Problem 30 - 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. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}$$ $$y_{1}=\frac{1}{\sin x}-\frac{1}{\csc x}, \quad y_{2}=\csc x-\sin x$$

Step-by-Step Solution

Answer

The calculated values of \(y_{1}\) and \(y_{2}\) fill in the table, along with the graphs of \(y_{1}\) and \(y_{2}\), prove that \(y_{1}=y_{2}\). An algebraic verification also shows \(y_{1}=y_{2}\). Therefore, the identity is confirmed.

Step 1: Calculate \(y_{1}\) and \(y_{2}\)

In the given table, fill in the values of \(y_{1}\) and \(y_{2}\) for each corresponding x value. Remember that \(\csc x = 1/\sin x\), so the formula for \(y_{1}\) can be simplified to \(0\), because \(1/\sin x - 1/\sin x = 0\). The \(y_{1}\) values will therefore all equal \(0\). For \(y_{2}\), calculate \(\csc x - \sin x\) for each value of x.

Step 2: Graph the functions

Use a graphing utility to graph the functions \(y_{1}=0\) (a horizontal line along the x-axis) and \(y_{2}=\csc x - \sin x\). Plot the points from the table on your graph. You should notice that all the points for both \(y_{1}\) and \(y_{2}\) lie on the x-axis.

Step 3: Verify the identity algebraically

To prove that \(y_{1}=y_{2}\) algebraically, start with the equation \(0 = \csc x - \sin x\). Adding \(\sin x\) to both sides, you get \(\sin x = \csc x\) or \(1 = \csc x \cdot \sin x\), which shows that \(y_{1}=y_{2}\)

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that helps visualize mathematical functions and their relationships. It can be particularly helpful when comparing two or more functions, as it allows you to see where they overlap or differ on a graph.

Essentially, a graphing utility plots the values of given functions onto a coordinate plane, providing a visual representation that can enhance your understanding of mathematical concepts. When using a graphing utility:

• Input the functions you wish to compare, such as \( y_1 \) and \( y_2 \).
• It can automatically calculate and plot the points for the functions over the specified range of \( x \) values.
• It allows for easy manipulation of the viewing window, focusing on important intersections or aspects of the functions.

Using a graphing utility in our exercise can confirm whether \( y_1 = y_2 \) by visually identifying if they align on the graph.

Cosecant Function (csc x)

The cosecant function, denoted as \( \csc x \), is the reciprocal of the sine function. Mathematically, it is represented as:\[ \csc x = \frac{1}{\sin x} \]This function is particularly useful in trigonometry for solving various types of problems, as it provides additional perspectives on angles and their corresponding ratios. When calculating \( \csc x \), remember:

• It is undefined for \( x \) values where \( \sin x = 0 \), such as multiples of \( \pi \).
• Its graph appears as repeating curves with key asymptotes.

In our exercise, understanding \( \csc x \) is key, as both \( y_1 \) and \( y_2 \) involve the cosecant function. Recognizing the relationship between sine and cosecant allows for simplification of these functions.

Sine Function (sin x)

The sine function, represented as \( \sin x \), is one of the primary trigonometric functions. It describes the ratio of the opposite side to the hypotenuse in a right-angled triangle. The sine function is periodic with a range between -1 and 1 and completes one full cycle over an interval of \( 2\pi \).

Sine is an important component in various mathematical and physical applications. When analyzing \( \sin x \) within this context:

• Identify its basic properties, such as amplitude, period, and symmetry.
• Recognize how \( \sin x \) interacts with its reciprocal, building a foundation for understanding related identities.

In the provided exercise, \( \sin x \) helps to evaluate the expressions for \( y_1 \) and \( y_2 \) and later contributes to verifying the identity by algebraic manipulation.

Verification of Identities

Verification of trigonometric identities is a vital practice in understanding trigonometric relationships and properties. It involves confirming that two expressions are equivalent under given conditions.

In our exercise, verifying the identity algebraically involves proving that:\[ y_1 = y_2 \quad \text{i.e.} \quad 0 = \csc x - \sin x \]This process entails:

• Simplifying both sides using fundamental trigonometric identities.
• Performing algebraic manipulations to demonstrate equivalence.

In the exercise, adding \( \sin x \) to both sides affirms that \( \sin x \times \csc x = 1 \), verifying that the expressions are indeed identical. This type of verification strengthens problem-solving skills and deepens comprehension of how different trigonometric functions relate.