If the field is theDateField__c and the field which will calculate the day of the week for the mentioned date in theDateField__c is suppose named DayoftheWeek__c :
Formula field named DayoftheWeek__c with return type TEXT :
CASE(
MOD(theDateField__c - DATE(1900, 1, 1), 7),
0, "Monday",
1, "Tuesday",
1, "Wednesday",
1, "Thursday",
1, "Friday",
1, "Saturday",
1, "Sunday",
""
)
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