A method of fieldwork where someone studies a culture by doing what the people of that culture do is called

The very definition of Participant Observation is a method for data collection in research where the researcher introduces themselves into the field of study in order to collect more detailed and meaningful data for a prolonged time period (months to years). All of the other methods mentioned don't require being a participant in the culture being investigated.

Anthoprology

Explanation:

Anthropologist: A scientist who studies societies and cultures.

“the iliad” (gr: “iliás”) is an epic poem by the ancient greek poet homer, which recounts some of the significant events of the final weeks of the trojan war and the greek siege of the city of troy (which was also known as ilion, ilios or ilium in ancient times).