Problem 52 - Algebra and Trigonometry

Question

The CN Tower in Toronto, Canada, is the tallest free-standing structure in North America. A woman on the observation deck, 1150 ft above the ground, wants to determine the distance between two landmarks on the ground below. She observes that the angle formed by the lines of sight to these two landmarks is \(43^{\circ} .\) She also observes that the angle between the vertical and the line of sight to one of the landmarks is \(62^{\circ}\) and to the other landmark is \(54^{\circ} .\) Find the distance between the two landmarks.

Step-by-Step Solution

Answer

The distance between the two landmarks is approximately 1686 feet.

Step 1: Identify Given Angles and Construct a Triangle

The woman observes the two landmarks from a height of 1150 ft. The angle between the lines of sight of the two landmarks (at her angle of view) is given as \(43^{\circ}\). The angle of declination from the observation deck to one landmark is \(62^{\circ}\) and to the other is \(54^{\circ}\). You can imagine these as forming a triangle on the ground with vertices at the two landmarks and a point directly below her on the ground.

Step 2: Calculate Base Angles at the Observation Point

The angles of declination given are relative to the vertical. However, a typical 2D geometric triangle on a flat surface uses a horizontal plane. Convert each declination angle to an angle with respect to the horizontal by subtracting them from \(90^{\circ}\). For the first landmark, the angle is \(28^{\circ} = 90^{\circ} - 62^{\circ}\). For the second landmark, the angle is \(36^{\circ} = 90^{\circ} - 54^{\circ}\).

Step 3: Apply the Law of Sines to Find Side Lengths

Given the angle measurements, apply the Law of Sines. You have the triangle ABC, where A and B are the landmarks on the ground and C is the point directly below the observation deck. Let \(AB\) be the distance between the landmarks we wish to find. Use the relationship: \[\frac{AB}{\sin 94^\circ} = \frac{1150}{\sin 43^\circ}\].

Step 4: Solve for Distance AB

Rearrange the equation from Step 3 to find \(AB\). First calculate \(\sin 94^{\circ} \approx 1\) as an approximation since the angle is near \(90^{\circ}\). Then use \(\sin 43^{\circ} \approx 0.682\). So: \[AB \approx \frac{1150}{0.682} \approx 1686.51\]. Thus, the distance between the two landmarks is approximately 1686 feet.

Key Concepts

Angle of Declination

The angle of declination is a measure used in trigonometry to describe the angle between the horizontal plane and a line of sight downward from a higher point. In the context of the CN Tower problem, this is the angle between the vertical (observation deck) and the line of sight to an object on the ground, like a landmark. For example, if you were standing on a high building looking down at the street, the angle formed between your direct line of sight and the horizontal line parallel to the ground is the angle of declination.

Understanding the angle of declination is crucial because it helps translate observations from a vertical starting point to angles used in horizontal geometric interpretations. Here, we convert the angles of declination from vertical to horizontal to make use of them in constructing triangles, which are part of geometric solutions:

• For the line of sight to the first landmark, the angle of declination was given as \(62^{\circ}\). Converting it to a horizontal angle requires subtracting from \(90^{\circ}\), giving \(28^{\circ}\).
• Similarly, for the second landmark, the angle of declination was \(54^{\circ}\), which translates to a horizontal angle of \(36^{\circ}\).

Converting angles in this manner allows us to easily apply trigonometric principles to solve real-world problems like calculating distances.

Distance Calculation

Calculating distances using angles involves using trigonometry principles such as the Law of Sines. The Law of Sines relates the lengths of sides of a triangle to the sines of its angles. In our given problem, we want to find the distance between two landmarks observed from the CN Tower's observation deck. The problem provides us angles formed by lines of sight that help us apply this method. Here are the steps:

First, a triangle is formed with vertices at the two landmarks and a point directly below the observer. The known height of the observation point serves as the opposite side, while the distance between the landmarks is the side we aim to find:

• Use the relationship from the Law of Sines: \(\frac{AB}{\sin 94^{\circ}} = \frac{1150}{\sin 43^{\circ}}\)
• Solving for \(AB\) involves finding the sine values: \(\sin 94^{\circ}\) is approximately \(1\) due to being close to a right angle, and \(\sin 43^{\circ}\) is approximately \(0.682\).
• By rearranging and calculating, we find \(AB \approx \frac{1150}{0.682} \approx 1686.51\).

Thus, the distance between the landmarks is approximately 1686 feet. Using trigonometry in this way provides a simple method to find distances when direct measurement is not feasible.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It is particularly useful in situations involving indirect measurements, such as determining distances and angles without physically measuring them.

In the scenario with the CN Tower, trigonometry is used to interpret the angles formed by lines of sight as a means to solve the distance problem presented. The Law of Sines is one such tool in trigonometry that helps relate the sides and angles in oblique (non-right angled) triangles by setting a proportion between the ratios of side lengths to their respective angle sines:

• In any triangle, if \(A\), \(B\), and \(C\) are angles opposite sides \(a\), \(b\), and \(c\), then \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).
• This allows us to solve for unknowns just by having enough angle and side data, transforming geometric interpretations into numeric calculations.

This foundational concept makes trigonometry a powerful tool both in academic settings and practical applications like navigation, architecture, and in solving problems associated with indirect measurement.