Find the distance between city A and city C. (Law of Cousines)

Question

The bearing from city A to city B is S E and the bearing from city B to city C is N E. (See picture below where  and . Click on the picture to see it more clearly).

It takes 1.6 hours for a car traveling at 56 miles per hour to go from A to B and 1.4 hours to go from B to C.

Find the distance between city A and city C.

Answer 1

You actually don't need the law of cosines because the angle between the legs of the trip is 90°. Just use Pythagorean Theorem to solve for A to C. The times are proportional to the distances for constant velocities, so just treat the values as weird distance units.

If the angle had been other than 90, you would have solved for c using c² = a² + b² - abcos(θ_c) where the angle indicated is the one opposite to c in the triangle.

Oh wait, the terms used were wrong if they have given angles α and β. In that case you use the cosine law with θ = α + β for angles on the diagram. (You can't solve this without those angles)