Hello, thedoge!

Your diagram must be off . . . We use the Law of Cosines.

A map suggests that Atlanta is 731 mi in a direction of 4.90° north of east from Dallas.

The same map shows that Chicago is 562 miles in a direction of 21.0° west of north from Atlanta.

Find the displacement from Dallas to Chicago. Code:

C * :
* :
* 562 :
* :
* 21° :
* :
69° * :
- - - - - - - * A
4.9° *
731 *
*
* 4.9°
D * - - - - - - - - - - - -

We have $\displaystyle \Delta CAD$ with: .$\displaystyle \angle CAD = 73.9^o,\:AC = 562,\:AD = 731$

Law of Cosines: .$\displaystyle DC^2\:=\:562^2 + 731^2 - 2(562)(731)\cos73.9^o \:=\:622,351.079$

Therefore: .$\displaystyle DC \;\approx\;788.89$ miles.