A Ladder Leans Against The Side Of A Building, Making A $70^{\circ}$ Angle With The Ground. If The Base Of The Ladder Is A Certain Distance From The Building, Use The Following Trigonometric Values:$\[ \begin{align*} \sin 70^{\circ}

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a powerful tool for solving problems in various fields, including physics, engineering, and architecture. In this article, we will use trigonometry to find the distance of the base of a ladder from a building, given that the ladder makes a $70^{\circ}$ angle with the ground.

The Problem

A ladder leans against the side of a building, making a $70^{\circ}$ angle with the ground. We are given that the height of the ladder against the building is 7 meters. We need to find the distance of the base of the ladder from the building.

Using Trigonometry to Solve the Problem

To solve this problem, we can use the sine function, which is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, the angle is $70^{\circ}$, and the side opposite the angle is the height of the ladder against the building, which is 7 meters.

We can write the sine function as:

$$\sin 70^{\circ} = \frac{7}{x}$$

where $x$ is the distance of the base of the ladder from the building.

Solving for x

To solve for $x$, we can rearrange the equation to isolate $x$:

$$x = \frac{7}{\sin 70^{\circ}}$$

Now, we can use a calculator to find the value of $\sin 70^{\circ}$, which is approximately 0.9397.

Substituting this value into the equation, we get:

$$x = \frac{7}{0.9397}$$

Simplifying the expression, we get:

$$x = 7.44$$

Therefore, the distance of the base of the ladder from the building is approximately 7.44 meters.

Discussion

In this article, we used trigonometry to find the distance of the base of a ladder from a building, given that the ladder makes a $70^{\circ}$ angle with the ground. We used the sine function to relate the angle, the height of the ladder, and the distance of the base of the ladder from the building.

This problem is a classic example of how trigonometry can be used to solve real-world problems. By using the sine function, we were able to find the distance of the base of the ladder from the building, which is a critical piece of information in many architectural and engineering applications.

Conclusion

In conclusion, trigonometry is a powerful tool for solving problems in various fields. By using the sine function, we can relate the angle, the height of the ladder, and the distance of the base of the ladder from the building. This problem is a classic example of how trigonometry can be used to solve real-world problems.

Trigonometric Values

The following trigonometric values are given:

Angle	Sine	Cosine	Tangent
$70^{\circ}$	0.9397	0.3420	2.744

Table of Contents

1. Introduction
2. The Problem
3. Using Trigonometry to Solve the Problem
4. Solving for x
5. Discussion
6. Conclusion
7. Trigonometric Values
8. Table of Contents

References

• [1] "Trigonometry" by Michael Corral, 2019.
• [2] "Mathematics for Engineers and Scientists" by Donald R. Hill, 2018.

Q&A: A Ladder Leans Against the Side of a Building

Q: What is the problem about?

A: The problem is about finding the distance of the base of a ladder from a building, given that the ladder makes a $70^{\circ}$ angle with the ground.

Q: What trigonometric function is used to solve the problem?

A: The sine function is used to solve the problem.

Q: What is the sine function?

A: The sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

Q: How is the sine function used to solve the problem?

A: The sine function is used to relate the angle, the height of the ladder, and the distance of the base of the ladder from the building.

Q: What is the height of the ladder against the building?

A: The height of the ladder against the building is 7 meters.

Q: What is the distance of the base of the ladder from the building?

A: The distance of the base of the ladder from the building is approximately 7.44 meters.

Q: How is the distance of the base of the ladder from the building calculated?

A: The distance of the base of the ladder from the building is calculated using the sine function:

$$x = \frac{7}{\sin 70^{\circ}}$$

Q: What is the value of $\sin 70^{\circ}$?

A: The value of $\sin 70^{\circ}$ is approximately 0.9397.

Q: How is the value of $\sin 70^{\circ}$ used to calculate the distance of the base of the ladder from the building?

A: The value of $\sin 70^{\circ}$ is used to calculate the distance of the base of the ladder from the building by substituting it into the equation:

$$x = \frac{7}{0.9397}$$

Q: What is the final answer?

A: The final answer is that the distance of the base of the ladder from the building is approximately 7.44 meters.

Frequently Asked Questions

• Q: What is the angle between the ladder and the ground?
  • A: The angle between the ladder and the ground is $70^{\circ}$.
• Q: What is the height of the ladder against the building?
  • A: The height of the ladder against the building is 7 meters.
• Q: What is the distance of the base of the ladder from the building?
  • A: The distance of the base of the ladder from the building is approximately 7.44 meters.

Conclusion

In this article, we used trigonometry to find the distance of the base of a ladder from a building, given that the ladder makes a $70^{\circ}$ angle with the ground. We used the sine function to relate the angle, the height of the ladder, and the distance of the base of the ladder from the building. The final answer is that the distance of the base of the ladder from the building is approximately 7.44 meters.

Trigonometric Values

The following trigonometric values are given:

Angle	Sine	Cosine	Tangent
$70^{\circ}$	0.9397	0.3420	2.744

Table of Contents

1. Introduction
2. The Problem
3. Using Trigonometry to Solve the Problem
4. Solving for x
5. Discussion
6. Conclusion
7. Trigonometric Values
8. Table of Contents

References

• [1] "Trigonometry" by Michael Corral, 2019.
• [2] "Mathematics for Engineers and Scientists" by Donald R. Hill, 2018.

Note: The references provided are for example purposes only and may not be actual references used in the article.