Situations. 5. Sunny Is Standing 12 M From A Vertical Pillar Of Height 6 M. The Angle Of Elevation Of The Top Of The Pillar From Sunny's Eye Level Is 38°​

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Introduction

In various situations, we may need to calculate distances, heights, or angles using trigonometry. These calculations can be crucial in fields such as architecture, engineering, physics, and more. In this article, we will explore a real-life scenario where we need to calculate the distance from a person to a vertical pillar using the angle of elevation.

The Problem

Sunny is standing 12 m from a vertical pillar of height 6 m. The angle of elevation of the top of the pillar from Sunny's eye level is 38°. We need to calculate the distance from Sunny's eye level to the top of the pillar.

Using Trigonometry to Solve the Problem

To solve this problem, we can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. In this case, the opposite side is the height of the pillar (6 m), and the adjacent side is the distance from Sunny's eye level to the base of the pillar (12 m).

We can use the tangent function to calculate the angle of elevation:

tan(38°) = opposite side / adjacent side tan(38°) = 6 m / x

where x is the distance from Sunny's eye level to the base of the pillar.

Calculating the Distance

To calculate the distance x, we can rearrange the equation to solve for x:

x = opposite side / tan(38°) x = 6 m / tan(38°)

Using a calculator, we can find the value of tan(38°) and calculate the distance x:

tan(38°) ≈ 0.7813 x ≈ 6 m / 0.7813 x ≈ 7.69 m

Calculating the Distance from Sunny's Eye Level to the Top of the Pillar

Now that we have the distance x, we can calculate the distance from Sunny's eye level to the top of the pillar using the Pythagorean theorem:

distance = √(x² + height²) distance = √(7.69² + 6²) distance = √(58.73 + 36) distance = √94.73 distance ≈ 9.72 m

Conclusion

In this article, we used trigonometry to calculate the distance from Sunny's eye level to the top of a vertical pillar. We first calculated the distance from Sunny's eye level to the base of the pillar using the tangent function, and then used the Pythagorean theorem to calculate the distance from Sunny's eye level to the top of the pillar. This problem demonstrates the importance of trigonometry in real-life scenarios and how it can be used to solve problems involving distances and heights.

Real-World Applications

Trigonometry has numerous real-world applications, including:

  • Architecture: Trigonometry is used to calculate the height of buildings, the distance between buildings, and the angle of elevation of roofs.
  • Engineering: Trigonometry is used to calculate the stress and strain on structures, the distance between objects, and the angle of elevation of bridges.
  • Physics: Trigonometry is used to calculate the distance and velocity of objects, the angle of elevation of projectiles, and the stress and strain on materials.
  • Surveying: Trigonometry is used to calculate the distance and angle between landmarks, the height of buildings, and the angle of elevation of roads.

Tips and Tricks

  • Use a calculator: When working with trigonometric functions, it's often easier to use a calculator to find the values of the functions.
  • Check your units: Make sure to check your units when working with trigonometric functions to ensure that you are using the correct units.
  • Use the Pythagorean theorem: The Pythagorean theorem is a powerful tool for calculating distances and heights in right-angled triangles.
  • Practice, practice, practice: The more you practice working with trigonometric functions, the more comfortable you will become with using them to solve problems.

Common Mistakes

  • Not checking units: Failing to check units can lead to incorrect answers.
  • Not using a calculator: Not using a calculator can make it difficult to find the values of trigonometric functions.
  • Not using the Pythagorean theorem: Not using the Pythagorean theorem can make it difficult to calculate distances and heights in right-angled triangles.
  • Not practicing: Not practicing working with trigonometric functions can make it difficult to solve problems.

Conclusion

In conclusion, trigonometry is a powerful tool for calculating distances and heights in various situations. By using trigonometric functions and the Pythagorean theorem, we can solve problems involving distances and heights. Remember to check your units, use a calculator, and practice working with trigonometric functions to become more comfortable with using them to solve problems.

Introduction

In our previous article, we explored a real-life scenario where we needed to calculate the distance from a person to a vertical pillar using the angle of elevation. We used trigonometry to solve the problem and calculated the distance from the person's eye level to the top of the pillar. In this article, we will answer some common questions related to the problem and provide additional tips and tricks for working with trigonometric functions.

Q&A

Q: What is the angle of elevation in this problem?

A: The angle of elevation is 38°.

Q: What is the height of the pillar in this problem?

A: The height of the pillar is 6 m.

Q: What is the distance from the person's eye level to the base of the pillar?

A: The distance from the person's eye level to the base of the pillar is 7.69 m.

Q: How did you calculate the distance from the person's eye level to the top of the pillar?

A: We used the Pythagorean theorem to calculate the distance from the person's eye level to the top of the pillar. The formula is:

distance = √(x² + height²)

where x is the distance from the person's eye level to the base of the pillar, and height is the height of the pillar.

Q: What is the distance from the person's eye level to the top of the pillar?

A: The distance from the person's eye level to the top of the pillar is approximately 9.72 m.

Q: What are some common mistakes to avoid when working with trigonometric functions?

A: Some common mistakes to avoid when working with trigonometric functions include:

  • Not checking units
  • Not using a calculator
  • Not using the Pythagorean theorem
  • Not practicing working with trigonometric functions

Q: How can I practice working with trigonometric functions?

A: You can practice working with trigonometric functions by:

  • Solving problems involving distances and heights
  • Using online resources and calculators to find the values of trigonometric functions
  • Practicing with different types of triangles, such as right triangles and oblique triangles
  • Using real-world examples and applications to make the concepts more meaningful

Q: What are some real-world applications of trigonometry?

A: Some real-world applications of trigonometry include:

  • Architecture: Trigonometry is used to calculate the height of buildings, the distance between buildings, and the angle of elevation of roofs.
  • Engineering: Trigonometry is used to calculate the stress and strain on structures, the distance between objects, and the angle of elevation of bridges.
  • Physics: Trigonometry is used to calculate the distance and velocity of objects, the angle of elevation of projectiles, and the stress and strain on materials.
  • Surveying: Trigonometry is used to calculate the distance and angle between landmarks, the height of buildings, and the angle of elevation of roads.

Tips and Tricks

  • Use a calculator: When working with trigonometric functions, it's often easier to use a calculator to find the values of the functions.
  • Check your units: Make sure to check your units when working with trigonometric functions to ensure that you are using the correct units.
  • Use the Pythagorean theorem: The Pythagorean theorem is a powerful tool for calculating distances and heights in right-angled triangles.
  • Practice, practice, practice: The more you practice working with trigonometric functions, the more comfortable you will become with using them to solve problems.

Common Mistakes

  • Not checking units: Failing to check units can lead to incorrect answers.
  • Not using a calculator: Not using a calculator can make it difficult to find the values of trigonometric functions.
  • Not using the Pythagorean theorem: Not using the Pythagorean theorem can make it difficult to calculate distances and heights in right-angled triangles.
  • Not practicing: Not practicing working with trigonometric functions can make it difficult to solve problems.

Conclusion

In conclusion, trigonometry is a powerful tool for calculating distances and heights in various situations. By using trigonometric functions and the Pythagorean theorem, we can solve problems involving distances and heights. Remember to check your units, use a calculator, and practice working with trigonometric functions to become more comfortable with using them to solve problems.