Determining The Distance Between Linda And Ali Using Trigonometry

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Introduction

In this mathematical exploration, we delve into a classic trigonometry problem involving distances and angles. Imagine Linda and Ali standing on opposite sides of a riverbank, their positions marked as points A and B, respectively. A house sits across the river at point C. We are given that Ali is 170 meters away from the house (BC = 170 meters), the angle at Linda's position (∠BAC) is 44°, and the angle at Ali's position (∠ABC) is 53°. Our mission is to determine the distance between Linda and Ali, which is the length of the side AB of triangle ABC. This problem exemplifies how trigonometric principles, specifically the Law of Sines, can be applied to solve real-world distance calculations. By understanding the relationships between angles and sides in a triangle, we can accurately measure distances that might otherwise be difficult to ascertain directly. This is not just a theoretical exercise; it has practical applications in fields like surveying, navigation, and engineering, where precise distance measurements are crucial.

Trigonometry serves as a powerful tool in solving geometric problems that involve angles and distances. In this particular scenario, we are presented with a triangle formed by Linda's position (A), Ali's position (B), and the location of the house (C). The information provided includes the distance from Ali to the house (BC), as well as the angles formed at Linda's position (∠BAC) and Ali's position (∠ABC). Our goal is to determine the distance between Linda and Ali, which corresponds to the side AB of the triangle. To accomplish this, we will leverage the Law of Sines, a fundamental concept in trigonometry that establishes a relationship between the sides of a triangle and the sines of their opposite angles. The Law of Sines is particularly useful when dealing with triangles that are not right-angled, as it allows us to calculate unknown side lengths or angles based on known information. Before we can apply the Law of Sines, we need to determine all the angles within the triangle. Since the sum of angles in a triangle is always 180°, we can calculate the angle at the house (∠ACB) by subtracting the known angles (∠BAC and ∠ABC) from 180°. Once we have all the angles, we can confidently apply the Law of Sines to find the distance between Linda and Ali.

Understanding the Problem

To effectively solve this problem, a clear understanding of the given information and the desired outcome is essential. Linda and Ali are positioned on a riverbank at points A and B, respectively, while a house stands across the river at point C. The distance between Ali and the house (BC) is 170 meters. The angle formed at Linda's position (∠BAC) is 44°, and the angle formed at Ali's position (∠ABC) is 53°. Our objective is to determine the distance between Linda and Ali, which is the length of the line segment AB. This problem falls under the domain of trigonometry, as it involves the relationships between angles and sides in a triangle. The key to solving this problem lies in recognizing that the given information forms a triangle (△ABC), and we can utilize trigonometric principles to find the unknown side length (AB). Specifically, the Law of Sines will be our primary tool. However, before we can apply the Law of Sines, we need to calculate the third angle of the triangle (∠ACB). This can be done using the fact that the sum of angles in a triangle is always 180°. By having all the angles and one side length, we can then confidently use the Law of Sines to determine the distance between Linda and Ali.

Visualizing the scenario can greatly aid in comprehending the problem. Imagine a triangle formed by Linda (A), Ali (B), and the house (C). The riverbank forms the base of the triangle (AB), with Linda and Ali positioned at its endpoints. The house is located across the river, forming the third vertex of the triangle. We know the distance from Ali to the house (BC), which is one side of the triangle. We also know the angles formed at Linda and Ali's positions (∠BAC and ∠ABC). The distance we want to find is the length of the base of the triangle (AB). By drawing a diagram, we can clearly see the relationships between the sides and angles of the triangle. This visual representation helps us to identify the appropriate trigonometric principles to apply. In this case, the Law of Sines is the most suitable approach, as it relates the sides of a triangle to the sines of their opposite angles. Before we can use the Law of Sines, we need to calculate the missing angle (∠ACB), which can be found by subtracting the known angles from 180°.

Applying the Law of Sines

The Law of Sines is a fundamental trigonometric principle that states the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides. In simpler terms, for any triangle ABC, the following relationship holds true:

a / sin(A) = b / sin(B) = c / sin(C)

where:

  • a, b, and c are the side lengths of the triangle
  • A, B, and C are the angles opposite to sides a, b, and c, respectively

In our problem, we want to find the distance between Linda and Ali, which is the length of side AB (let's call it 'c'). We know the length of side BC (a = 170 meters), angle A (44°), and angle B (53°). To apply the Law of Sines, we first need to find angle C. Since the sum of angles in a triangle is 180°, we can calculate angle C as follows:

C = 180° - A - B
C = 180° - 44° - 53°
C = 83°

Now that we have all three angles and one side length, we can use the Law of Sines to find side c (AB). We can set up the following proportion:

c / sin(C) = a / sin(A)

Substituting the known values:

c / sin(83°) = 170 / sin(44°)

To solve for c, we multiply both sides by sin(83°):

c = (170 * sin(83°)) / sin(44°)

Using a calculator, we find that sin(83°) ≈ 0.9925 and sin(44°) ≈ 0.6947. Plugging these values into the equation:

c ≈ (170 * 0.9925) / 0.6947
c ≈ 242.7 meters

Therefore, the distance between Linda and Ali is approximately 242.7 meters. This calculation demonstrates the power of the Law of Sines in solving triangle problems where we have information about angles and side lengths. By setting up the correct proportions and using trigonometric functions, we can accurately determine unknown distances.

