Using Trigonometry to Find the Length of the Shadow

Oct 13, 2025 by leolavanderia.com.br 29 views

Introduction

When the sun is at a certain angle above the horizon, it casts a shadow on the ground. The length of the shadow depends on the height of the object casting the shadow and the angle of the sun. In this article, we will explore how to find the length of the shadow cast by a building given the height of the building and the angle of the sun above the horizon.

Understanding the Problem

To solve this problem, we need to use trigonometry, specifically the tangent function. The tangent function relates the angle of the sun above the horizon to the ratio of the height of the building to the length of the shadow. The formula for the tangent function is:

tan(θ) = opposite side / adjacent side

In this case, the opposite side is the height of the building (75 ft), and the adjacent side is the length of the shadow (which we want to find). The angle θ is the angle of the sun above the horizon (58°).

Using Trigonometry to Find the Length of the Shadow

We can use the tangent function to set up an equation to solve for the length of the shadow. We know that:

tan(58°) = 75 ft / x

where x is the length of the shadow. To solve for x, we can multiply both sides of the equation by x:

x * tan(58°) = 75 ft

Next, we can divide both sides of the equation by tan(58°) to isolate x:

x = 75 ft / tan(58°)

Calculating the Length of the Shadow

Now that we have the equation, we can plug in the value of tan(58°) to find the length of the shadow. Using a calculator, we get:

tan(58°) ≈ 1.60

Now we can plug this value into the equation:

x = 75 ft / 1.60 x ≈ 46.88 ft

Rounding the Answer to the Nearest Tenth

The problem asks us to round our answer to the nearest tenth. Therefore, we round 46.88 ft to 46.9 ft.

Conclusion

In this article, we used trigonometry to find the length of the shadow cast by a building given the height of the building and the angle of the sun above the horizon. We set up an equation using the tangent function and solved for the length of the shadow. We then calculated the value of the length of the shadow and rounded it to the nearest tenth.

Real-World Applications

This problem has real-world applications in various fields such as architecture, engineering, and surveying. For example, architects and engineers need to calculate the length of shadows to design buildings and structures that are in harmony with their surroundings. Surveyors use trigonometry to measure distances and angles between objects, which is essential in mapping and navigation.

Tips and Tricks

When using trigonometry to solve problems, make sure to identify the opposite and adjacent sides of the angle.
Use a calculator to find the values of trigonometric functions such as sine, cosine, and tangent.
Always round your answers to the nearest tenth or hundredth, depending on the problem.

Common Mistakes

Failing to identify the opposite and adjacent sides of the angle.
Not using a calculator to find the values of trigonometric functions.
Not rounding the answer to the nearest tenth or hundredth.

Final Thoughts

In conclusion, finding the length of the shadow cast by a building given the height of the building and the angle of the sun above the horizon is a classic problem in trigonometry. By using the tangent function and solving for the length of the shadow, we can find the answer to this problem. This problem has real-world applications in various fields and is an essential concept in mathematics.

Introduction

In our previous article, we explored how to find the length of the shadow cast by a building given the height of the building and the angle of the sun above the horizon. In this article, we will answer some frequently asked questions related to this problem.

Q: What is the formula for finding the length of the shadow?

A: The formula for finding the length of the shadow is:

x = 75 ft / tan(58°)

where x is the length of the shadow.

Q: What is the tangent function?

A: The tangent function is a trigonometric function that relates the angle of the sun above the horizon to the ratio of the height of the building to the length of the shadow. The formula for the tangent function is:

tan(θ) = opposite side / adjacent side

In this case, the opposite side is the height of the building (75 ft), and the adjacent side is the length of the shadow (which we want to find).

Q: How do I use a calculator to find the value of tan(58°)?

A: To find the value of tan(58°) using a calculator, follow these steps:

Press the "tan" button on your calculator.
Enter the angle 58° using the "angle" button.
Press the "=" button to get the value of tan(58°).

Q: What is the value of tan(58°)?

A: The value of tan(58°) is approximately 1.60.

Q: How do I round my answer to the nearest tenth?

A: To round your answer to the nearest tenth, follow these steps:

Look at the hundredth place digit (the second digit after the decimal point).
If the hundredth place digit is 5 or greater, round up the tenth place digit.
If the hundredth place digit is less than 5, round down the tenth place digit.

Q: What is the length of the shadow cast by a building 75 ft tall if the sun is 58° above the horizon?

A: The length of the shadow cast by a building 75 ft tall if the sun is 58° above the horizon is approximately 46.9 ft.

Q: What are some real-world applications of this problem?

A: This problem has real-world applications in various fields such as architecture, engineering, and surveying. For example, architects and engineers need to calculate the length of shadows to design buildings and structures that are in harmony with their surroundings. Surveyors use trigonometry to measure distances and angles between objects, which is essential in mapping and navigation.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

Failing to identify the opposite and adjacent sides of the angle.
Not using a calculator to find the values of trigonometric functions.
Not rounding the answer to the nearest tenth or hundredth.

Q: What is the importance of trigonometry in mathematics?

A: Trigonometry is an essential concept in mathematics that has numerous real-world applications. It is used to solve problems involving right triangles, circles, and waves. Trigonometry is also used in various fields such as physics, engineering, and computer science.

Q: How can I practice solving trigonometry problems?

A: You can practice solving trigonometry problems by:

Using online resources such as Khan Academy and Mathway.
Working on practice problems from your textbook or online resources.
Joining a study group or finding a tutor to help you practice.

Conclusion

In this article, we answered some frequently asked questions related to finding the length of the shadow cast by a building given the height of the building and the angle of the sun above the horizon. We also discussed the importance of trigonometry in mathematics and provided some tips and tricks for practicing trigonometry problems.