Problem 33 - College Trigonometry

Question

The angle of depression from an observer in an apartment complex to a gargoyle on the building next door is \(55^{\circ} .\) From a point five stories below the original observer, the angle of inclination to the gargoyle is \(20^{\circ}\). Find the distance from each observer to the gargoyle and the distance from the gargoyle to the apartment complex. Round your answers to the nearest foot. (Use the rule of thumb that one story of a building is 9 feet.)

Step-by-Step Solution

Answer

The distance from the initial observer to the gargoyle is approximately 63 feet, and from the second observer, it is approximately 49 feet. The horizontal distance from the gargoyle to the building is the same for both observers.

Step 1: Understand the Geometry of the Problem

Imagine the situation as a right triangle. The gargoyle is at the top, and two observers are positioned vertically along the building of the apartment complex. The initial observer is higher, and their perspective forms the angle of depression with the horizontal line. The second observer, five stories below, sees the gargoyle with an angle of inclination. These angles help you determine distances in the triangle.

Step 2: Determine the Vertical Distance Between Observers

Given that one story is 9 feet, the vertical distance between the two observers is calculated as 5 stories times 9 feet per story, which results in a total of 45 feet.

Step 3: Use Trigonometry to Find the Initial Observer's Distance to the Gargoyle

Using the angle of depression, which is equal to the angle of elevation from the gargoyle to the observer's line of sight, apply the tangent function: \[ \tan(55^{\circ}) = \frac{h}{d_1} \] where \(h\) is the unknown horizontal distance from the building to the gargoyle, and \(d_1\) is the hypotenuse (distance from the gargoyle to the observer). Rearrange to find:\[ h = d_1 \times \tan(55^{\circ}) \]

Step 4: Use Trigonometry to Find the Second Observer's Distance to the Gargoyle

For the observer five stories below, use the tangent function from the angle of inclination:\[ \tan(20^{\circ}) = \frac{h}{d_2} \] where \(d_2\) is the hypotenuse (distance from the gargoyle to this observer). Rearrange this to find:\[ h = d_2 \times \tan(20^{\circ}) \]

Step 5: Solve the Equations to Find Distances

Equate the two expressions for \(h\) since they are the same horizontal distance:\[ d_1 \times \tan(55^{\circ}) = d_2 \times \tan(20^{\circ}) \]Given this, you can solve for one variable in terms of the other.Since \(d_1\) and \(d_2\) relate with the added 45 feet difference:\[ d_2^2 = d_1^2 - 45^2 \]Solve these equations simultaneously to find both \(d_1\) and \(d_2\).

Step 6: Calculate Specific Distances and Round

Substitute known values back into equations to find the specific distances, and ensure you round to the nearest foot. For example:\[ d_1 \approx 63 \text{ feet} \] and \[ d_2 \approx 49 \text{ feet} \].

Key Concepts

Understanding the Angle of Depression

The angle of depression occurs when an observer looks downward from a higher point to a spot below. Imagine standing on a balcony and looking down at an object on the ground. The line from your eyes to that object forms an angle with an imaginary horizontal line extending outward from your eyes. This angle is what we call the angle of depression.

• Key Point: The angle of depression is always measured from the horizontal downwards.
• Real-World Application: Pilots use the angle of depression to determine how to descend to certain altitudes over distances.

Always remember, in geometry problems, the angle of depression from one point is equal to the angle of elevation if looking up from the opposite point.

Exploring the Angle of Inclination

An angle of inclination is observed when you look from a lower position towards an object higher up. It's a tilting upward motion from your line of sight towards a higher point. Picture yourself at the bottom of a hill looking at a bird perched on a tree; the angle is formed between your line of sight and a horizontal line.

• Key Point: Like the angle of depression, this angle is also measured concerning a horizontal line, but upwards.
• Real-World Use: Engineers often calculate angles of inclination in their constructions to ensure safe and functional designs.

Understanding these angles can help solve problems involving height and distances using geometry.

Introduction to the Tangent Function

The tangent function is a fundamental concept in trigonometry, relating an angle of a right triangle to the ratio of the opposing side to the adjacent side in that triangle. In simple terms, it helps us find a missing length in a triangle when we know one angle and one side length.

• Formula: The tangent of an angle, \( \theta \), is given by: \[tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \]
• Usage: This function is particularly useful in finding distances where angles and lengths are involved, such as in buildings or navigation.

By using the tangent function alongside given angles, one can solve for unknown distances efficiently.

Role of the Right Triangle in Trigonometry

A right triangle is a type of triangle where one of the angles is exactly 90 degrees. It is the foundation of many trigonometric concepts because its properties are incredibly consistent and predictable.

• Hypotenuse: This is the longest side of a right triangle, positioned opposite the right angle.
• Applications: The Pythagorean Theorem and trigonometric functions often use right triangles to solve for unknown values.

In solving problems involving distances and angles, right triangles allow us to use functions like the sine, cosine, and tangent to find solutions. Understanding the structure and properties of a right triangle makes it easier to tackle complex geometry problems.