I have the answer, and the steps to do it, but I don't understand it.

Here are the steps:

East displacement from Manhattan to Lincoln:
(147km)sin85 + (106km)sin167 + (166km)sin235 = 34.3 km

North displacement:
(147km)cos85 + (106km)cos167 + (166km)cos235 = -185.7

SqRt(34.3^2 + -185.7 ^2) = 189km

Direction relative to north, arctan (34.3/-185.7) = -10.46 = 349.5 degrees
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I think the main thing I don't get is, how did they cut up that information into right triangles so that they could use sin and cos. I don't see it at all. What I would have thought for example the first length, would be was sin(5 degrees ) = x/147km since it is at 85 degrees or 5 degrees above east which is 90 degrees. Anybody have any insight or tips on seeing this?

Just think of the cities as points on a Cartesian x,y plane. The components of each vector showing the displacement from one point to the other are simply the difference in x and y co-ordinates. The angle is simply arctan y/x.