent the emotions for a while before trying to cool him/her down. If you start speaking while they are still fuming, you will only encourage them to speak, in fact shout louder. The key to diffusing an angry person is to become a listener while they are talking.
Irrespective of what you think about the point of view of an angry person, you should acknowledge what they are saying and that you indeed understand how they feel. Your non-verbal skills will play an important role in this.
Occasionally repeat what they have said and nod your head to show your understanding of their words and the reason for their anger. Keep your voice sympathetic, i.e. soft and quiet, while talking to an angry person, as it will encourage them to bring the volume of their voice down as well.
Though not often practiced, mimicking body language is an effective way of diffusing an angry person. Try to connect with them by imitating their body language. If they are sitting with their arms crossed, you should also sit down and cross your arms. If their body is positioned in a certain manner, try to match that position. Be careful while imitating their body language though, as you do not want them to think that you are making fun of them.
While you would be tempted to lie to an angry person to calm them down, do not do that. Be honest with them and stick to the truth instead of changing the facts just to please them.
Do not promise anything that you might not be able to deliver later. If you are genuinely interested in doing something for them, just promise them that you will try your best.