Problem 26 - Precalculus: Functions and Graphs

Question

From a point 15 meters above level ground, a surveyor measures the angle of depression of an object on the ground at \(68^{\circ}\). Approximate the distance from the object to the point on the ground directly beneath the surveyor.

Step-by-Step Solution

Answer

The distance is approximately 6.06 meters.

Step 1: Understanding the Problem

We need to find the horizontal distance from the object to the base of a point where the surveyor is situated, which is 15 meters above the ground. The angle of depression from the surveyor to the object on the ground is given as \(68^{\circ}\).

Step 2: Setting Up the Triangle

Imagine a right triangle formed by the horizontal line from the object to the point directly beneath the surveyor (ground level), the vertical line from the surveyor to the ground (15 meters), and the line of sight from the surveyor to the object. The angle of depression \(68^{\circ}\) is equal to the angle of elevation from the object to the surveyor's line of sight.

Step 3: Using Trigonometry

Use the tangent function, which relates the opposite side (15 meters) to the adjacent side (the distance we need to find), using the angle of elevation or depression. The tangent function is given by \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

Step 4: Applying the Tangent Function

Since \( \theta = 68^{\circ} \), apply the formula: \[ \tan(68^{\circ}) = \frac{15}{d} \] where \(d\) is the distance we want to find.

Step 5: Solving for Distance

Rearrange the equation to solve for \(d\): \[ d = \frac{15}{\tan(68^{\circ})} \].

Step 6: Calculating the Result

Calculate \(d\) using a calculator: \[ \tan(68^{\circ}) \approx 2.475 \]. Thus, \[ d = \frac{15}{2.475} \approx 6.06 \]. So, the distance is approximately 6.06 meters.

Key Concepts

Angle of Depression

In trigonometry, the angle of depression is defined as the angle formed by the horizontal line and the line of sight from an observer to an object below them. This angle is crucial in solving problems involving heights and distances, especially when determining unseen dimensions.

Let's break it down: Imagine you are standing at a high point, looking down at an object on the ground. The line of sight from your eyes to the object forms the hypotenuse of a right triangle, while the horizontal line is the base of this triangle. The angle of depression is the angle formed between this horizontal line and your line of sight.

Why is it termed as 'depression'? Because it indicates a downward direction from a higher point to a lower point, just like feeling a downward mood can be depressing. Keep this imagery in mind to easily recall this concept!

In practical problems, the angle of depression is equal to the angle of elevation from the object on the ground to the observer at a height. This is because horizontal lines are parallel to each other, reflecting principles from geometry.

Tangent Function

The tangent function is an essential part of trigonometry, relating the angles and sides of a right triangle. Specifically, the tangent of an angle in a right triangle is the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle.

This can be understood through the mathematical expression:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

Here, \( \theta \) is the angle in question. The side opposite the angle is simply the vertical distance or height, and the side adjacent is the horizontal distance or base of the triangle.

The tangent function is widely used because, with the knowledge of an angle and one side, you can find the other side. In this exercise, the surveyor used the known height (15 meters) as the opposite side and the angle of depression (68 degrees) to find the unknown horizontal distance.

This relationship is why trigonometry is a powerful tool in solving real-world problems.

Trigonometric Ratios

Trigonometric ratios refer to the relations of the sides of right triangles concerning their angles. The primary trigonometric ratios are sine, cosine, and tangent, which each use different parts of the triangle.

To understand these ratios:

• Sine ( \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \) ) is the ratio of the length of the opposite side to the length of the hypotenuse.
• Cosine ( \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \) ) is the ratio of the length of the adjacent side to the hypotenuse.
• Tangent ( \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \) ) is the ratio of the opposite side to the adjacent side.

These ratios provide a way to calculate the dimensions of triangles when certain angles and side lengths are known. In right triangles, these ratios are specifically useful for indirect measurements.

Understanding and memorizing these relationships through practice will help greatly when you're required to solve problems involving distances and angles, just like the exercise above.