Determine The Height Of A Telephone Pole Using Trigonometry

by ADMIN 60 views

Have you ever wondered how to determine the height of a towering structure without physically climbing it? Trigonometry, the branch of mathematics that deals with the relationships between the sides and angles of triangles, provides us with the tools to solve such real-world problems. In this article, we'll embark on a journey to uncover the height of a telephone pole, utilizing the principles of trigonometry and the concept of the angle of elevation.

The Scenario A Geometric Puzzle

Imagine a person standing precisely 36 feet away from the base of a telephone pole. As the person gazes upwards towards the top of the pole, their line of sight forms an angle with the horizontal ground. This angle, known as the angle of elevation, is measured to be 30 degrees. Our mission, should we choose to accept it, is to determine the height of the telephone pole, employing our knowledge of trigonometry.

Setting the Stage Right Triangles and Trigonometric Ratios

To tackle this problem effectively, we need to visualize the scenario as a right triangle. The telephone pole forms the vertical side of the triangle, the distance between the person and the pole represents the horizontal side, and the line of sight from the person to the top of the pole acts as the hypotenuse. This right triangle provides the perfect framework for applying trigonometric ratios.

Trigonometric ratios, such as sine, cosine, and tangent, establish relationships between the angles and sides of a right triangle. In our case, the tangent ratio proves to be the most useful. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. In our scenario, the opposite side corresponds to the height of the telephone pole (which we aim to find), and the adjacent side is the distance between the person and the pole (36 feet).

The Tangent Tango Applying the Trigonometric Ratio

Let's denote the height of the telephone pole as 'h'. Using the tangent ratio, we can write the following equation:

tan(30°) = h / 36

Now, we need to determine the value of tan(30°). From our knowledge of trigonometric values for special angles, we know that tan(30°) = 1/√3. Substituting this value into the equation, we get:

1/√3 = h / 36

To solve for 'h', we can multiply both sides of the equation by 36:

h = 36 / √3

Rationalizing the Denominator A Touch of Elegance

While the expression h = 36 / √3 provides the correct answer, it's customary to rationalize the denominator, eliminating the radical from the denominator. To do this, we multiply both the numerator and denominator by √3:

h = (36 * √3) / (√3 * √3)

Simplifying the expression, we get:

h = (36 * √3) / 3

Further simplification yields:

h = 12√3

Therefore, the height of the telephone pole is 12√3 feet, which corresponds to option B.

A Closer Look at the Solution

Our journey to determine the height of the telephone pole highlights the power and elegance of trigonometry. By applying the concept of the angle of elevation and the tangent ratio, we were able to solve a real-world problem without resorting to physical measurements. The solution, 12√3 feet, represents the exact height of the pole, expressed in a concise and mathematically sound manner.

It's worth noting that the angle of elevation plays a crucial role in this problem. A larger angle of elevation would imply a greater height for the pole, while a smaller angle would suggest a shorter pole. The distance between the person and the pole also influences the calculated height. A greater distance would necessitate a taller pole to achieve the same angle of elevation.

Alternative Approaches Exploring Different Perspectives

While we successfully employed the tangent ratio to solve this problem, it's insightful to consider alternative approaches. We could have used the sine or cosine ratio if we knew the length of the hypotenuse (the line of sight). However, since the hypotenuse was not provided, the tangent ratio proved to be the most direct route to the solution.

Another approach involves recognizing the 30-60-90 right triangle formed in the scenario. In a 30-60-90 triangle, the sides are in a specific ratio: 1 : √3 : 2. The side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is √3 times the length of the side opposite the 30-degree angle. By applying these ratios, we could have arrived at the same solution, 12√3 feet, albeit through a slightly different path.

Real-World Applications Trigonometry in Action

The problem we've tackled, determining the height of a telephone pole, is just one example of the myriad applications of trigonometry in the real world. Trigonometry plays a vital role in various fields, including:

  • Surveying: Surveyors use trigonometric principles to measure distances, angles, and elevations, creating accurate maps and land surveys.
  • Navigation: Navigators rely on trigonometry to determine the position and course of ships and aircraft, ensuring safe and efficient travel.
  • Engineering: Engineers employ trigonometry in the design and construction of bridges, buildings, and other structures, ensuring stability and structural integrity.
  • Astronomy: Astronomers utilize trigonometry to measure the distances to stars and planets, unraveling the mysteries of the cosmos.
  • Physics: Physicists use trigonometric functions to describe the motion of waves, projectiles, and other physical phenomena.

