At first, get the first day of current month (for example):
NSDate *today = [NSDate date];
NSCalendar *gregorian = [[NSCalendar alloc] initWithCalendarIdentifier:NSGregorianCalendar];
NSDateComponents *components = [gregorian components:(NSEraCalendarUnit | NSYearCalendarUnit | NSMonthCalendarUnit) fromDate:today];
components.day = 1;
NSDate *firstDayOfMonth = [gregorian dateFromComponents:components];
Then use NSDateFormatter to print it as a weekday:
NSDateFormatter *dateFormatter = [[NSDateFormatter alloc] init];
[dateFormatter setDateFormat:@"EEEE"];
NSLog(@"%@", [dateFormatter stringFromDate:firstDayOfMonth]);
P.S. also take a look at Date Format Patterns