Problem 67 - Precalculus : Building Concepts and Connections

Question

In this set of exercises, you will use inverse trigonometric functions to study real-world problems. Round all answers to four decimal places. A 15 -foot pole is to be stabilized by two wires of equal length, one on each side of the pole. One end of each wire is to be attached to the top of the pole; the other end is to be staked to the ground at an acute angle \(\theta\) with respect to the horizontal. Because of considerations, the ratio of the length of either wire to the height of the pole is to be no more than \(\frac{4}{3} .\) What is the limiting value of \(\theta\) in degrees? Is this limiting value a maximum value of \(\theta\) or a minimum value of \(\theta ?\) Explain.

Step-by-Step Solution

Answer

The limiting value of \(\theta\) is its maximum value, which can be calculated by taking the inverse cosine of the ratio \(\frac{3}{4}\). This value cannot be exceeded without violating the conditions of the problem, i.e., the ratio of the length of the wire to the height of the pole.

Step 1: Setup Ratio According to Problem Statement

According to the problem, the ratio of the length of the wire to the height of the pole is \(\frac{4}{3}\). As the wire and pole make a right angle triangle with the ground, this ratio can be interpreted as the hypotenuse / adjacent or cosine of \(\theta\). Hence, \(\cos(\theta) = \frac{3}{4}\)

Step 2: Calculate the Angle \(\theta\)

In order to obtain the value of \(\theta\), we need to use the inverse cosine function: \(\theta = \cos^{-1}(\frac{3}{4})\). By solving this, the angle will be obtained.

Step 3: Determine Whether the Value of \(\theta\) is Maximum or Minimum

Because the wires are to be staked in the ground at an acute angle \(\theta\), the limiting value of \(\theta\) is the maximum value. This is because if \(\theta\) increases beyond this value, the ratio of the length of the wire to the height of the pole would exceed the maximum ratio of \(\frac{4}{3}\), thereby contradicting the constraints of the problem.

Key Concepts

Understanding Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the angles and sides of triangles. It is especially focused on right triangles, where one angle is always 90 degrees.
Trigonometric functions, like sine, cosine, and tangent, are used to relate these angles and sides, making them extremely useful in various calculations.
Whether you're building structures, analyzing waves, or navigating, trigonometry provides the tools to solve problems by understanding the geometric properties of shapes.

Exploring the Right Triangle

A right triangle is a triangle with one 90-degree angle. The other two angles in the triangle are always less than 90 degrees, known as acute angles.
The sides of a right triangle have special names:

• The side opposite the right angle is the hypotenuse, the longest side.
• The adjacent side is the one next to the angle of interest.
• The opposite side is the one across from the angle of interest.

Knowing these relationships helps in setting up equations that involve trigonometric functions, giving us the ability to solve for unknown lengths or angles.

Focusing on Cosine Function

The cosine function is fundamental in trigonometry. It relates the angle in a right triangle to the ratio of the length of the adjacent side to the hypotenuse.
For an angle \(\theta\), it is defined as:
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
In inverse trigonometric functions, like inverse cosine, we find the angle when given a ratio. For example, knowing that \(\cos(\theta) = \frac{3}{4}\), you can find \(\theta\) using:
\[ \theta = \cos^{-1}\left(\frac{3}{4}\right) \]
This allows us to compute the angle in degrees or radians, which is essential for applying trigonometry to real-world problems.

Applying to Real-World Situations

Trigonometry isn't just for theoretical problems; it has many practical applications. From engineering to physics and even daily tasks like construction, these principles guide us in finding solutions.
In our example, a pole stabilized by wires requires calculations to ensure the wires are the right length, holding the pole securely without exceeding material constraints.

• Safety standards are maintained by controlling angles and lengths.
• Efficiency is ensured by optimizing material use.

Real-world applications of trigonometry help us design systems that are safe, effective, and reliable, illustrating its importance beyond the classroom.