Pre-calculus Problem

Question

A ship is sailing due north, parallel to the coast. The captain wants to determine how far the ship is from the coastline. To do this, he makes two sightings of a lighthouse next to the shore. At the first sighting, the lighthouse is 72° north of due east. Then he sails north for a distance of 4 miles and makes another sighting of the lighthouse. This time it is 25° south of due east. What is the distance between the ship's path and the coast? (this problem is for "Right and Non-Right Triangle Applications" for additional context)

Answer 1

There is another way to solve. We have a triangle with the two paths from the ship to the lighthouse and the path of the ship. The triangle has angles 18°, 65°, and 180° - 65° - 18° = 97°.

We can use the sin law to get the distance of the first sighting F

F = 4 sin 65°/sin 97°.

Then, there is another triangle where F is the hypotenuse, one side is from the ship's first location to land, and one side is along the shore to the lighthouse. As 72° is the angle between adjacent, the side from the ship to land, and hypotenuse, we have the distance to land = F cos 72° = 4 sin 65°cos 72° /sin 97°.

Note that if we replace cos 72° with sin 18°, we have the satisfying solution 4 sin 65° sin 18° /sin 97°, which includes the sin of the three angles of the triangle and would be the same if we solved using the other side.

Answer 2