Identify the right triangle

Oct 13, 2025 by magicalceylontours.com 18 views

Introduction

In trigonometry, the angle of depression is the angle between the horizontal and a line from the observer's eye to the point of interest. This concept is crucial in solving problems involving right triangles, especially when dealing with heights and distances. In this article, we will explore how to find the distance of a boat from the base of a cliff given the angle of depression and the height of the cliff.

Understanding the Problem

We are given a vertical cliff that is 30 meters high. From the top of the cliff, the angle of depression of a boat is 25 degrees. Our goal is to find the distance of the boat from the base of the cliff. To solve this problem, we can use the concept of right triangles and trigonometric ratios.

Using Trigonometry to Solve the Problem

Let's break down the problem step by step:

Identify the right triangle: We can draw a right triangle with the cliff as the vertical leg (opposite side), the line from the top of the cliff to the boat as the hypotenuse, and the line from the base of the cliff to the boat as the adjacent side.
Identify the angle: We are given the angle of depression, which is 25 degrees. This angle is between the horizontal and the line from the top of the cliff to the boat.
Use trigonometric ratios: We can use the sine, cosine, or tangent ratio to solve the problem. Since we are given the angle and the opposite side (height of the cliff), we can use the sine ratio.

Applying the Sine Ratio

The sine ratio is defined as:

sin(θ) = opposite side / hypotenuse

In this case, the opposite side is the height of the cliff (30 meters), and the angle is 25 degrees. We can plug in the values and solve for the hypotenuse (distance from the top of the cliff to the boat).

sin(25°) = 30 / hypotenuse

To solve for the hypotenuse, we can multiply both sides by the hypotenuse and then divide both sides by sin(25°):

hypotenuse = 30 / sin(25°)

Calculating the Hypotenuse

Using a calculator, we can find the value of sin(25°) and then calculate the hypotenuse:

sin(25°) ≈ 0.4226

hypotenuse ≈ 30 / 0.4226 hypotenuse ≈ 71.43 meters

Finding the Distance from the Boat to the Base of the Cliff

Now that we have the hypotenuse (distance from the top of the cliff to the boat), we can use the Pythagorean theorem to find the distance from the boat to the base of the cliff. The Pythagorean theorem states that:

a² + b² = c²

where a and b are the legs of the right triangle, and c is the hypotenuse. In this case, the hypotenuse is the distance from the top of the cliff to the boat (71.43 meters), and one of the legs is the height of the cliff (30 meters). We can plug in the values and solve for the other leg (distance from the boat to the base of the cliff):

a² + 30² = 71.43²

a² + 900 = 5093.33

a² = 4193.33

a ≈ √4193.33 a ≈ 64.55 meters

Conclusion

In this article, we used trigonometry to solve a problem involving the angle of depression and the height of a cliff. We found the distance from the top of the cliff to the boat using the sine ratio and then used the Pythagorean theorem to find the distance from the boat to the base of the cliff. The final answer is approximately 64.55 meters.

Additional Examples and Applications

This problem can be extended to other scenarios, such as:

Finding the distance from a building to a point on the ground given the angle of elevation and the height of the building.
Calculating the height of a mountain given the angle of depression and the distance from the base of the mountain to the point of observation.
Solving problems involving right triangles in physics, engineering, and other fields.

Final Thoughts

Trigonometry is a powerful tool for solving problems involving right triangles. By understanding the concepts of angles, sides, and trigonometric ratios, we can solve a wide range of problems in mathematics, physics, engineering, and other fields. In this article, we used the sine ratio and the Pythagorean theorem to solve a problem involving the angle of depression and the height of a cliff. We hope that this article has provided a clear and concise explanation of the concepts involved and has inspired readers to explore the world of trigonometry.

Introduction

In our previous article, we explored how to find the distance of a boat from the base of a cliff given the angle of depression and the height of the cliff. We used trigonometry to solve the problem and found the distance from the boat to the base of the cliff. In this article, we will answer some frequently asked questions related to the problem and provide additional examples and applications.

Q&A

Q1: What is the angle of depression, and how is it related to the problem?

A1: The angle of depression is the angle between the horizontal and a line from the observer's eye to the point of interest. In this problem, the angle of depression is 25 degrees, and it is related to the problem because it helps us find the distance from the boat to the base of the cliff.

Q2: Why do we use the sine ratio to solve the problem?

A2: We use the sine ratio to solve the problem because it relates the angle of depression to the height of the cliff and the distance from the top of the cliff to the boat. The sine ratio is defined as sin(θ) = opposite side / hypotenuse, where θ is the angle of depression, the opposite side is the height of the cliff, and the hypotenuse is the distance from the top of the cliff to the boat.

Q3: How do we find the distance from the boat to the base of the cliff?

A3: To find the distance from the boat to the base of the cliff, we use the Pythagorean theorem, which states that a² + b² = c², where a and b are the legs of the right triangle, and c is the hypotenuse. In this case, the hypotenuse is the distance from the top of the cliff to the boat, and one of the legs is the height of the cliff. We can plug in the values and solve for the other leg (distance from the boat to the base of the cliff).

Q4: What are some real-world applications of trigonometry?

A4: Trigonometry has many real-world applications, including:

Navigation: Trigonometry is used in navigation to find distances and directions between two points.
Physics: Trigonometry is used in physics to describe the motion of objects and the forces acting on them.
Engineering: Trigonometry is used in engineering to design and build structures, such as bridges and buildings.
Astronomy: Trigonometry is used in astronomy to calculate the positions and distances of celestial objects.

Q5: How can I practice trigonometry problems?

A5: You can practice trigonometry problems by:

Using online resources, such as Khan Academy and Mathway.
Working with a tutor or teacher to practice problems.
Using a trigonometry textbook or workbook to practice problems.
Creating your own problems and solutions to practice.

Additional Examples and Applications

Example 1: Finding the Distance from a Building to a Point on the Ground

Suppose we want to find the distance from a building to a point on the ground given the angle of elevation and the height of the building. We can use the sine ratio to solve the problem:

sin(θ) = opposite side / hypotenuse

where θ is the angle of elevation, the opposite side is the height of the building, and the hypotenuse is the distance from the building to the point on the ground.

Example 2: Calculating the Height of a Mountain

Suppose we want to calculate the height of a mountain given the angle of depression and the distance from the base of the mountain to the point of observation. We can use the sine ratio to solve the problem:

sin(θ) = opposite side / hypotenuse

where θ is the angle of depression, the opposite side is the height of the mountain, and the hypotenuse is the distance from the base of the mountain to the point of observation.

Conclusion

In this article, we answered some frequently asked questions related to the problem and provided additional examples and applications. We hope that this article has provided a clear and concise explanation of the concepts involved and has inspired readers to explore the world of trigonometry.