Problem 39 - Precalculus Mathematics for Calculus

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) \(\theta(h) = \tan^{-1}(\frac{h}{2})\).

Step 1: Understanding the Problem

The task involves expressing the relationship between the height of a space shuttle and the angle at which it is viewed from a fixed distance during its launch. We need two expressions: one for height as a function of the angle (part a), and the other for the angle as a function of height (part b).

Step 2: Setting Up the Triangle

The scenario can be modeled as a right triangle, where the horizontal leg represents the distance from the observer to the launch pad, the vertical leg represents the height of the shuttle, and the hypotenuse represents the line of sight from the observer to the shuttle.

Step 3: Defining Trigonometric Function for Part (a)

For part (a), we need to express the height of the shuttle as a function of the angle \(\theta\). The height \(h\) is the opposite side to the angle \(\theta\) in the right triangle, and the distance from the observer is the adjacent side. Therefore, using the tangent function, we have \[\tan(\theta) = \frac{h}{2}\].

Step 4: Solving for Height in Terms of Angle

To express \(h\) as a function of \(\theta\), rearrange the equation: \(h = 2 \tan(\theta)\). So, \(h(\theta) = 2 \tan(\theta)\).

Step 5: Defining Inverse Trigonometric Function for Part (b)

For part (b), we need to express the angle \(\theta\) as a function of the height \(h\). Using the inverse tangent function, we start with \(\theta = \tan^{-1}(\frac{h}{2})\).

Step 6: Expressing the Angle as a Function of Height

Thus, the angle \(\theta\) can be expressed as \(\theta(h) = \tan^{-1}(\frac{h}{2})\).

Key Concepts

Right Triangle Trigonometry

Right triangle trigonometry is a fundamental concept that helps us understand relationships between the sides and angles of right triangles. Right triangles have one angle measuring 90 degrees. This characteristic makes it easy to apply trigonometric functions.
In our scenario, the observer's line of sight, the distance to the launch pad, and the height of the shuttle form a right triangle.
Here:

• The horizontal leg of the triangle represents the distance (2 miles) from the observer to the launch pad.
• The vertical leg represents the height of the shuttle.
• The hypotenuse is the direct line from the observer to the shuttle in the sky.

By applying right triangle trigonometry, we can determine unknown lengths or angles using known values in these right-angle triangle setups.

Tangent Function

The tangent function, often denoted as \( \tan \), is one of the primary trigonometric functions. It's especially useful in right triangles when dealing with angles of elevation. To find the tangent of an angle \( \theta \), you divide the length of the opposite side by the length of the adjacent side.
In our exercise, the tangent function is applied as follows:

• Opposite side: Height of the shuttle \( h \).
• Adjacent side: Distance from the observer to the launch pad, which is \( 2 \) miles.

Therefore, the tangent function for this right triangle setup is given by \( \tan(\theta) = \frac{h}{2} \).
From this expression, we can derive the height of the shuttle as a function of the angle of elevation: \( h(\theta) = 2 \tan(\theta) \). This equation allows us to calculate how high the shuttle is based on the angle from which the observer views it.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential when you need to find an angle from known side lengths. In this problem, we use the inverse tangent function \( \tan^{-1} \), also known as arctan, to find the angle of elevation \( \theta \) as a function of height.
The inverse tangent function works by reversing the process of the tangent. If the tangent of \( \theta \) is \( \frac{h}{2} \), then \( \theta \) is found by the inverse tangent of \( \frac{h}{2} \). This relationship is expressed as:

• \( \theta = \tan^{-1}(\frac{h}{2}) \).

This equation shows us the angle of elevation based on the shuttle's height.
Inverse trigonometric functions expand our ability to solve trigonometric problems by working backward from the side ratios to determine specific angles.