Textbook Question
Determine an equation for each graph.
Ch. 4 - Graphs of the Circular Functions
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 5, Problem 4.35
Decide whether each statement is true or false. If false, explain why.
The graph of y = sec x in Figure 37 suggests that sec(-x) = sec x for all x in the domain of sec x.
Verified step by step guidance1
Recall that the secant function is defined as \( \sec x = \frac{1}{\cos x} \).
Consider the property of cosine: \( \cos(-x) = \cos x \), which indicates that cosine is an even function.
Using the property of cosine, substitute into the secant function: \( \sec(-x) = \frac{1}{\cos(-x)} = \frac{1}{\cos x} = \sec x \).
Since \( \sec(-x) = \sec x \), the secant function is also an even function.
Therefore, the statement \( \sec(-x) = \sec x \) is true for all \( x \) in the domain of \( \sec x \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Secant Function
The secant function, denoted as sec(x), is defined as the reciprocal of the cosine function: sec(x) = 1/cos(x). It is important to understand its properties, including its domain and range, as well as its periodic nature. The secant function is undefined wherever the cosine function is zero, leading to vertical asymptotes in its graph.
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Graphs of Secant and Cosecant Functions
Even and Odd Functions
A function is classified as even if f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. Conversely, a function is odd if f(-x) = -f(x), indicating symmetry about the origin. Recognizing whether a function is even or odd is crucial for determining the behavior of functions like secant in relation to their inputs.
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Graphical Interpretation
Graphical interpretation involves analyzing the visual representation of a function to understand its properties and behaviors. For the secant function, examining its graph can reveal symmetries and periodicity. In this case, observing the graph of y = sec(x) helps to confirm whether sec(-x) equals sec(x), which is essential for validating the statement in the question.
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Related Practice
Textbook Question
Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0, for the given graph. See Example 6.
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Textbook Question
Graph each function over a one-period interval.
y = csc((1/2)x - π/4)
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Textbook Question
A rotating beacon is located at point A, 4 m from a wall. The distance a is given by
a = 4 |sec 2πt|,
where t is time in seconds since the beacon started rotating. Find the value of a for each time t. Round to the nearest tenth if applicable.
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t = 1.24
Textbook Question
Fill in the blank(s) to correctly complete each sentence.
The graph of y = -3 sin x is obtained by stretching the graph of y = sin x by a factor of ________ and reflecting across the ________-axis.
Textbook Question
Graph each function over a one-period interval.
y = csc (x - π/4)
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