A big challenge can occur when a customer's frustration connects you to the problem. It doesn't matter whether or not you caused it, the customer can't get over their anger.
One way to overcome this issue is the good cop, bad cop technique. The customer is angry at you, which makes you the bad cop in this situation. All you have to do is introduce a co-worker or a supervisor to take over the interaction who can act as the good cop.
I've seen this get great results time and time again. A new person on the scene helps the customer instantly calm down and accept the assistance being offered.
This is a tough technique for some people because they confuse being the bad cop with being bad at service. This isn't true at all. Using the good cop, bad cop technique takes an advanced professional who is able to help a customer feel better even if it means getting someone else involved.