Problem 6 - Precalculus : Building Concepts and Connections

Question

Use your knowledge of vertical translations to graph at least two cycles of the given functions. $$g(x)=\csc x-\frac{1}{2}$$

Step-by-Step Solution

Answer

The function \(g(x) = \csc x - \frac{1}{2}\) is a graph of the cosecant function, shifted downwards by \(\frac{1}{2}\) units. The vertical asymptotes occur at \(x = n\pi\), where \(n\) is an integer, and the period of the function is \(2\pi\).

Step 1: Understanding the cosecant function

The cosecant function, denoted as \(\csc(x)\), is the reciprocal of the sine function. It is defined as \(\csc(x) = \frac{1}{\sin(x)}\). The period of the cosecant function is \(2\pi\) and it has vertical asymptotes at \(x = n\pi\), where \(n\) is an integer.

Step 2: Understanding vertical translation

A vertical translation involves shifting the graph of a function up or down without changing its shape or orientation. The function \(g(x) = \csc x - \frac{1}{2}\) is a vertical translation of the graph of the cosecant function, shifted downwards by \(\frac{1}{2}\) units.

Step 3: Graphing the function

To graph the function \(g(x) = \csc x - \frac{1}{2}\), start by graphing the cosecant function for at least two periods, say from \(x = -2\pi\) to \(x = 2\pi\). Then, shift each point on the graph down by \(\frac{1}{2}\) units. Remember that the vertical asymptotes remain the same, while the maximum and minimum values of the function are affected by the vertical shift.

Key Concepts

Cosecant Function

The cosecant function, symbolized as \(\csc(x)\), forms an essential part of trigonometry. This function is essentially the reciprocal of the sine function, so it’s defined as \(\csc(x) = \frac{1}{\sin(x)}\). To understand the cosecant function better, consider the graph of the sine function. Wherever the sine function has a value of zero, the cosecant function will have a vertical asymptote. This is because division by zero is undefined. Thus, the cosine function exhibits vertical asymptotes at points \(x = n\pi\), where \(n\) is any integer. The graph of the cosecant function displays a series of U-shaped or inverted U-shaped curves, repeating every \(2\pi\), which is also the function's period.The behavior of these U-shaped curves is due to the reciprocal nature of the sine values. As \(\sin(x)\) approaches zero, \(\csc(x)\) values approach infinity, hence the sharp peaks and troughs observed in its graph. Understanding these points is crucial for manipulating and graphing the function accurately.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions include cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). They serve as the counterparts to the primary trigonometric functions:

• The cosecant is the reciprocal of sine: \(\csc(x) = \frac{1}{\sin(x)}\)
• The secant is the reciprocal of cosine: \(\sec(x) = \frac{1}{\cos(x)}\)
• The cotangent is the reciprocal of tangent: \(\cot(x) = \frac{1}{\tan(x)}\)

Understanding these relationships is vital for interpreting their graphs and determining where they are undefined. Just as sine, cosine, and tangent reach specific points, the reciprocals turn these values upside down. Wherever these primary functions hit zero, the reciprocal functions have undefined or infinite values, resulting in vertical asymptotes on their graphs. Vertical asymptotes mark essential breaks in these graphs, indicating points where the function is not defined. This is particularly vivid for cosecant, as it mirrors the behavior of the sine function by flipping its extremes.

Periodicity in Trigonometric Functions

Periodicity is a striking feature of trigonometric functions, signifying that they repeat their values in regular intervals. This repetitive nature is known as the period. For sine and cosine, this period is \(2\pi\), which means their values repeat every \(2\pi\) units on the x-axis. Similarly, the cosecant function, being directly related to the sine function, also shares this \(2\pi\) period. This means that every \(2\pi\) units, the graph of \(\csc(x)\) will exhibit the same set of values and behavior again.Visualizing this, you’d see the cosecant graph as a sequence of repeating hills and valleys across the x-axis. Furthermore, this characteristic offers predictability; once you understand one cycle, you know how the function behaves indefinitely.Recognizing periodicity in trigonometric functions like \(\csc(x)\) makes it easier to work with equations, simplifying problems and helping you predict future values across different cycles. It serves as a map, guiding your understanding of these functions in trigonometry.