Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2} .\) Then verify the identity algebraically. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}$$ $$y_{1}=\left(1+\tan ^{2} x\right) \sin ^{2} x, \quad y_{2}=\tan ^{2} x$$

Step-by-Step Solution

Answer

Upon graphing and solving numerically, we see that for all values of \( x \) in our table, \( y_1=y_2 \). After substitution and simplification, the algebraic verification also confirms the identity and thus all three methods confirm that \( y_1=y_2 \).

Step 1- Filling up the Table

Use the graphing utility to complete the table. Fill in the values for \( y_1 \) and \( y_2 \) given the values of \( x \) on the table by plugging in the \( x \) values into \( y_1 \) and \( y_2 \) equations.

Step 2- Graphing

Graph the functions \( y_1=\left(1+\tan ^{2} x\right) \sin ^{2} x \) and \( y_2=\tan ^{2} x \) on the graphing utility. Make sure to use the same viewing window for both equations so the comparison can be made visually.

Step 3 - Analyzing the Table and Graph

Check the table and the graph to see if \( y_1 = y_2 \) for all values of \( x \). The calculated values in the table should match for both functions at those specific \( x \) values, and the graph of the functions should look identical.

Step 4 - Verifying Algebraically

Now, verify the identity algebraically. Start with \( y_1 = y_2 \) and manipulate the left side until you get to \( y_2 \). We know that \( \sin^2x = \tan^2x/(1+ \tan^2x) \). Substitute this into \( y_1 \) and simplify to get \( y_2 \) equation.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that allows students to visualize mathematical functions easily. When tackling problems that involve multiple functions, such as those with trigonometric identities, visualizing the graphs can help tremendously in understanding their behavior and how they relate to each other.

By inputting the equations for both functions, such as:

• \( y_1 = (1 + \tan^2 x) \sin^2 x \)
• \( y_2 = \tan^2 x \)

...into a graphing utility, you can plot these functions on the same viewing window. This gives a clear picture of whether and where the functions coincide over the range of x-values. In this example, you would enter different values of x to compute the corresponding y-values for both functions, effectively filling the missing slots in the table provided in the original exercise.

Algebraic Verification

Verifying an identity algebraically involves manipulating one side of an equation to match the form of the other. This is a crucial skill in mathematics, especially when dealing with trigonometric identities.

In the given task, you start by assuming that:

• \( y_1 = y_2 \)

For algebraic verification, you break down \( y_1 = (1 + \tan^2 x) \sin^2 x \) by substituting known identities.

Recall that \( \sin^2x = \frac{\tan^2x}{1+\tan^2x} \). Substitute this into \( y_1 \), and you can simplify to demonstrate that \( y_1 \) reduces to \( y_2 \), thereby confirming the equality.

This step-by-step simplification is important to show the equivalence algebraically, and it provides a solid foundation for understanding how the identities relate.

Function Comparison

Function comparison is all about analyzing the similarities and differences between two or more functions. In this exercise, you need to compare \( y_1 \) and \( y_2 \) to determine if they are equal.

After filling the table using the graphing utility, make a direct comparison of the y-values at each listed x-value. Are they the same? This numerical check establishes a preliminary check of their equivalence.

Next, examine the graphed functions. When graphed, if both lines overlap perfectly across all relevant x-values, it hints strongly at their equality. However, remember that verifying visually is not enough. The algebraic verification solidifies what you observe on the graph.

This combination of graphical and numerical comparison offers a full-spectrum approach to understanding how the functions behave and agree.

Tangent and Sine Functions

The tangent and sine functions are fundamental trigonometric functions that often appear together in identities.

Understanding how they interact is crucial. The tangent function is defined as:

• \( \tan x = \frac{\sin x}{\cos x} \)

This relationship connects the tangent and sine functions intrinsically. In the task at hand, you are exploring how these functions interact by setting expressions of sine squares and tangent squares equivalent to each other.

The identity used in algebraic verification builds on these fundamental definitions and relationships. Recognizing that \( \sin^2x = \frac{\tan^2x}{1 + \tan^2x} \) allows you to manipulate one function form to another, a skillset that is invaluable for trigonometric identity solutions and proofs across mathematics.