Given with a = 22, b = 17, and c = 19, find . round the cosine value to the nearest thousandth and answer to the nearest whole degree.

Assuming all angles are to be calculated.
Using the cosine rule:
a^2=b^2+c^2-2bc(cos(A))
=>
cos(A)=(b^2+c^2-a^2)/(2bc)
substitute values,
cos(A)=(17^2+19^2-22^2)/(2*17*19)=83/323=0.25697
A=75.11 degrees
similarly
cos(B)=(19^2+22^2-17^2)/(2*19*22)=139/209=0.66507
B=48.31 degrees
cos(C)=(22^2+17^2-19^2)/(2*22*17)=103/187=0.55080
C=56.58 degrees
Check: (75.11+48.31+56.58)=180.00 degrees, ok.

Note:
1. Better accuracy than required is calculated for a more accurate check.  You need to round answers to the specified accuracy.
2. In fact, not all tables are made from formulas.  Some tables are compiled by experimentation and require hundreds and thousands of man-years to complete (e.g. steam tables).