Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to complete the table and graph the two 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. $$y_{1}=\cos (x+\pi) \cos (x-\pi), \quad y_{2}=\cos ^{2} x$$

Step-by-Step Solution

Answer

By using the graphing utility to create the graphs and table, and comparing the results, it can be observed that \(y_{1}=\cos (x+\pi) \cos (x-\pi)\) and \(y_{2}=\cos^{2} x\) have the same y-values for all x-values graphed. Additionally, the algebraic manipulation confirms that these two functions are equivalent.

Step 1: Graph the functions

Use a graphing utility to graph the functions \(y_{1}=\cos (x+\pi) \cos (x-\pi)\) and \(y_{2}=\cos^{2} x\) on the same set of axes. Make sure to use the same range and domain for both functions.

Step 2: Create the table

Use the graphing utility to create a table of values for both functions. The values in the table should show the y-values for a range of x-values. Make sure to use the same x-values for both functions.

Step 3: Compare the graph and the table

Look at the graph and the table. The y-values for \(y_{1}\) and \(y_{2}\) should be the same at each x-value, which shows that \(y_{1}=y_{2}\).

Step 4: Verify the identity algebraically

Solve the algebraic expressions to verify that \(y_{1}=\cos (x+\pi) \cos (x-\pi)\) is equivalent to \(y_{2}=\cos ^{2} x\). The formula \(\cos(a)\cos(b) = 0.5[cos(a+b) + cos(a-b)]\) can be used. For \(y_{1}\), \(a\)=x+\(\pi\) and \(b\)=x-\(\pi\). Those sum up to \(2x\) and subtract to \(2\pi\). The final simplified expression for \(y_{1}\) is \(\cos^{2} x\) which shows \(y_{1} = y_{2}\).

Key Concepts

Trigonometric Functions Graphing

Graphing trigonometric functions is a fundamental skill in trigonometry that helps students visualize the behavior of these functions. Trigonometric graphs often exhibit repeating patterns known as periodicity. For example, the cosine and sine functions have wave-like shapes that repeat every \(2\pi\) radians.

When graphing \(y_1 = \cos(x + \pi)\cos(x - \pi)\) alongside \(y_2 = \cos^2 x\), students must pay attention to key features such as amplitude, period, phase shift, and vertical shift. By using the same range and domain for both functions, one can effectively compare their graphs. This visual comparison can provide preliminary evidence for the equality of \(y_1\) and \(y_2\), before proceeding with algebraic verification. Trigonometric functions graphing is also an excellent way to predict the behavior of composite trigonometric functions and their transformations.

Algebraic Verification of Identities

Algebraic verification of identities involves showing that two different trigonometric expressions are equivalent by simplifying one or both expressions. Trigonometric identities, such as angle sum and difference formulas, are often used in this process. To verify the identity \(y_1 = \cos(x + \pi)\cos(x - \pi)\) and \(y_2 = \cos^2 x\) algebraically, you can apply these formulas to rewrite and simplify the expressions.

In this case, the product-to-sum identity \(\cos(a)\cos(b) = \frac{1}{2}[\cos(a+b) + \cos(a-b)]\) is instrumental in transforming the product of two cosine terms into a format that can be easily compared to \(y_2\). Through such algebraic manipulations, we can demonstrate the equality of two trigonometric expressions and thus verify the identity beyond graphical evidence.

Use of Graphing Utilities in Trigonometry

Graphing utilities are invaluable tools in trigonometry that can provide students with numerical and visual insights into trigonometric functions. These utilities allow for precise plotting of functions, generation of tables of values, and observation of the intersection points of different functions. In exercises like the one provided, students can use graphing utilities to observe the overlap of graphs for \(y_1\) and \(y_2\), indicating they are indeed the same function.

The ability to generate a table of values supports numerical evidence when x-values result in the same y-values for both functions. Furthermore, the use of these utilities assists students in understanding the practical applications of trigonometry, by facilitating the visualization and comparison of complex trigonometric expressions.

Cosine Function Properties

The cosine function, one of the primary trigonometric functions, has several key properties that are important to understand when working with trigonometric identities. These properties include its periodic nature, with a period of \(2\pi\) radians, and its even symmetry, meaning that \(\cos(-x) = \cos(x)\).

Additionally, the cosine function has a range between -1 and 1, and its graph is a wave that starts at a maximum value of 1 when x is 0. The property of \(\cos(x + \pi) = -\cos(x)\) is particularly useful when dealing with functions involving a shift by \(\pi\), as seen in the given exercise. The understanding of such properties is crucial when simplifying trigonometric expressions and verifying identities as these intrinsic characteristics can provide shortcuts and insights into the behavior of the cosine function.