Problem 137 - Algebra and Trigonometry Real Mathematics, Real People

Question

Aeronautics An airplane flying at an altitude of 6 miles is on a flight path that passes directly over an observer (see figure). Let \(\theta\) be the angle of elevation from the observer to the plane. Find the distance from the observer to the plane when (a) \(\theta=30^{\circ}\) (b) \(\theta=90^{\circ},\) and \((\mathrm{c}) \theta=120^{\circ}\).

Step-by-Step Solution

Answer

The distance from the observer to the plane when (a) \(\theta=30^{\circ}\) is 12 miles, (b) \(\theta=90^{\circ}\) is 6 miles, and (c) \(\theta=120^{\circ}\) is 7 miles.

Step 1: Model the situation

The situation can be described with a right triangle, where the observer, the airplane, and the point directly beneath the airplane on the ground form the vertices. The altitude of the airplane forms the opposite side to angle \(\theta\), and the distance from the observer to the airplane is the hypotenuse. We know that the altitude is 6 miles. We are supposed to calculate the hypotenuse for three different angles: \(\theta = 30^{\circ}, 90^{\circ}, 120^{\circ}\).

Step 2: Use the definition of tangent to solve for (a) \(\theta=30^{\circ}\)

In our right angle triangle, \(\tan(\theta)= \frac{opposite}{adjacent}\). We can transform the equation to find the length of hypotenuse, which is \(\frac{opposite}{\tan(θ)}\). Substituting the values, the hypotenuse when \(\theta=30^{\circ}\) will be \(\frac{6}{\tan(30)} = 12\) miles.

Step 3: Solve for (b) \(\theta=90^{\circ}\)

When \(\theta=90^{\circ}\), the observer is looking straight up at the plane, so the line of sight and the ground form a straight line, and the distance from the observer to the plane is equal to the altitude of the plane, which is 6 miles.

Step 4: Apply the sine rule to solve for (c) \(\theta=120^{\circ}\)

We know that \(\sin(\theta)= \frac{opposite}{hypotenuse}\). In our triangle, when \(\theta=120^{\circ}\), the line of sight to the plane is the hypotenuse. We can use the alternate version of the sine function to find hypotenuse, which is \(\frac{opposite}{\sin(θ)}\). Substituting the values, the hypotenuse will be \(\frac{6}{\sin(120)} = 7\) miles.

Key Concepts

Angles of Elevation

In trigonometry, the angle of elevation is the angle between the horizontal line of sight and the line of sight up to an object. Imagine standing outside, looking up at an airplane flying overhead.
The angle your eyes make with the horizontal ground when you look at the airplane is the angle of elevation. The angle of elevation is important when determining distances for objects at various heights. Here are some key things to remember about angles of elevation:

• It is always measured from the horizontal upward.
• In the exercise, the angle of elevation helps you determine how far away the airplane is from the observer.
• The angle changes depending on the position of the observer relative to the object.

Knowing the angle of elevation allows us to use trigonometric functions to solve real-life problems involving right triangles.

Trigonometric Functions

Trigonometric functions are essential tools for solving problems involving angles and distances, especially in right triangles. The three primary functions you should be familiar with are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)).For right triangles, each function relates the angles to the lengths of the sides:

• Tangent relates the opposite side to the adjacent side: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• Sine relates the opposite side to the hypotenuse: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
• Cosine relates the adjacent side to the hypotenuse: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).

In the exercise, the tangent function was used for calculating distances by manipulating the function to solve for the hypotenuse.Understanding how these functions work can help you determine unknown side lengths and angles when given some of the sides. Remember that these functions are based on the proportions of the sides relative to the angles within the triangle.

Right Triangle Properties

Right triangle properties are fundamental in trigonometry and help simplify complex problems. A right triangle consists of two legs and a hypotenuse, with one angle being exactly 90 degrees.Some essential features of right triangles are:

• The hypotenuse is always the longest side and is opposite the right angle.
• The legs are the other two shorter sides forming the right angle.
• One can use the Pythagorean theorem, which states that \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are legs and \( c \) is the hypotenuse, to find missing sides.

In our exercise problem, because the airplane, observer, and ground form a right triangle, we can utilize special triangle relationships and properties to solve for the hypotenuse for given angles.Understanding right triangle properties helps simplify the application of trigonometric functions and the calculation of side lengths, making it easier to resolve problems in two-dimensional space.