Problem 41 - Algebra 2

Question

Indirect Measurement A communications tower has wires anchoring it to the ground. Each wire is attached to the tower at a height 20 \(\mathrm{ft}\) above the ground. The length \(y\) of the wire is modeled with the function \(y=20\) csc \(\theta,\) where \(\theta\) is the measure of the angle formed by the wire and the ground. a. Graph the function. b. Find the length of wire needed to form an angle of \(45^{\circ} .\) c. Find the length of wire needed to form an angle of \(60^{\circ} .\) d. Find the length of wire needed to form an angle of \(75^{\circ} .\)

Step-by-Step Solution

Answer

The graph is a course cosine graph stretch by a factor of 20. The length of the wire needed to form an angle of \(45^{\circ}\), \(60^{\circ}\), and \(75^{\circ}\) are approximately \(28.28\) feet, \(23.09\) feet, and \(20.64\) feet, respectively.

Step 1: Graph Function

The graph of csc \(\theta\) would be similar to a reciprocal sine graph but with undefined points at \(\theta\) multiples of \(\pi\). To sketch the graph of \(y = 20 \) csc \(\theta\), stretch the csc \(\theta\) graph by a factor of 20.

Step 2: Calculate Wire Length for \(\theta = 45^{\circ}\)

To calculate the length of the wire that forms an angle of \(45^{\circ}\) with the ground, plug \(\theta = 45^{\circ}\) into the length function \(y = 20 \) csc \(\theta\). The csc \(\theta\) function is equivalent to 1/sin \(\theta\). Therefore, \(y = 20/\) sin \(45^{\circ}\). Following the calculation, you get \(y \approx 28.28\) feet.

Step 3: Calculate Wire Length for \(\theta = 60^{\circ}\)

To calculate the length of the wire that forms an angle of \(60^{\circ}\) with the ground, plug \(\theta = 60^{\circ}\) into the length function \(y = 20 \) csc \(\theta\). Therefore, \(y = 20/\) sin \(60^{\circ}\). Following the calculation, you get \(y \approx 23.09\) feet.

Step 4: Calculate Wire Length for \(\theta = 75^{\circ}\)

To calculate the length of the wire that forms an angle of \(75^{\circ}\) with the ground, plug \(\theta = 75^{\circ}\) into the length function \(y = 20 \) csc \(\theta\). Therefore, \(y = 20/\) sin \(75^{\circ}\). Following the calculation, you get \(y \approx 20.64\) feet.

Key Concepts

Cosecant Function

The cosecant function, usually denoted as \( \csc \theta \), is the reciprocal of the sine function. In mathematical terms, it is defined as \( \csc \theta = \frac{1}{\sin \theta} \). This function is important because it tells us how the length of the hypotenuse (in a right triangle) relates to the opposite side of the angle \( \theta \).

This function is particularly useful in real-world applications, such as measuring the length of an anchored wire, as seen in our exercise. Here, the cosecant function helps model the length of the wire in relation to the angle it makes with the ground. The expression \( y = 20 \csc \theta \) indicates that the wire length is directly proportional to \( \csc \theta \), multiplied by 20.

• The larger the angle \( \theta \), the smaller the \( \csc \theta \) value, and vice versa.
• This is because, as \( \theta \) increases, sine values increase towards 1, causing their reciprocals to decrease.

Indirect Measurement

Indirect measurement is a technique that allows us to determine distances or lengths without measuring them directly. This can be particularly useful when dealing with tall structures like towers or buildings, where direct measurement is impractical.

In the context of our problem, indirect measurement is applied using the cosecant function. By knowing the angle the wire makes with the ground, and the height at which it is attached, we can calculate the wire's length. We use the equation \( y = 20 \csc \theta \), utilizing trigonometric properties without needing to physically measure the wire.

This approach provides a convenient and efficient way to solve real-world problems using mathematical models, ensuring safety and accuracy without the need for complex equipment.

• Trigonometry-based indirect measurements are widely used in engineering and architecture.
• They help in determining heights, lengths, and distances that are otherwise difficult to measure directly.

Angle of Elevation

The angle of elevation is the angle formed by the line of sight from an observer looking upward to an object. It is measured from the horizontal upward. In this exercise, the angle of elevation \( \theta \) is the angle between the ground and the wire.

This angle is crucial in determining the length of the wire. As \( \theta \) changes, the needed wire length changes, illustrating how the angle of elevation influences the model \( y = 20 \csc \theta \).

• An increasing angle of elevation implies a shorter wire length, as it approaches the vertical attachment point.
• A smaller angle results in a longer required wire length.
• Understanding the angle of elevation allows for effective planning in construction and design projects.

Graphing Trigonometric Functions

Graphing trigonometric functions provides a visual understanding of how these functions behave. When graphing \( y = 20 \csc \theta \), you would see a transformation of the \( \csc \theta \) graph, which stretches vertically by a factor of 20.

Since \( \csc \theta \) is undefined where \( \sin \theta = 0 \) (at multiples of \( \pi \)), its graph has vertical asymptotes at these points. This characteristic helps in predicting how the length of the wire varies as \( \theta \) changes.

Plotting these graphs can:

• Show how different angles affect the wire length visually.
• Help anticipate the behavior of the equation under various conditions.
• Make abstract concepts more tangible and easier to grasp.

Graphs provide a powerful tool for interpreting and predicting real-world applications of trigonometry.