Help me, please!!
The bearing of a lighthouse sighted by the navigator of a ship was found to be 118°15′. After the
ship traveled 12 km on a course of 25°10′, the navigator found the bearing of the lighthouse to be
160°.
a. Find the distance between the ship and the lighthouse at the time of the first sighting.
b. Find the distance between the ship and the lighthouse at the time of the second sighting.

You want the distances from a ship to a lighthouse, given the bearing of the lighthouse is initially 118°15', and is 160° after the ship moves 12 km on a bearing of 25°10'.

Triangle

The attachment shows a representation of the geometry of the problem. The internal angles of the triangle are ...

A = 118°15' -25°10' = 93°5'

B = 180° -(160° -25°10') = 20° +25°10' = 45°10'

C = 160° -118°15' = 41°45'

Law of sines

We can use the law of sines to find the desired distances.

The distance from the first sighting is found from ...

AC/sin(B) = AB/sin(C)

AC = AB·sin(B)/sin(C) = 12·sin(45°10')/sin(41°45') ≈ 12·0.709161/0.665882

AC ≈ 12.7799 ≈ 12.78 . . . . km

a. The distance to the lighthouse at first sighting is about 12.78 km.

The distance from the second sighting is found from ...

BC/sin(A) = AB/sin(C)

BC = AB·sin(A)/sin(C) = 12·sin(93°5')/sin(41°45') ≈ 12·0.998552/0.665882

BC ≈ 17.9951 ≈ 18.00 . . . . km

b. The distance to the lighthouse at second sighting is about 18.00 km.