Problem 29 - 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. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}$$ $$y_{1}=\frac{1}{\tan x}+\frac{1}{\cot x} ; \quad y_{2}=\tan x+\cot x$$

Step-by-Step Solution

Answer

After graphing the given functions \(y_{1} = \frac{1}{\tan x} + \frac{1}{\cot x}\) and \(y_{2} = \tan x + \cot x\), and placing the appropriate 'y' values into the table for each 'x' value, it's proven both visually and numerically that \(y_{1} = y_{2}\). The algebraic verification confirms that \(y_{1} = \frac{1}{\tan x} + \frac{1}{\cot x}\) is identical to \(y_{2} = \tan x + \cot x\), thereby proving the original equation.

Step 1: Graphing and Completing the Table

First, use a graphing utility to plot the graph of functions \(y_{1} = \frac{1}{\tan x} + \frac{1}{\cot x}\) and \(y_{2} = \tan x + \cot x\). Then, fill in the table with the corresponding 'y' values for each given 'x' value. The graphing utility will help you in getting the corresponding 'y' values.

Step 2: Comparing the Graphs and Tables

Next, compare the graphs of the two functions \(y_{1}\) and \(y_{2}\). They should overlap each other if they are identical. Also, for each 'x' value in the table, the 'y' values for \(y_{1}\) and \(y_{2}\) should match.

Step 3: Verifying Algebraically

Now, verify the identity algebraically. Given \(y_{1} = \frac{1}{\tan x} + \frac{1}{\cot x}\), which can be written as \(\cot x + \tan x\). This is identical to \(y_{2} = \tan x + \cot x\). Thus, we can confidently say that \(y_{1} = y_{2}\).

Key Concepts

Graphing Utility

Graphing utilities are powerful tools that allow you to visualize mathematical functions and their relationships. When working with trigonometric identities, like in our exercise, a graphing utility helps us plot the functions of interest on a coordinate plane. This offers a visual representation which can make understanding complex identities much easier.
In this particular exercise, the functions to be graphed are \(y_{1} = \frac{1}{\tan x} + \frac{1}{\cot x}\) and \(y_{2} = \tan x + \cot x\). A graphing utility will plot these functions over a specified range of \(x\) values, which you typically set based on the problem requirements or your investigation needs.

• Ensure your graphing utility is set to the correct mode for trigonometric functions, typically radians.
• Input the trigonometric functions accurately into the utility.
• Use the utility to calculate 'y' values for the given 'x' values, as shown in the table from the exercise.
• Observe the graphs: they should lie on top of each other if the identities hold true, illustrating equality visually.

These visual checks are your first step in verifying that the two expressions are equivalent, before moving on to algebraic verification.

Function Comparison

When comparing functions, especially in trigonometry, you want to see how their values behave across different inputs. In our exercise, function comparison is done both through graph visualizations and table comparisons.
First, after using a graphing utility to plot the functions, visually inspect whether the graphs overlap. If they do, this is an indication that the functions possibly represent the same entity and, hence, support their algebraic equivalence.
Second, check the calculated 'y' values for each 'x' in the table. Ideally, for every \(x\) value given, the output \(y_{1}\) and \(y_{2}\) should be equal. This numerical agreement reinforces the hypothesis drawn from the graphical overlap.

• Review the plots closely: even tiny deviations might suggest a scale or calculation error.
• Focus on the role of x-values: choose values that are within the valid domain for both tan and cot.
• Take note of any exceptions or undefined areas.

While such comparisons provide strong evidence, confirming equivalence algebraically is equally important.

Algebraic Verification

Algebraic verification involves manipulating equations to show that two expressions are equivalent. This is the most reliable method to confirm that two trigonometric identities truly represent the same function.
Let's dive into the identity verification in our example: start with \(y_{1} = \frac{1}{\tan x} + \frac{1}{\cot x}\).

• The expression \( \frac{1}{\tan x} \) is equivalent to \( \cot x \), since \( \tan x = \frac{1}{\cot x} \).
• Similarly, \( \frac{1}{\cot x} \) becomes \( \tan x \).

Thus, simplifying \(y_{1}\) gives us \( \cot x + \tan x \), which is exactly \(y_{2}\). By performing these simple algebraic steps, you confirm the identity: \(y_{1} = y_{2}\).
Algebraic proof is robust because it demonstrates the equivalence without reliance on graphical representations, which can sometimes be limited by precision issues or domain restrictions. It's always good to support graphical observations with this algebraic confirmation, ensuring your solution is comprehensive and rigorous.