Math Help - Need help finding an angle in degrees

1. Need help finding an angle in degrees

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. Modeling the Earth as flat, use this information to find the displacement from Dallas to Chicago.
magnitude

Now I've drawn this out and I get distance between dallas/chicago = 774 miles.

Now as far as the angles inside the triangle I used the law of sines.

sin(a)/A = sin(b)/B = sin(c)/C I know the answer is between 41.2 and 48.6 but I think I made a calculation error somewhere because I keep slightly getting the wrong answer.

Ive got less than one hour.

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

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

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

3. Too late to help on my homework, but you still helped me to understand the problem. Thank you.

I had figured my initial distance was incorrect, but it was graded correctly.