A major reason for the formation of groups like the National Congress of American Indians was to show the Whites that lived East of them that they were not the "savages" that they were as depicted by the newspapers, and that they can adapt and become like the whites. On top of that, it was to give them a structural government to rely on in case the Americans wanted to deport them west. Of course, this did not work, as most, if not all, Native American tribes were deported west.
Talking to a person you do not know, an email to a business representative, and similar would be situations where someone would have to use formal language.
