A ranger in tower A spots a fire at a direction of 302 degre

A ranger in tower A spots a fire at a direction of 302 degree. A ranger in tower B, located 45 mi at a direction of 56 degree from tower A, spots the fire at a direction of 280 degree. How far from tower A is the fire? How far from tower B?

Solution

Directions are given using North as 0/360º
Points A, B and the fire (point F) form a triangle. Side AB is 45 miles
Angle A is 90 - 58 = 32º
Angle B is 10º (280-270)
Angle F is 138º (180 - 32 - 10)
Use the law of sines to find AF
AF/sin(10) = 45/sin(138)
AF =~ 11.68 miles