These are just a few examples of how trigonometry permeates our world, enabling us to solve complex problems and understand the relationships between angles, distances, and heights.

Conclusion A Triumph of Trigonometry

In this article, we successfully determined the height of a telephone pole using the principles of trigonometry. By visualizing the scenario as a right triangle, applying the tangent ratio, and performing some algebraic manipulations, we arrived at the solution: 12√3 feet. This exercise demonstrates the power of trigonometry in solving real-world problems and highlights its importance in various fields.

As we conclude our trigonometric expedition, let's remember that mathematics is not just a collection of abstract formulas and equations. It's a powerful tool that enables us to understand the world around us, solve problems, and make informed decisions. By embracing the principles of trigonometry, we can unlock a deeper understanding of the relationships between angles, distances, and heights, paving the way for new discoveries and innovations.

FAQs About Trigonometry

  1. What is trigonometry? Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It is particularly useful for solving problems involving right triangles and angles of elevation and depression.

  2. What are the basic trigonometric ratios? The three basic trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). These ratios relate the angles of a right triangle to the lengths of its sides. Sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.

  3. How is trigonometry used in real life? Trigonometry has numerous real-life applications, including surveying, navigation, engineering, astronomy, and physics. It is used to measure distances, angles, and elevations, and to design and construct structures. Trigonometry is essential in fields such as architecture, where precise measurements and angles are crucial for creating stable and aesthetically pleasing buildings.

  4. What is the angle of elevation? The angle of elevation is the angle formed between the horizontal line and the line of sight when an observer looks upward to a point above. It is commonly used in trigonometry to solve problems involving heights and distances. For instance, determining the height of a building or a tree can be easily done using the angle of elevation and trigonometric ratios.

  5. What is the angle of depression? The angle of depression is the angle formed between the horizontal line and the line of sight when an observer looks downward to a point below. It is the counterpart to the angle of elevation and is used in similar trigonometric problems where the observer is looking down from a height.

  6. How do you solve a trigonometric problem? To solve a trigonometric problem, start by drawing a diagram to visualize the situation. Identify the given information, such as angles and side lengths. Determine which trigonometric ratio (sine, cosine, or tangent) is appropriate for the problem based on the given and required information. Set up an equation using the trigonometric ratio and solve for the unknown. Always double-check your answer to ensure it makes sense in the context of the problem.

  7. What are special right triangles and why are they important? Special right triangles are triangles with specific angle measures that allow for easy calculation of side lengths. The two most common special right triangles are the 30-60-90 triangle and the 45-45-90 triangle. These triangles have side length ratios that can be memorized or easily derived, making them useful in simplifying trigonometric problems. For example, in a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2, which helps in finding unknown side lengths quickly.

  8. How can a calculator help in solving trigonometric problems? A calculator can be a valuable tool for solving trigonometric problems, especially when dealing with angles that are not standard angles (such as 30, 45, and 60 degrees). Calculators can quickly compute trigonometric functions (sine, cosine, tangent, and their inverses) for any angle measure. Make sure your calculator is set to the correct mode (degrees or radians) before performing calculations.

  9. What are some common mistakes to avoid when solving trigonometric problems? Common mistakes include using the wrong trigonometric ratio, incorrectly labeling the sides of a triangle, and not setting the calculator to the correct mode (degrees or radians). Always double-check your work and ensure that your answer makes sense in the context of the problem. Another common mistake is not drawing a clear diagram, which can lead to misidentification of angles and sides.

  10. Where can I learn more about trigonometry? You can learn more about trigonometry through textbooks, online courses, educational websites, and tutoring services. Khan Academy, Coursera, and other online platforms offer comprehensive courses on trigonometry. Additionally, math textbooks and study guides often provide detailed explanations and practice problems. Engaging with these resources will help you build a solid foundation in trigonometry and its applications.