Problem 53 - Algebra and Trigonometry

Question

An observer views the space shuttle from a distance of 2 miles 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) \( h = 2 \tan(\theta) \); (b) \( \theta = \arctan\left(\frac{h}{2}\right) \).

Step 1: Define the Problem

We're given that an observer is 2 miles away from the shuttle's launch pad. We need to find expressions for the height of the space shuttle in terms of the angle of elevation \( \theta \) and vice versa. This forms a right triangle with the observer watching the shuttle.

Step 2: Express the Height as a Function of the Angle of Elevation

In the right triangle formed: the base is 2 miles, the opposite side is the height of the space shuttle \( h \), and the hypotenuse is the line of sight. Using the tangent function, \( \tan(\theta) = \frac{h}{2} \). Solving for \( h \), we get \( h = 2 \tan(\theta) \).

Step 3: Express the Angle of Elevation as a Function of the Height

Again using the tangent function for the right triangle, \( \tan(\theta) = \frac{h}{2} \). To express \( \theta \) in terms of \( h \), use the inverse tangent function, or arctan. Thus, \( \theta = \arctan\left(\frac{h}{2}\right) \).

Key Concepts

Right Triangle

A right triangle is a triangle that has one angle measuring exactly 90 degrees. This type of triangle is fundamental in trigonometry as it provides a simple relationship between the angles and sides. When dealing with right triangles, we often use the three main trigonometric functions: sine, cosine, and tangent. In a right triangle, the side opposite the right angle is called the hypotenuse. The other two sides are known as the "legs" of the triangle. In our exercise, imagine a right triangle created by the observer, the launch pad, and the space shuttle as the shuttle climbs skyward. Here:

• The base (adjacent side) is the horizontal distance from the observer to the launch pad, 2 miles.
• The opposite side is the height of the space shuttle, which varies as the shuttle moves.
• The hypotenuse would be the direct sightline or the diagonal line seen from the observer to the shuttle.

Understanding these elements of a right triangle helps establish relationships using trigonometric functions, particularly in solving problems related to angles and side lengths.

Angle of Elevation

The angle of elevation is the angle formed between a horizontal line of sight and the line of sight up to an object. In our scenario with the space shuttle, the observer's sightline starts at the launch pad level and angles upwards towards the shuttle. When calculating the angle of elevation in problems, you typically consider the base as a reference point, often using horizontal lines such as the ground level or the observer's eye level. This angle becomes crucial when utilizing trigonometric functions to ascertain height or distance.For instance, if the observer views the shuttle making an angle \(\theta\) with the horizontal plane, this angle helps us find:

• The height of the shuttle using the tangent function: \(\tan(\theta) = \frac{h}{2}\), where \(h\) is the height of the space shuttle.

Thus, expressing the height in terms of this angle is key to solving real-world problems related to measurement and projection.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when the values of the trigonometric functions are known. In typical scenarios, you take ratios derived from triangles and revert them to angles with these inverse functions.In the case of the space shuttle and observer, when the height \(h\) is known, but you need to determine the angle of elevation \(\theta\), the inverse trigonometric function comes into play. Specifically, the inverse tangent function, denoted as \(\arctan\), is utilized. This method is useful for recovering angle measures from known tangents:

• From \(\tan(\theta) = \frac{h}{2}\), solve for \(\theta\) by computing \(\theta = \arctan\left(\frac{h}{2}\right)\).

Inverse functions thus bridge the gap between knowing side ratios and determining precise angle measures, offering a powerful tool for varied calculations in trigonometry.