Vector Maps (Trig): Solving Flight Path from Lincoln to Manhattan

[PascalPanther]

Map: http://xs206.xs.to/xs206/06362/map.gif

On a training flight, a student pilot flies from Lincoln, NE to Clarinda, IA, then to St. Joseph, MO, and then to Manhattan, KS. The directions are shown relative to north: 0 degrees is north, 90 degrees is east, 180 degrees is south, and 270 degrees is west. Use the method of components to find:
1. the distance she has to fly from Manhattan to get back to Lincoln
2. the direction (relative to north) she must fly to get there.

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?

 

[Andrew Mason]

Map: http://xs206.xs.to/xs206/06362/map.gif

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.

AM