What is the ruling on saying "You are divorced" to a non-Muslim wife in a state of extreme anger, then retracting it and admitting the mistake? Does the divorce take effect in this case, and what must be done for the wife to return to the husband's bond?

If the husband utters a divorce while being volitional, aware of what he is saying, and understanding its meaning, then it takes effect, whether it is in Arabic or a foreign language. If it is the first or second divorce, the husband can reconcile with his wife as long as she is within her waiting period (iddah) by saying: "I have reconciled with my wife." Her knowledge or consent is not required. However, if it is the third divorce, the wife becomes irrevocably divorced (baynunah kubra), and she is not lawful for him unless she marries another husband willingly, consummates the marriage with him, and then he divorces her or dies, and her waiting period from him expires.