Is a wife considered divorced after her husband utters the word of divorce twice in a state of extreme anger, knowing that this is the third divorce?

Anger does not prevent a divorce from occurring unless it leads the person to a state of complete loss of awareness and comprehension. If your anger did not reach this level, then the divorce is binding, and the wife becomes irrevocably divorced and forbidden to you until she marries another husband, according to the Almighty's saying: "Divorce is twice. Then, either keep [her] in kindness or [send her forth] with good treatment," and "And if he has divorced her [for the third time], then she is not lawful to him afterward until she marries another husband." However, if the divorce occurred in a state of complete loss of awareness and comprehension (Ighlaq), then it does not take effect, based on the Prophet's (peace be upon him) saying: "There is no divorce and no emancipation in Ighlaq." Therefore, you must assess your situation and present the matter to the Sharia courts or to people of knowledge.