The research method in cultural anthropology that involves living in a culture for an extended period while gathering data is participant observation the inductive method archival research life history

Participant Observation

Explanation:

According to my research on Anthropological Research methods, I can say that based on the information provided within the question this type of research method is called Participant Observation. This type of methodology is usually used among Anthropologists, and is defined as a method where researchers choose to live in the lives of the group or species that they are studying.

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