setting up a trig equation and solving it. either using law of cosines or law of sines.

Question

An air traffic controller is tracking a plane 3.4 miles due north of the radar tower. A second plane is located 2.1 miles from the tower at a heading of 72 degrees north of west. To the nearest tenth of a mile, how far apart are the two planes?

Answer 1

This is a Law of Cosines problem, since we do not have an angle and its° opposing side to work with. The angle between the two sides of the triangle is 90° - 72° = 18°, since the "72 degrees north of west" describes an exterior angle that is complimentary to the one inside our triangle at the location of the radar tower. Using the Law of Cosines :

c² = a² + b² - 2ab * cos(C)

gives us

c² = (3.4)² + (2.1)² - 2 * 3.4 * 2.1 * cos(18°) = 2.388912947

so the planes are

c = √2.388912947 = 1.565610865 ≈ 1.57 miles apart