Jake is on the Appalachian Trial hiking. He hikes 8 miles due north and then turns and hikes an additional 10 miles at a bearing of N 67°W at which point he has reached a vista, eats lunch, and decides to bushwhack directly back to the trailhead. How far is he now from the trailhead? What bearing must he take to get back to the trailhead?
Other than the fact that you can't bushwhack on the Appalachian Trail, we have a clear cut case of using the cosine law:
let the distance back be x miles
x^2 = 8^2 + 10^2 - 2(8)(10)cos113°
= 273.40...
x = 16.534.. (don't round off your calculator answer)
now by the sine law:
let the angle opposite the 10 side be θ
sinθ/10 = sin113/16.534...
sinθ = .556702...
θ = 33.8°
so he should aim for S 33.8° E