Problem 60 - Algebra and Trigonometry

Question

The angle of elevation from the top of a house to a jet flying 2 miles above the house is \(x\) radians. If \(d\) represents the horizontal distance, in miles, of the jet from the house, express \(d\) in terms of a trigonometric function of \(x .\) Then graph the function for \(0

Step-by-Step Solution

Answer

The horizontal distance, \(d\), of the jet from the house can be expressed in terms of a trigonometric function of the angle of elevation, \(x\), as \(d =\frac{2}{\tan(x)}\). This function can be graphed in the interval \(0 < x < \pi\).

Step 1: Understand and visualize the problem

This step involves understanding the problem and visualizing it. Visualize a right triangle, where the house and the jet form the two non-right angles of the triangle. The height of the jet above the house is the side opposite the angle of elevation and the horizontal distance between the jet and the house is the side adjacent to the angle.

Step 2: Formulate the equation

With the concept of trigonometry, it is known that the tangent of any angle is the ratio of the length of the side opposite the angle to the length of the adjacent side. Therefore, in the right triangle that represents the scenario in the problem, the tangent of the angle of elevation (\(x\)) is equal to the ratio of the height of the jet from the house (2 miles) to the horizontal distance of the jet from the house (\(d\)). Therefore, \(\tan(x) =\frac{2}{d}\).

Step 3: Express \(d\) in terms of \(x\)

By rearranging the equation obtained in the previous step, it can be expressed \(d\) in terms of \(x\). That is \(d =\frac{2}{\tan(x)}\).

Step 4: Graph the function

A the function \(d(x) =\frac{2}{\tan(x)}\) is a reciprocal trigonometric function, which need to be graphed in the interval \(0 < x < \pi\). This interval excludes the values of \(x\) that make the denominator zero. The graph of the function will show how the horizontal distance of the jet from the house changes as the angle of elevation changes within the specified interval.

Key Concepts

Angle of Elevation

When you look up at an object from a lower point, the angle you tilt your head is known as the angle of elevation. It's an angle formed between the line of sight from your eyes to the object and the horizontal line from your eyes. The angle of elevation is crucial in problems involving measurements from a lower point to a higher point.
For example:

• You are standing outside, looking up at a plane. The angle your line of sight makes with the horizontal is the angle of elevation.
• You need to note that the angle of elevation is measured from the horizontal line upward to the object.

Understanding this angle helps in breaking down complex problems into simpler ones by focusing on right triangles used in these scenarios.

Right Triangle

Right triangles are a critical component in understanding geometry and trigonometry. A right triangle is a triangle that has one angle equal to 90 degrees. This feature allows us to use the Pythagorean theorem and basic trigonometric functions.

• The legs of a right triangle are the two sides that form the right angle.
• The hypotenuse is the longest side opposite the right angle.
• Problems involving right triangles frequently involve calculating the lengths of sides or the measures of angles using trigonometric functions.

In our exercise, the right triangle consists of:

• The vertical side, which is the height of the jet above the house (2 miles).
• The horizontal side, which is the distance we aim to calculate, represented as "d" in the equation.
• This arrangement nicely fits into using trigonometric functions to solve for unknown distances.

Tangent Function

Trigonometric functions help relate the angles and sides of a triangle. The tangent function, often abbreviated as "tan," is particularly useful in right triangles. The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

• If you know the angle and one side, you can use the tangent function to find the other side.
• Mathematically, it is expressed as: \( \tan(x) = \frac{\text{opposite}}{\text{adjacent}} \).

In our problem concerning the jet and house,:

• \( \tan(x) = \frac{2}{d} \), where 2 is the height from the house to the jet.
• To find the horizontal distance \( d \), we can rearrange this equation: \( d = \frac{2}{\tan(x)} \).

This function shows how changes in the angle of elevation directly affect the distance "d" you are trying to calculate.