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

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The discussion focuses on calculating the flight path from Manhattan back to Lincoln using vector components. The student pilot's journey involves displacements calculated using sine and cosine functions to determine east and north components. The total distance back to Lincoln is found to be 189 km, with a direction of approximately 349.5 degrees relative to north. A key point of confusion is how to visualize the problem in terms of right triangles and Cartesian coordinates. Understanding the displacement as differences in x and y coordinates simplifies the process of applying trigonometric functions.
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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?
 
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PascalPanther said:
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
 
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