91Ó°ÊÓ

In trigonometry, the concept of distance calculation is crucial when analyzing the positions of different points in a plane. The distance formula helps us find the straight-line distance between two points in a coordinate plane. This formula is similar to the Pythagorean theorem and is given by:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.\]

• Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
• The formula calculates the hypotenuse of the right triangle formed by the differences in the x and y coordinates as the two legs.

For example, to find how far George is from John, we calculate the distance between John's starting position at \((0, 0)\) and George's position, derived from transforming Paul's location by the given directions. Understanding how to place these points correctly in a coordinate system is essential for proper computation.

A coordinate system is a framework for specifying positions in a plane through coordinates. The most common system is the Cartesian coordinate system, which uses x and y axes to define locations. The origin \((0, 0)\) is the point where these axes intersect, serving as a reference point.In this exercise, John is positioned at the origin, simplifying calculations as his coordinates are \((0,0)\). Paul is positioned 14 meters west of John, translating to coordinates \((-14, 0)\). Placing individuals in this grid allows for straightforward application of trigonometric functions and distance formulas.Knowing how to effectively place and interpret points in a coordinate system is vital for:

• Executing trigonometric calculations such as determining the direction and distance.
• Understanding the position of each person relative to known reference points.

This knowledge simplifies solving complex spatial problems and helps visualize movements and positions.

Vector analysis involves using vectors to describe and solve problems related to direction and magnitude. In the context of this exercise, we deal with vectors to understand the movement from Paul's position to George's position.Knowing the angle and direction of George relative to Paul (37° south of east at 36.0 m), we use vectors to compute George's precise location. Using vector decomposition, we break the movement into:

• The horizontal component: \(\Delta x = 36.0 \cdot \cos(37°)\).
• The vertical component: \(\Delta y = -36.0 \cdot \sin(37°)\).

These components reflect the physical movements on the coordinate grid. Adding these vector components to Paul's coordinates gives George's exact location. Applying vector analysis helps us find:

• The resultant vector showing overall movement from Paul to George.
• The new coordinates for better distance measurement from John.
• The direction angle \(\theta\), illustrating the bearing of George from John using trigonometric functions like \(\tan^{-1}\).

Understanding vectors simplifies calculations and offers a clear view of directional changes and distances in a plane.