Problem 26 - 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. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}$$ $$y_{1}=\sec x-\cos x, \quad y_{2}=\sin x \tan x$$

Step-by-Step Solution

Answer

Upon graphing the functions \(y_{1}=\sec x-\cos x\) and \(y_{2}=\sin x \tan x\), completing the table for the given \(x\) values, and verifying the algebraic identity would demonstrate that \(y_{1}\) is indeed equal to \(y_{2}\). It's advised to use a good graphing utility and knowledge of trigonometric identities for accurate solutions.

Step 1: Calculate values

First, use a graphing utility to calculate \(y_{1}\) and \(y_{2}\) values corresponding to the \(x\) values in the given table. Plug each \(x\) value into the functions \(y_{1}=\sec x-\cos x\) and \(y_{2}=\sin x \tan x\) to get the respective \(y_{1}\) and \(y_{2}\) values.

Step 2: Complete the table

Once the \(y_{1}\) and \(y_{2}\) values are calculated, complete the table filling in the respective values. The table will serve as the evidence for \(y_{1}\) = \(y_{2}\) when graphed.

Step 3: Graph the functions

Now, graph the functions \(y_{1}\) and \(y_{2}\) in the same viewing window using the calculated \(y_{1}\) and \(y_{2}\) values and the given \(x\) values. The graph will visually prove the identity \(y_{1}\)= \(y_{2}\).

Step 4: Verify the identity algebraically

Finally, to confirm the results, verify algebraically that \(y_{1}=\sec x-\cos x\) is equal to \(y_{2}=\sin x \tan x\). Use trigonometric identities to simplify the equations, until they match each other.

Key Concepts

Graphing Trigonometric Functions

When graphing trigonometric functions like secant (\text{sec}) and cosine (\text{cos}), or sine (\text{sin}) and tangent (\text{tan}), it's important to begin with understanding the basic shapes of their graphs. For instance, sine and cosine functions have a wave-like pattern known as a sinusoidal shape, with peaks and troughs. The tangent function, however, has a repeating pattern of vertical asymptotes and secant has a similar pattern but reflected over the cosine curve.

Graphing utilities are valuable tools for plotting these functions with precision, catering to various x-values and allowing us to view the behavior of the function over a chosen interval. This visual representation can be crucial to reinforcing the relationships and transformations happening within trigonometrical identities. When tasked with verifying an identity like \(y_{1} = y_{2}\), visual evidence combined with an algebraic proof offers a compelling confirmation of the identity's validity.

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that provide relationships between different trigonometric functions. These identities are equations that hold true for all values of the variable involved. They are used to simplify expressions, solve trigonometric equations, and to prove other mathematical concepts.

An example of a trigonometric identity is \(1 + \tan^2 x = \text{sec}^2 x\), which shows a relationship between tangent and secant functions. Such identities are crucial for solving complex trigonometric problems. When you are verifying an identity, such as \(y_{1}=\text{sec} x - \text{cos} x = y_{2}=\text{sin} x \tan x\), the goal is to transform one side of the equation to make it look exactly like the other side using known identities.

Secant and Cosine

Understanding the relationship between secant (\text{sec}) and cosine (\text{cos}) is essential for working with trigonometric equations and identities. The secant function is the reciprocal of the cosine function, which is mathematically denoted as \(\text{sec} x = \frac{1}{\text{cos} x}\). This identity suggests that for any angle \(x\), the value of \(\text{sec} x\) is the multiplicative inverse of \(\text{cos} x\).

Visualizing the Relationship

When graphing, \text{cos} x exhibits a wave pattern, while \text{sec} x will have the same pattern but will also include vertical lines (asymptotes) where the cosine value equals zero because division by zero is undefined. Hence, understanding how \text{sec} and \text{cos} coexist on the unit circle and how their graphs relate can significantly aid in verifying identities involving these functions.

Sine and Tangent

The sine (\text{sin}) and tangent (\text{tan}) functions are two of the six fundamental trigonometric functions. They are related through the trigonometric identity \(\tan x = \frac{\text{sin} x}{\text{cos} x}\). This means that the tangent of an angle in a right-angled triangle can be found by dividing the length of the side opposite the angle (\text{sin} x) by the length of the adjacent side (\text{cos} x).

In the context of verifying identities, remembering their relationship can simplify complex expressions. For example, the identity \(\text{sec} x - \text{cos} x\) could be expressed in terms of \text{sin} and \text{tan} by utilizing this relationship as shown in the original exercise solution. By breaking down these concepts, students can tackle challenging problems one piece at a time and gradually build a complete understanding of how trigonometric functions interconnect.