see the diagram.
then look below for explanations.
the points are labeled as:
barter island = A
crater atoll = B
dream island = C
the bearing from A to C is 60 degrees.
that would be from barter island to crater atoll.
the bearing from C to B is 140 degrees.
that would be from crater atoll to dream island.
the lines AC and CB are the route she took.
the line AB is the route she should have taken.
draw a vertical line through point C.
since the bearing from C to B is 140, then the supplementary angle is 40 degrees as shown on the diagram.
draw a vertical line through point A.
the line AC forms a transversal of the vertical lines through point A and point C.
since both these lines are vertical, they are also parallel.
the 60 degrees shown at point A and the 60 degrees shown at point C are alternate interior angles of those parallel lines.
this makes then equal to each other.
within triangle ACB, you have just found the value of the angle at point C of the triangle.
that would be angle ACB which is equal to 100 degrees.
since you know the distances between A and C, and C and B, and you know the value of the included angle C of the triangle, you can use the law of cosines to find the distance between A and B.
using the law of cosines, you will find that the distance from A to B is equal to 125.23 kilometers.
that's the distance she should have flown.
the distance she actually flew is 160 km.
therefore she had to fly 160 - 125.23 = 34.77 kilometers out of her way.
now that you know the value of the line AB, you can use the law of sines or the law of cosines to find the value of angles A and B of the triangle.
the information shown on the diagram is all the information you needed to solve this problem as far as i can see.
the rest of the information was superfluous as far as i can tell.
the key was finding the angle at point C of the triangle.
from there, the distance between point A and B could be found using the law of cosines.
once you got that, the law of cosines or sines could be used to find the rest of the angles of the triangle.
this, i believe, is the solution to this problem.
here's a reference that i found useful.
https://cosmolearning.org/video-lectures/trigonometry-bearing-problems-4-examples/
here's another one.
https://www.youtube.com/watch?v=8rGa18-SJhA
there's lots more on the web.
just do a search for trigonometry bearing type problems and they should pop up.