Problem 90 - Precalculus : Building Concepts and Connections

Question

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. The Washington Monument in Washington, DC, is 555 feet high. If the angle of elevation of the top of the monument from a certain point on the ground is \(60^{\circ},\) how far is that point from the center of the base? Consider the base of the monument to be on the ground.

Step-by-Step Solution

Answer

The distance from the observation point to the base of the monument is approximately 320.208 feet.

Step 1: Understand the problem

The problem provides the height of the Washington Monument (555 feet) and the angle of elevation (60 degrees). The task is to find the distance from the observation point to the base of the monument (the adjacent side of a right triangle). The tangent function will be used to solve the problem, represented as: \(\tan(\Theta) = \frac{opposite}{adjacent}\). Rearranging, we get \(\text{adjacent} = \frac{opposite}{\tan(\Theta)}\).

Step 2: Insert given values into the formula

We substitute the given values into the equation: \(\text{adjacent} = \frac{555}{\tan(60)}\).

Step 3: Calculate the adjacent side

Next, we calculate the value of the equation. The tangent of 60 degrees is \(sqrt{3}\), so the equation simplifies to \(\text{adjacent} = \frac{555}{\sqrt{3}}\). By doing the division, we find the adjacent side (i.e., the distance from the observation point to the monument) approximately to be 320.208 feet (rounded to four decimal places).

Key Concepts

Angle of Elevation

The angle of elevation is a concept often encountered in both trigonometry and everyday situations. It refers to the angle above the horizontal line from an observer's line of sight to an object. Imagine standing some distance away from the Washington Monument and looking up to its peak. The angle your line of sight makes with respect to the ground level is what we term the angle of elevation.

Understanding this angle is crucial when solving for unknown distances or heights in right triangle scenarios, as it helps us set up our problem in terms of trigonometric ratios. For instance, knowing the angle of elevation can directly aid in determining the height of an object or the distance to it if some other dimension of a right triangle is known.

Tangent Function

The tangent function is one of the fundamental ratios in trigonometry, deeply connected to the properties of right triangles. It's defined as the ratio of the side opposite to a given acute angle to the side adjacent to that angle within a right triangle. Formally, for an angle \( \Theta \), it is expressed as \(\tan(\Theta) = \frac{opposite}{adjacent}\).

In our Washington Monument example, we used this exact trigonometric function to solve for the distance from a point on the ground to the monument by rearranging the formula: \(\text{adjacent} = \frac{opposite}{\tan(\Theta)}\). By inserting the height of the monument for the opposite side and the angle of elevation for \( \Theta \), the tangent function effectively calculates the distance (adjacent side) for us. Remember that a solid understanding of how to manipulate this function can be the key to solving many real-life geometry problems.

Trigonometry in Real-World Problems

Trigonometry is not just an abstract area of mathematics, it's a powerful tool that applies to a multitude of real-world problems, enabling us to measure distances and heights that would be troublesome or outright impossible to measure directly. In the case of the Washington Monument, trigonometry allows us to determine its distance from a certain point on the ground without physically measuring it across the space.

This principle is used in a variety of fields including engineering, architecture, astronomy, and navigation. Trigonometry's application extends to the design of structures, investigation of celestial bodies, and even in the creation of maps. By mastering trigonometric functions like the tangent, sine, and cosine, as well as understanding concepts like the angle of elevation and depression, students equip themselves with the skills necessary to solve practical challenges and gain a deeper appreciation for the mathematics that underpins our world.