Answer
The distance between the lighthouses is 5856.1 meters
Work Step by Step
Let $C$ be the location of the ship. The points A, B, and C form a triangle.
The angle $A = 180^{\circ}-129^{\circ}43' = 50^{\circ}17'$
The angle $B = 39^{\circ}43'$
The angle $C = 180^{\circ}- 50^{\circ}17'-39^{\circ}43' = 90^{\circ}$
Let $b = 3742~m$ and let $c$ be the side opposite the angle $C$. Note that $c$ is the distance between the two lighthouses.
We can use the law of sines to find $c$:
$\frac{c}{sin~C} = \frac{b}{sin~B}$
$c = \frac{b~sin~C}{sin~B}$
$c = \frac{(3742~m)~sin~90^{\circ}}{sin~39^{\circ}43'}$
$c = 5856.1~m$
The distance between the lighthouses is 5856.1 meters
