Problem 39 - Algebra and Trigonometry

Question

Height of the Space Shuttle An observer views the space shuttle from a distance of 2 \(\mathrm{mi}\) from the launch pad. (a) Express the height of the space shuttle as a function of the angle of elevation \(\theta\) . (b) Express the angle of elevation \(\theta\) as a function of the height \(h\) of the space shuttle.

Step-by-Step Solution

Answer

(a) \( h(\theta) = 2 \tan(\theta) \); (b) \( \theta(h) = \arctan\left(\frac{h}{2}\right) \).

Step 1: Analyze the Problem

We are asked to express the height of the space shuttle as a function of the angle of elevation and vice versa. This requires understanding the relationship between the height, the distance from the observer to the launch pad, and the angle of elevation. We will use right triangle trigonometry to model this scenario.

Step 2: Use Right Triangle Trigonometry (Part a)

The problem forms a right triangle where the observer's distance to the launch pad is the adjacent side and the shuttle’s height is the opposite side. The tangent function relates these two sides in trigonometry: \( \tan(\theta) = \frac{h}{2} \). Therefore, the height of the Space Shuttle as a function of the angle of elevation is given by \( h(\theta) = 2 \tan(\theta) \).

Step 3: Express the Angle of Elevation (Part b)

Using the equation \( \tan(\theta) = \frac{h}{2} \), we solve for \( \theta \) to express it as a function of height. The angle of elevation is given by \( \theta(h) = \arctan\left(\frac{h}{2}\right) \). This formula relates the angle of elevation to the height of the space shuttle.

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. In our exercise, the scenario involves an observer, the launch pad, and the space shuttle forming such a triangle. Here’s why understanding right triangles is important:

• They are simple geometric shapes that help us model scenarios in trigonometry.
• They have specific properties such as one right angle, which allows us to use trigonometric functions like sine, cosine, and tangent.
• The sides of a right triangle are referred to as the hypotenuse, opposite, and adjacent, based on their relationship to a particular angle.

Understanding the layout of a right triangle is crucial. In this case, the observer's distance from the launch pad is the adjacent side, the height of the shuttle is the opposite side, and the line from the observer to the shuttle would be the hypotenuse if it were considered. Right triangles are foundation stones in trigonometry and lead us to further explore how we relate angles and side lengths.

Tangent Function

The tangent function is one of the primary trigonometric functions and plays a vital role in solving right triangle problems. It is particularly useful when dealing with angles and side lengths. In trigonometry, for any given angle in a right triangle:

• The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• In the context of our exercise, the tangent of the angle of elevation relates the shuttle’s height to the observer's distance.
• The function can help find unknown sides or angles when certain dimensions are known.

For this exercise, we used the relation \( \tan(\theta) = \frac{h}{2} \) to express the shuttle’s height as a function of the angle of elevation, resulting in the formula \( h(\theta) = 2\tan(\theta) \). Understanding how to manipulate and solve equations involving the tangent function is fundamental in right triangle trigonometry.

Angle of Elevation

The angle of elevation is an important concept in trigonometry. It refers to the angle formed between a horizontal line and the line of sight to an object above the horizontal line. Here’s why it’s key in problems like our shuttle scenario:

• It allows us to describe and calculate how high the shuttle is in relation to an observer.
• In our problem, this angle forms with the imaginary horizontal line from the observer's eye level to the top of the shuttle.
• Utilizing the tangent function, we can express the height in terms of the angle of elevation and vice-versa.

We know that the angle of elevation \( \theta \) can change depending on how high the shuttle ascends. To find \( \theta \) when we know the height of the shuttle \( h \), we use \( \theta(h) = \arctan\left(\frac{h}{2}\right) \). Calculating the angle of elevation is often a critical step when determining heights and distances in real-world applications using trigonometry.

Right Triangle Trigonometry

Right triangle trigonometry is the study of relationships in right triangles between the angles and the lengths of their sides. This branch of trigonometry is particularly useful in practical applications where we need to model real-world problems using simple geometric principles.

• At its core, right triangle trigonometry involves three main functions: sine, cosine, and tangent, which link angles to side ratios.
• Using these functions, we can solve for unknown sides or angles, given certain measurements.
• In our exercise, the tangent function was pivotal in relating the height of the space shuttle to the angle of elevation, demonstrating a classic use of right triangle trigonometry.

Part of what makes right triangle trigonometry so powerful is its ability to simplify complex real-world problems into manageable equations and solutions. By understanding and applying these principles, such as in the shuttle exercise, one can deduce unknown quantities with straightforward calculations and logical steps.