How far is the yacht from each lookout station? How far is the yacht from the coast?

Question

Two lookout stations, which are 25 miles apart along the coast on a north-south line, spot an approaching yacht. One lookout station measures the direction to the yacht at N33degrees E, and the other station measures the direction of the yacht at S62degrees E. How far is the yacht from each station? How far is the yacht from the coast?

Answer 1

Let A be the North station with a the side opposite.

Let B be the South station with b the side opposite.

Let C be the yacht with 25 miles the side opposite.

Angle A = 62

Angle B = 33

Angle C = 180 - 62 - 33 = 85

a = 25sin62/sin85 ≈ 22.158 mi.

b = 25sin33/sin85 ≈ 13.67 mi.

d = 13.67sin62 ≈ 12.07 mi.