How would you deal with a client who is very angry with you on the phone?

Any company customer/client courtesy is priority and management are trained to deal with angry situations. When talking use calm, slow voice,

sympathetic tone listening to their problem and their concerns and gather more information. Ask questions of specifics on direct points of their issue. Asking for specific information tells them that you are getting right on their issue. These two things stop escalation of the anger in its tracks. This will calm most but there will always be an exception and all you can do is use the same calm and respectful courtesy even with them and never forget to keep professional. Also, you might note that people from some areas speak in a very direct, loud tone as a matter of course and it just sounds angry to those of us in other areas of the country.