Using a Trigonometric Table

Oct 13, 2025 by miodeya.com 57 views

Introduction

In this article, we will delve into the world of trigonometry and explore how to calculate the angle at which the sun is shining on a building given its height and the length of its shadow. This problem is a classic example of a right-angled triangle, where the building's height is the opposite side, the shadow's length is the adjacent side, and the angle we want to find is the angle between the building and the ground.

Understanding the Problem

To solve this problem, we need to use the concept of trigonometric ratios, specifically the tangent function. The tangent function 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 building's height (62 feet), and the adjacent side is the shadow's length (21 feet).

Calculating the Angle

To calculate the angle, we can use the inverse tangent function (arctangent) to find the angle whose tangent is equal to the ratio of the opposite side to the adjacent side. In this case, we want to find the angle whose tangent is equal to 62/21.

Using a Calculator

We can use a calculator to find the angle by entering the following values:

Opposite side: 62 feet
Adjacent side: 21 feet

Using a calculator, we get:

tan(θ) = 62/21 ≈ 2.95

To find the angle, we can use the inverse tangent function:

θ = arctan(2.95) ≈ 71.57°

Using a Trigonometric Table

Alternatively, we can use a trigonometric table to find the angle. A trigonometric table is a table that lists the values of trigonometric functions for various angles. In this case, we can look up the value of tan(71.57°) in the table and find that it is approximately equal to 2.95.

Using a Formula

We can also use a formula to find the angle. The formula for the tangent function is:

tan(θ) = opposite side / adjacent side

We can rearrange this formula to solve for θ:

θ = arctan(opposite side / adjacent side)

Substituting the values, we get:

θ = arctan(62/21) ≈ 71.57°

Conclusion

In this article, we have shown how to calculate the angle at which the sun is shining on a building given its height and the length of its shadow. We have used the concept of trigonometric ratios, specifically the tangent function, to find the angle. We have also used a calculator, a trigonometric table, and a formula to find the angle. The angle is approximately 71.57°.

Applications

This problem has many real-world applications, such as:

Architecture: Architects use trigonometry to design buildings and ensure that they are structurally sound.
Surveying: Surveyors use trigonometry to measure the angles and distances between landmarks and buildings.
Physics: Physicists use trigonometry to calculate the angles and distances between objects in motion.

Tips and Tricks

Use a calculator: A calculator can save you time and effort when calculating trigonometric functions.
Use a trigonometric table: A trigonometric table can be a useful reference when calculating trigonometric functions.
Use a formula: A formula can be a useful tool when calculating trigonometric functions.

References

Mathematics: Mathematics is the study of numbers, quantities, and shapes.
Trigonometry: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.
Right-angled triangles: Right-angled triangles are triangles with one right angle (90°).

Glossary

Tangent function: The tangent function is a trigonometric function that is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle.
Inverse tangent function: The inverse tangent function is a function that returns the angle whose tangent is equal to a given value.
Right-angled triangle: A right-angled triangle is a triangle with one right angle (90°).

Introduction

In our previous article, we explored how to calculate the angle at which the sun is shining on a building given its height and the length of its shadow. In this article, we will answer some frequently asked questions related to this problem.

Q&A

Q: What is the formula to calculate the angle of the sun?

A: The formula to calculate the angle of the sun is:

θ = arctan(opposite side / adjacent side)

Where θ is the angle, opposite side is the height of the building, and adjacent side is the length of the shadow.

Q: What is the opposite side and adjacent side in this problem?

A: In this problem, the opposite side is the height of the building (62 feet), and the adjacent side is the length of the shadow (21 feet).

Q: Can I use a calculator to find the angle?

A: Yes, you can use a calculator to find the angle. Simply enter the values of the opposite side and adjacent side, and the calculator will give you the angle.

Q: Can I use a trigonometric table to find the angle?

A: Yes, you can use a trigonometric table to find the angle. Look up the value of tan(71.57°) in the table, and you will find that it is approximately equal to 2.95.

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

A: The angle of the sun in this problem is approximately 71.57°.

Q: Can I use this formula to calculate the angle of the sun for any building?

A: Yes, you can use this formula to calculate the angle of the sun for any building, as long as you know the height of the building and the length of its shadow.

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

A: Some real-world applications of this problem include:

Architecture: Architects use trigonometry to design buildings and ensure that they are structurally sound.
Surveying: Surveyors use trigonometry to measure the angles and distances between landmarks and buildings.
Physics: Physicists use trigonometry to calculate the angles and distances between objects in motion.

Q: What are some tips and tricks for solving this problem?

A: Some tips and tricks for solving this problem include:

Use a calculator: A calculator can save you time and effort when calculating trigonometric functions.
Use a trigonometric table: A trigonometric table can be a useful reference when calculating trigonometric functions.
Use a formula: A formula can be a useful tool when calculating trigonometric functions.

Conclusion

In this article, we have answered some frequently asked questions related to the problem of calculating the angle of the sun given the height of a building and the length of its shadow. We have also provided some tips and tricks for solving this problem.

References

Mathematics: Mathematics is the study of numbers, quantities, and shapes.
Trigonometry: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.
Right-angled triangles: Right-angled triangles are triangles with one right angle (90°).

Glossary

Tangent function: The tangent function is a trigonometric function that is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle.
Inverse tangent function: The inverse tangent function is a function that returns the angle whose tangent is equal to a given value.
Right-angled triangle: A right-angled triangle is a triangle with one right angle (90°).