Problem 22 - Algebra 2

Question

Suppose the architect in Example 3 reduces the length of the base of the triangle to 100 ft. The function that models the height of the triangle becomes \(y=50 \tan \theta\) a. Graph the function on a graphing calculator. b. What is the height of the triangle when \(\theta=16^{\circ} ?\) c. What is the height of the triangle when \(\theta=22^{\circ} ?\)

Step-by-Step Solution

Answer

The graph of the function should look like increasing as the angle increase. The height of the triangle when \(\theta=16^{\circ}\) and \(\theta=22^{\circ}\) can be found by substituting the angles into the function y and calculating the resulting value.

Step 1: Understanding the function

First, understand that the function \(y=50 \tan \theta\) is to calculate height of the triangle. In this function, 50 is the constant factor to the tangent of angle \(\theta\). The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Therefore, reducing the base of the triangle changes the function to what we have now.

Step 2: Graphing the function

To graph the function, plot the angle \(\theta\) along the x-axis and the height along the y-axis. Plug in various values for the angle \(\theta\) to get corresponding values of y, and plot those points on the graph. Connect those points to draw the function's graph. Remember that the angle should be in radians when calculating the tangent value.

Step 3: Calculating height when \(\theta=16^{\circ}\)

For finding the height at \(\theta = 16^{\circ}\), substitute \(16^{\circ}\) in the given function. Make sure to convert degrees to radians because the tangent function in calculators uses radian measure. After substituting: \(y = 50 \tan (16 \times \frac{\pi}{180})\). Now calculate the value of y on a calculator.

Step 4: Calculating height when \(\theta=22^{\circ}\)

Similar to step 3, for finding the height at \(\theta = 22^{\circ}\), substitute \(22^{\circ}\) in the given function. Make sure to convert degrees to radians: \(y = 50 \tan (22 \times \frac{\pi}{180})\). Now calculate the value of y on a calculator.

Key Concepts

Tangent Function

The tangent function, \(\tan \theta\), is a trigonometric function that relates the angles of a right triangle to the ratio of two of its sides. Specifically, in a right triangle, the tangent of an angle \(\theta\) is defined as the ratio of the length of the opposite side to the length of the adjacent side. This can be expressed mathematically as:

• \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\)

This function is very useful in various fields, such as architecture and engineering, because it provides a straightforward method to determine the height or distance when one of the sides, and an angle, of a right triangle is known.
In the context of the exercise, the function \(y = 50 \tan \theta\) models the height of the triangle, making use of the tangent of angle \(\theta\) and depicting how the height changes with different angles.

Right Triangle

A right triangle is a type of triangle that has one angle measuring 90 degrees. It is characterized by its sides: the hypotenuse, opposite, and adjacent sides. The hypotenuse is the side opposite the right angle and is the longest side. The opposite side is the one across from the angle of interest, while the adjacent side touches the angle and the right angle.
In trigonometry, right triangles are fundamental because they are used to define the trigonometric ratios: sine, cosine, and tangent.

• The opposite side is used to define the sine and tangent.
• The adjacent side is used for cosine and tangent.
• The hypotenuse is involved in sine and cosine.

Understanding these relationships is crucial for solving problems involving right triangles, such as finding unknown side lengths or angles.

Graphing Calculator

A graphing calculator is a powerful tool used to visualize mathematical functions, including trigonometric functions like the tangent. It allows you to input functions and see their graphs, which can help in understanding the behavior and characteristics of these functions. When working with trigonometric functions, using a graphing calculator involves:

• Entering the function correctly, paying attention to angle units (degrees vs. radians).
• Setting the axes to appropriate ranges to display the graph effectively.
• Observing how changes in the function or angles affect the graph.

For example, in the exercise, graphing \(y = 50 \tan \theta\) on a graphing calculator helps in visualizing how the height of the triangle varies with angle \(\theta\). It provides a visual representation that can clarify how steep the slope gets as the angle increases.

Angle Conversion

Angle conversion is a fundamental concept when working with trigonometric functions. Trigonometric functions on calculators generally use radians, but many problems, including those in real life, use degrees. Converting between these two measurements is crucial.

• The conversion from degrees to radians is necessary for calculating trigonometric functions like tangent within calculator settings that default to radian mode.

The conversion formula is:

• \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)

In the provided exercise, to calculate the height at angles such as \(16^\circ\) and \(22^\circ\), you must first convert these angles to radians using the formula above. This ensures that when you input them into trigonometric functions like \(\tan\), the results will be accurate. Understanding and mastering angle conversion is essential for solving trigonometric problems effectively.