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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblem 26
Chapter 2, Problem 26

Use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.


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Verified step by step guidance
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Identify the original trigonometric function from the given graph. It will be either sine or cosine, as their reciprocal functions are cosecant and secant respectively.
Recall that the reciprocal function of sine is cosecant, defined as \(\csc x = \frac{1}{\sin x}\), and the reciprocal function of cosine is secant, defined as \(\sec x = \frac{1}{\cos x}\).
Analyze the original graph to find where the function crosses the x-axis (zeros). These points correspond to vertical asymptotes in the reciprocal function's graph because division by zero is undefined.
Sketch the reciprocal function by plotting the reciprocal values of the original function's points, noting that the reciprocal function will have vertical asymptotes where the original function is zero, and will approach zero where the original function has large magnitude.
Write the equation of the reciprocal function based on the original function identified: if the original is \(y = \sin x\), then the reciprocal is \(y = \csc x\); if the original is \(y = \cos x\), then the reciprocal is \(y = \sec x\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Reciprocal Trigonometric Functions

Reciprocal functions like cosecant (csc) and secant (sec) are defined as the reciprocals of sine and cosine, respectively. Specifically, csc(x) = 1/sin(x) and sec(x) = 1/cos(x). Understanding these relationships is essential to transform sine or cosine graphs into their reciprocal counterparts.
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Graphing Reciprocal Functions

To graph cosecant or secant, start with the sine or cosine graph and identify points where the original function is zero, as these correspond to vertical asymptotes in the reciprocal graph. The reciprocal graph has branches that approach these asymptotes and reflect the peaks and troughs of the original function inversely.
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Equation Identification from Graphs

Determining the equation from a graph involves recognizing the function type, amplitude, period, phase shift, and vertical shift. For reciprocal functions, these parameters come from the original sine or cosine function before taking the reciprocal, allowing you to write the exact equation of the cosecant or secant graph.
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Related Practice
Textbook Question

In Exercises 21–28, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, find the following: a. the maximum displacement b. the frequency c. the time required for one cycle. d = 1/3 sin 2t

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Textbook Question

In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = −2 sin(2x + π/2)

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Textbook Question

In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. sin⁻¹ 0.3

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Textbook Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.

y = 3 sin(πx + 2)

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Textbook Question

In Exercises 25–28, use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.

Textbook Question

In Exercises 1–26, find the exact value of each expression. _ sec⁻¹ (−√2)