Calculating the Distance Traveled Using Trigonometry: A Travel Example

Trigonometry is a powerful tool for solving real-world problems, particularly those involving distances and angles. In this article, we will use the Trigonometric Law of Cosines to determine the final distance of a traveler from their starting point. Let's explore a scenario where a person travels 1 km due north and then 1 km at an angle of 30° east of north.

Problem Statement

A person travels 1 km due north from their starting point and subsequently travels 1 km in a direction that is 30° east of north. We need to determine the distance from the starting point to the current position of the person.

Plan and Solution

To solve this problem, we will use the Trigonometric Law of Cosines. Given the angle between the lines is 150°, we can apply the formula:

c^2 a^2 b^2 - 2ab CosC

where:

a 1 km b 1 km C 150°

Let's solve for c:

c^2 1^2 1^2 - 2(1)(1) Cos150°

Since Cos150° -Cos30° -√3/2

c^2 1 1 - 2(1)(1)(-√3/2)

c^2 2 √3

c^2 ≈ 3.732

c ≈ √3.732 ≈ 1.93 km

The distance from the starting point is approximately 1.93 km.

Alternative Solutions

Another approach involves using a 30/60/90 degree triangle:

Given the second leg of the journey is 8 km at 30° east of north, we can break this into its components:

Adj 8 * cos(30°) 8 * (√3/2) 4√3 km Opp 8 * sin(30°) 8 * (1/2) 4 km

The total horizontal and vertical distance traveled are:

Total vertical distance 5 km 4 km 9 km Total horizontal distance 4√3 km

The distance from the starting point can then be calculated using the Pythagorean theorem:

Distance √(9^2 (4√3)^2) √(81 48) √129 ≈ 12.58 km

Using the direction, we can determine that the man is 12.58 km from his starting position, 71.46° North of East, or 18.54° East of North.

Conclusion

Through the application of trigonometric principles, we can effectively calculate distances and angles in various scenarios. Whether using the Law of Cosines or breaking down leg lengths using trigonometry, the underlying principles remain the same, providing a robust method for solving travel-related problems.