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Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is an essential tool in vector addition, which is why we often use trigonometric functions like sine and cosine when breaking down vectors into Cartesian components.

In the given exercise, you walked at certain angles from a specified direction. To determine how far you've walked in each horizontal (x) and vertical (y) direction, we need trigonometry. When you walk 35 km at a 25-degree angle, you can split this movement into two parts: one along the east-west direction and the other along the north-south direction.
The cosine function helps determine the east-west component, while the sine function helps with the north-south component.

• The equation for the x-component using cosine is: \(x = ext{distance} \times \cos( ext{angle})\)
• The equation for the y-component using sine is: \(y = ext{distance} \times \sin( ext{angle})\)

We used these trigonometric functions for both the 35 km and 22 km walks, thus allowing us to later calculate the resultant displacement.

Cartesian components refer to how we express a vector in terms of its horizontal (x) and vertical (y) components, especially when dealing with vectors on an xy-plane. This is very useful in simplifying calculations involved in vector addition.

In the exercise, each walk was not in a straight line due north or east, so we need to convert them. Using trigonometry, we convert the magnitude of a vector and the angle into x and y components using the wand direction to calculate the components. By breaking each segment of your walk into x and y components:

• The first walk is split into:
  • \(x_1 = 35 \cos(25^{\circ})\)
  • \(y_1 = 35 \sin(25^{\circ})\)
• The second walk is:
  • \(x_2 = 22 \cos(230^{\circ})\)
  • \(y_2 = 22 \sin(230^{\circ})\)

Once these components are calculated, adding them together will provide the total x and y components, which can then be converted back into a single resultant vector. This method simplifies finding the overall direction and length of displacement. Cartesian components provide a clearer understanding of complex movement patterns.

Resultant displacement represents the net distance and direction you end up at from your starting point after moving along different paths. It's what you get when you add two or more vectors together. After breaking the vectors into Cartesian components, you can find the total displacement.

The step-by-step solution combines the Cartesian components to find the resultant displacement.
The displacement's magnitude, which is the straight-line directed path from your starting position to your final position, is calculated using the Pythagorean theorem:

• \(R = \sqrt{x_{\text{total}}^2 + y_{\text{total}}^2}\)

In this case, it equals approximately 17.71 km. Resultant displacement not only considers how far you've traveled overall but also the exact direction you're facing relative to a standard direction, like east.
Finally, the overall direction of the resultant displacement is found using the inverse tangent function, giving the angle relative to the specified axis, which rounds to 353 degrees. This means you're almost headed directly south-east relative to the starting point. Resultant displacement provides a comprehensive understanding of your overall movement after a series of directional changes.