Question

A map suggests that Atlanta is d1 = 729 mi in a direction of θ1 = 5.15° north of east from Dallas. The same map shows that Chicago is d2 = 562 miles in a direction of θ2 = 20.8° west of north from Atlanta. Modeling the Earth as flat, use this information to find the displacement from Dallas to Chicago.
magnitude miles
direction ° north of east of Dallas

Answer 1

dDCeast = dDAeast + dACeast = 729 cos 5.15° – 562 sin 20.8° = 526.6 miles.
dDCnorth = dDAnorth + dACnorth = 729 sin 5.15° + 562 cos 20.8° = 590.4 miles.
By the Pythagorean theorem
d=√((d_DCeast )^2+(d_DCnorth )^2 )

d=√((526.6)^2+(590.4)^2 )=791.13 miles

Then
tan⁡〖θ=d_DCnorth/d_DCeast 〗

tan⁡〖θ=590.4/526.6〗=1.12

θ = 48.2°
Thus, Chicago is 791.13 miles at 48.2° north east of Dallas