Step-by-Step Solution

To provide a clear and concise solution, let's break down the steps involved in determining the distance between Linda and Ali:

  1. Identify the Given Information: We are given the distance BC = 170 meters, angle A (∠BAC) = 44°, and angle B (∠ABC) = 53°.
  2. Determine the Unknown: We need to find the distance AB, which we'll denote as 'c'.
  3. Calculate Angle C: Since the sum of angles in a triangle is 180°, we can find angle C (∠ACB) using the formula: C = 180° - A - B. Substituting the given values, we get C = 180° - 44° - 53° = 83°.
  4. Apply the Law of Sines: The Law of Sines states that a / sin(A) = b / sin(B) = c / sin(C). We can use the proportion c / sin(C) = a / sin(A) to find 'c'.
  5. Substitute Values: Plug in the known values into the proportion: c / sin(83°) = 170 / sin(44°).
  6. Solve for 'c': Multiply both sides of the equation by sin(83°) to isolate 'c': c = (170 * sin(83°)) / sin(44°).
  7. Calculate Sine Values: Use a calculator to find the sine of 83° and 44°. We get sin(83°) ≈ 0.9925 and sin(44°) ≈ 0.6947.
  8. Compute 'c': Substitute the sine values into the equation: c ≈ (170 * 0.9925) / 0.6947.
  9. Final Calculation: Perform the calculation to find the approximate value of 'c': c ≈ 242.7 meters.

Therefore, the distance between Linda and Ali is approximately 242.7 meters. This step-by-step solution provides a clear roadmap for solving the problem, highlighting the key concepts and calculations involved. By following these steps, anyone can confidently tackle similar trigonometry problems involving triangles and the Law of Sines.

Practical Applications

The problem we've solved, determining the distance between two points using angles and one known distance, has numerous practical applications in various fields. This isn't just a theoretical exercise; it's a technique used in real-world scenarios where direct measurement is difficult or impossible. One of the most common applications is in surveying. Surveyors use trigonometry to measure land, create maps, and establish property boundaries. They might need to determine the distance between two points across a river, a ravine, or any other obstacle. By setting up a triangle and measuring angles and one side length, they can accurately calculate the desired distance using the principles we've discussed. Similarly, in navigation, sailors and pilots use trigonometry to determine their position and the distance to their destination. They might use landmarks or celestial bodies to create triangles and calculate distances and bearings. This is particularly important in situations where GPS or other electronic navigation systems are unavailable.

Engineering also relies heavily on trigonometry for various tasks. When building bridges, tunnels, or other structures, engineers need to make precise measurements. They might use triangulation techniques to calculate distances and elevations, ensuring that the structure is built correctly and safely. In forestry, foresters use trigonometry to estimate the height of trees and the area of forests. This information is crucial for managing timber resources and assessing forest health. By measuring angles and distances, they can calculate tree heights without having to climb each tree. Even in astronomy, the principles of trigonometry are used to measure the distances to stars and other celestial objects. By observing the apparent shift in a star's position as the Earth orbits the Sun (a phenomenon called parallax), astronomers can create triangles and calculate the star's distance. These examples demonstrate the broad applicability of trigonometry in solving real-world problems. The ability to calculate distances using angles and one known side length is a valuable skill in many professions and contributes to our understanding of the world around us. The Law of Sines, which we used to solve our problem, is a powerful tool in this regard.

Conclusion

In conclusion, we successfully determined the distance between Linda and Ali, who were standing on a riverbank, by applying the Law of Sines. Given the distance from Ali to a house across the river (170 meters) and the angles formed at Linda's and Ali's positions (44° and 53°, respectively), we calculated the distance between them to be approximately 242.7 meters. This problem exemplifies the power of trigonometry in solving real-world distance calculations. By understanding the relationships between angles and sides in a triangle, we can accurately measure distances that might otherwise be difficult to ascertain directly. The Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant, was instrumental in our solution. We first calculated the third angle of the triangle using the fact that the sum of angles in a triangle is 180°. Then, we applied the Law of Sines to set up a proportion and solve for the unknown distance. This approach highlights the importance of visualizing the problem, identifying the relevant trigonometric principles, and applying them systematically.

The practical applications of this type of problem extend far beyond the classroom. Surveyors, navigators, engineers, foresters, and even astronomers rely on trigonometry to make accurate measurements and calculations. Whether it's determining property boundaries, navigating ships or airplanes, constructing bridges, estimating timber resources, or measuring the distances to stars, the principles of trigonometry are essential tools. The ability to calculate distances using angles and one known side length is a valuable skill in many professions and contributes to our understanding of the world around us. This problem serves as a reminder that mathematics is not just an abstract subject; it has practical relevance and can be used to solve real-world challenges. By mastering trigonometric concepts like the Law of Sines, we can gain a deeper appreciation for the power and versatility of mathematics. The solution we've presented provides a clear and concise roadmap for tackling similar problems, emphasizing the importance of careful analysis, accurate calculations, and a solid understanding of trigonometric principles.