Precalculus Question (Law of Sines)?

Question

Tom Hanks, the navigator of a ship at sea, spots two lighthouses that he knows to be 2 miles apart along a straight shoreline. He determines that the angles formed between two line-of-sight observations of the lighthouse and the line from the ship directly to shore are 12º and 30º.
(a) How far is the ship from lighthouse A?
(b) How far is the ship from lighthouse B?
(c) How far is the ship from shore?

I'll try to draw a little illustration... \ is 12º, | is ship from shore straight away, and / is 30º
A -------2miles--------B
\| /
\| /
\| /
\| /
\ | /
\ | /
\12º | 30º/
\ BOAT /

This answer was correct, but it doesn't logically make sense to me.
sin 42º = sin 60º
2 miles x
x = 2.588 miles (a)

then I found b = 2.923 miles
and c = 2.534 miles
The #'s that I'm using don't logically match up with the picture though...

Well A, and B, make more sense than C to me... what am I supposed to do for C?

Answer 1

yes it does.

the three angles of the triangle are 78, 42 and 60
so sin(42)/2 = sin(60)/b and sin(78)/a