Physics help?

Question

A map shows Chicago to be 560 miles in a direction 21.0 degrees west of north from Atlanta, and Dallas to be 730 miles in a direction 5.0 degrees south of west from Atlanta. Assuming a flat earth, find the displacement from Dallas to Chicago.

Answer 1

Let  Atlanta, Chicago, Dallas by 3 points A, C, D.

DC =  DA + AC
DA: 5° N of E = 5° from x-axis
AC: 21°W of N = 90°+21° = 111° from x-axis
DA: (730 cos 5°, 730 sin 5°) = (727.22, 63.624)
AC: (560cos111°. 560sin111°) = (-200.69, 522.8)
DC = DA + AC

     =  (526.53, 586.424) 
Distance from Dallas to Chicago
|DC| = v(526.53)² + (586.424)²

       =788.12 miles

Direction from Dallas to Chicago:
θ = arctan[586.424/526.53]

   = 48.1°