A man is looking at a flag pole that is 4m away and 12m tall. What angle should his head be at so that he is staring at the top of it?

There are going to be the technical people who are going to want to know how far up his eyes are and blah blah blah...  They're going to over-complicate a problem that's trying to teach you some basics.  They've confused obfuscation with cleverness.

Think of a right triangle where the opposite side to the man is 12 meters and the adjacent side is 4 meters.  We need to know the angle from where he is.

tan(t) = opposite / adjacent = 12/4 = 3
tan(t) = 3
t = tan-1(3) = arctan(3)

That is, we need to know the angle "t" that has a tangent of 3.  Make sure the calculator is in degree mode and you'll get:

‭71.565051177077989351572193720453‬...

72 degrees, roughly.

That depends on his height but the angle of elevation is tan^-1(12/4) = 71.565 degrees to 3 decimal places

To get an exact answer, you need his height from ground to eyes. Let's say 1.7m
12-1.7 = 10.3m from eye horizontal to top of pole
4m from pole
tan A = 10.3/4 = 2.575
about 68.8 degrees