Trigonometry in Real Life: A 200m Tower and a 60 Degrees Angle of Elevation
This article explores how basic trigonometric principles can be applied to solve real-world problems. Specifically, we will use trigonometric functions, such as the tangent, to find the distance from a man to the foot of a 200-meter high tower given an angle of elevation of 60 degrees.
The Problem and Solution
A man observes a 200-meter high tower and measures the angle of elevation to the top of the tower as 60 degrees. The question is: what is the distance of the man from the foot of the tower?
Step-by-Step Solution
We can use the tangent function to solve this problem. Remember that the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
Step 1: Identify the Given Values and the Required Distance
From the problem, we have two key pieces of information:
- The height of the tower (opposite side) 200 meters
- The angle of elevation (θ) 60 degrees
The distance of the man from the foot of the tower (adjacent side) is what we need to find. We can use the tangent function for this.
Using Trigonometry to Solve for Distance
Using the definition of the tangent function, we have:
[tan(60^circ) frac{text{opposite}}{text{adjacent}} frac{200}{x}]
Since (tan(60^circ) sqrt{3}), the equation becomes:
[sqrt{3} frac{200}{x}]
Solving for (x), we get:
[x frac{200}{sqrt{3}}]
[x frac{200}{1.732}]
[x ≈ 115.47] meters
Conclusion
To summarize, using the principles of trigonometry, particularly the tangent function, we determined that the distance of the man from the foot of the 200-meter high tower, with an angle of elevation of 60 degrees, is approximately 115.47 meters.
Relevance of Angle of Elevation in Real Life
The concept of angle of elevation is not just theoretical or academic; it is used in various fields such as architecture, engineering, surveying, and even in the military for determining the range of targets. Understanding how to measure and calculate such angles is crucial for many practical applications in these fields.
Further Reading
If you are interested in learning more about trigonometry and its applications, consider exploring additional resources such as textbooks, online courses, or practical guides. Each of these can provide more in-depth explanations and examples to strengthen your understanding of trigonometric concepts and how they apply to real-world scenarios.