Calculating the Height of a Building Using Trigonometry

The angle of depression is a concept often encountered in trigonometry, particularly in real-world applications where distances and heights need to be determined indirectly. In this article, we will explore how to calculate the height of a building based on the given angle of depression and the distance from the observer to the ground. We will demonstrate the solution with a step-by-step approach, using both trigonometric functions and the properties of right triangles.

Problem Statement

Suppose you are standing at the top of a building and you notice a car on the ground. The angle of depression to the car is 60 degrees, and the distance from the base of the building to the car is 540 feet. What is the height of the building?

Step-by-Step Solution

Diagram and Setup

Notations: h - represents the height of the building. Angle of depression 60°. Distance from the base of the building to the car 540 feet.

Right Triangle: The angle of depression corresponds to the angle of elevation from the car to the top of the building, which is also 60°. We can use the tangent function, which is the ratio of the opposite side to the adjacent side in a right triangle.

Using Tangent

tan 60° opposite side / adjacent side.

In this case, the opposite side is the height of the building h, and the adjacent side is the distance from the building to the car, which is 540 feet.

Setting Up the Equation

tan 60° h / 540

We know that tan 60° √3.

Solving for h

√3 h / 540

h 540 × √3

Using the approximation √3 ≈ 1.732:

h ≈ 540 × 1.732 ≈ 934.68 ft

Alternative Method: Using Properties of a 30°-60°-90° Right Triangle

Another way to solve this problem is by using the properties of a 30°-60°-90° right triangle. In this type of triangle, the sides are in the ratio 1:√3:2. Here, the distance from the base of the building to the car is the side opposite the 30° angle, and the height of the building is the side opposite the 60° angle.

Opposite side 540 ft.

Adjacent side (base of the triangle) 540 / √3 311.8 ft.

Height of the building 540 × √3 540 × 1.73 934.2 ft.

Double Check and Verification

To ensure the solution is reasonable, we can verify using the Pythagorean theorem:

(540)^2 (h)^2 (Base)^2

(540)^2 (934.68)^2 ≈ (1080)^2

Conclusion

The height of the building is approximately 934.68 feet.