Answer the following:

Question

Two lighthouses are located on a north-south line. From lighthouse A, the bearing of a ship 3742 meters away is 129°43’. From lighthouse B, the bearing of the ship is 39°43’. Find the distance between the lighthouses.

Answer 1

When using compass bearings, your angle begins at 0 degrees from North and goes clockwise.

step 1 - convert arc minutes to degrees

divide the arc minutes by 60 to get your decimal

43/60 = 0.72

angle A = 129.72

angle B = 39.72

step 2 - set up triangles

Lighthouse A - starting from north, turn 129.72 degrees clockwise. 90 degrees leaves you facing East and another 39.72 degrees reaches the boat. You end up facing 39.72 degrees southeast or 39.72 below the x axis. We know this hypotenuse length is 3742 meters.

Triangle for lighthouse A - a right triangle with one leg facing East and another facing south, with an angle of 39.72 degrees in between them. The hypotenuse is 3742. Let’s find the lengths of the East and south legs:

East leg = 3742cos(39.72) = 2878.26

South leg = 3742sin(39.72) = 2391.27

Lighthouse B - starting from north, turn 39.72 degrees clockwise. We end up 90-39.72 degrees above the x axis. Our angle is 50.28 degrees northeast/above the x axis. We don’t know the hypotenuse of the triangle, but since these lighthouses share the same north-south line, the East component of each triangle must be the same. We can find the north component using tangent since we have the East leg.

East leg = 3742cos(39.72) = 2878.26

tangent(50.28) = North leg/ East leg

North leg = tangent(50.28)*East leg

North leg = tangent(50.28)*2878.26 = 3464.42

Starting from Lighthouse A and going south 2391.27m and starting from Lighthouse B and going north 3464.42m lands you at the same spot, where these two right triangles meet.

Therefore, the distance between the 2 lighthouses is the sum of these distances.

2391.27 + 3464.42 = 5855.69 m distance from Lighthouse A to Lighthouse B