Solving for the Height of the Lighthouse

Oct 13, 2025 by miodeya.com 52 views

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will explore how trigonometry can be used to solve real-life problems, specifically finding the height of a lighthouse using the angle of elevation and the distance from the boat to the lighthouse.

Understanding the Problem

The problem states that the distance from the boat to the lighthouse is 115 meters, and the angle of elevation is $42^{\circ}$ . We are asked to find the height of the lighthouse using one of the given equations. To approach this problem, we need to understand the concept of trigonometric ratios and how they can be used to solve for unknown sides of a triangle.

Trigonometric Ratios

Trigonometric ratios are used to describe the relationships between the sides and angles of a triangle. The three basic trigonometric ratios are:

Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

In this problem, we are given the angle of elevation and the distance from the boat to the lighthouse. We can use the tangent ratio to solve for the height of the lighthouse.

Using the Tangent Ratio

The tangent ratio is defined as:

\tan \theta = \frac{\text{opposite}}{\text{adjacent}}

In this case, the angle of elevation is $42^{\circ}$ , and the distance from the boat to the lighthouse is 115 meters. We can use the tangent ratio to solve for the height of the lighthouse (opposite side).

\tan 42^{\circ} = \frac{y}{115}

where $y$ is the height of the lighthouse.

Solving for the Height of the Lighthouse

To solve for the height of the lighthouse, we can rearrange the equation to isolate $y$ .

y = 115 \times \tan 42^{\circ}

Using a calculator, we can find the value of $\tan 42^{\circ}$ and multiply it by 115 to find the height of the lighthouse.

y = 115 \times 0.9004 = 103.846

Therefore, the height of the lighthouse is approximately 103.85 meters.

Conclusion

In this article, we used trigonometry to solve for the height of a lighthouse using the angle of elevation and the distance from the boat to the lighthouse. We applied the tangent ratio to find the height of the lighthouse and used a calculator to find the value of $\tan 42^{\circ}$ . This problem demonstrates the practical application of trigonometry in real-life scenarios and highlights the importance of understanding trigonometric ratios in solving problems involving right triangles.

Real-Life Applications of Trigonometry

Trigonometry has numerous applications in various fields, including:

Navigation: Trigonometry is used in navigation to determine distances and directions between two points.
Physics: Trigonometry is used in physics to describe the motion of objects and to calculate forces and energies.
Engineering: Trigonometry is used in engineering to design and build structures, such as bridges and buildings.
Computer Science: Trigonometry is used in computer science to create 3D graphics and to simulate real-world phenomena.

Common Trigonometric Formulas

Here are some common trigonometric formulas that are used to solve problems involving right triangles:

Pythagorean Theorem: $a^2 + b^2 = c^2$
Sine Formula: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
Cosine Formula: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
Tangent Formula: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Tips and Tricks

Here are some tips and tricks for solving problems involving trigonometry:

Use a calculator: A calculator can be a powerful tool for solving trigonometric problems.
Draw a diagram: Drawing a diagram can help you visualize the problem and identify the relevant trigonometric ratios.
Check your units: Make sure to check your units when solving trigonometric problems.
Use the correct formula: Make sure to use the correct formula for the problem you are trying to solve.

Conclusion

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will answer some frequently asked questions about trigonometry, covering topics such as trigonometric ratios, right triangles, and real-life applications.

Q: What is trigonometry?

A: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It involves the study of trigonometric ratios, such as sine, cosine, and tangent, and their applications in solving problems involving right triangles.

Q: What are the three basic trigonometric ratios?

A: The three basic trigonometric ratios are:

Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Q: What is the Pythagorean Theorem?

A: The Pythagorean Theorem is a fundamental concept in trigonometry that states:

a^2 + b^2 = c^2

where $a$ and $b$ are the lengths of the legs of a right triangle, and $c$ is the length of the hypotenuse.

Q: How do I use trigonometry to solve problems involving right triangles?

A: To use trigonometry to solve problems involving right triangles, you need to:

Identify the relevant trigonometric ratio (sine, cosine, or tangent) based on the given information.
Use the trigonometric ratio to set up an equation involving the unknown side or angle.
Solve the equation using algebraic manipulations and trigonometric identities.
Check your solution by plugging it back into the original equation.

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

A: Trigonometry has numerous applications in various fields, including:

Navigation: Trigonometry is used in navigation to determine distances and directions between two points.
Physics: Trigonometry is used in physics to describe the motion of objects and to calculate forces and energies.
Engineering: Trigonometry is used in engineering to design and build structures, such as bridges and buildings.
Computer Science: Trigonometry is used in computer science to create 3D graphics and to simulate real-world phenomena.

Q: How do I choose the correct trigonometric ratio for a given problem?

A: To choose the correct trigonometric ratio for a given problem, you need to:

Identify the given information (angles, sides, or both).
Determine the relationship between the given information and the unknown side or angle.
Choose the trigonometric ratio that best describes the relationship between the given information and the unknown side or angle.

Q: What are some common trigonometric formulas?

A: Here are some common trigonometric formulas that are used to solve problems involving right triangles:

Pythagorean Theorem: $a^2 + b^2 = c^2$
Sine Formula: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
Cosine Formula: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
Tangent Formula: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Q: How do I use a calculator to solve trigonometric problems?

A: To use a calculator to solve trigonometric problems, you need to:

Enter the given information (angles, sides, or both) into the calculator.
Choose the correct trigonometric function (sine, cosine, or tangent) based on the problem.
Use the calculator to evaluate the trigonometric function and find the solution.

Conclusion

In conclusion, trigonometry is a powerful tool for solving problems involving right triangles. By understanding trigonometric ratios and applying them to real-life scenarios, we can solve problems that involve the height of objects, distances, and directions. This article answered some frequently asked questions about trigonometry, covering topics such as trigonometric ratios, right triangles, and real-life applications.