SOLUTION: 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’.

Algebra ->  Trigonometry-basics -> SOLUTION: 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’.      Log On


   

Question 1198572: 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 by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

x = distance between the two lighthouses
A & B = the two lighthouses
C = location of the ship
D = helper point to label the upper-most red angle

The diagram was made with GeoGebra.

The red angles are given.
The blue angle is computed like this:
(angle DAC)+(angle CAB) = 180
angle CAB = 180 - (angle DAC)
angle CAB = 180 - (129°43’)
angle CAB = 179°60’ - (129°43’)
angle CAB = (179°-129°)+(60’-43’)
angle CAB = 50°17’

Then notice how
(angle CAB)+(angle CBA) = (50°17’)+(39°43’)
(angle CAB)+(angle CBA) = (50°+39°)+(17’+43’)
(angle CAB)+(angle CBA) = 89°60’
(angle CAB)+(angle CBA) = 90°
Which shows that triangle ABC is a right triangle. The 90 degree angle is located at point C.

This allows us to use the trig ratio sine to say
sin(angle) = opposite/hypotenuse
sin(angle ABC) = AC/AB
sin(39°43’) = 3742/x
sin(39+43/60) = 3742/x
sin(39.716667) = 3742/x

I'll let you finish up.
Make sure your calculator is in degree mode.