Two fire-lookout stations are 15 miles apart, with station A directly east of station B. Both stations spot a fire. The angular direction of the fire from station B is N52°E and the angular direction of the fire from station A is N36°W. How far is the fire from station A?
(Use law of sines)
so, if we label the fire as F, then in ∆ABF, we have
∡A = 90-36 = 54°
∡B = 90-52 = 38°
so ∡C = 180-(A+B) = 88°
we want to find AF (side b), so the law of sines says
b/sinB = f/sinF
b/sin38° = 15/sin88°
b = 9.24 mi