Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 27
a. Graph \(y=\cos x\) and \(y=\sec x\) together for \(-3 \pi / 2 \leq x\) \(\leq 3 \pi / 2\) . Comment on the behavior of sec \(x\) in relation to the signs and values of \(\cos x\) b. Graph \(y=\sin x\) and \(y=\csc x\) together for \(-\pi \leq x \leq 2 \pi\) Comment on the behavior of \(\csc x\) in relation to the signs and values of \(\sin x .\)
Short Answer
Expert verified
\( \sec x \) and \( \csc x \) mirror \( \cos x \) and \( \sin x \), respectively, with reciprocal behavior, and are undefined where the originals are zero.
Step by step solution
01
Understanding the Functions
The function \( y = \cos x \) is the cosine function, which oscillates between -1 and 1 with a period of \( 2\pi \). The function \( y = \sec x \) is the secant function, which is the reciprocal of the cosine function. For \( y = \sec x \), it is undefined where \( \cos x = 0 \), which occurs at odd multiples of \( \pi/2 \).
02
Graphing \( y = \cos x \) and \( y = \sec x \)
Graph \( y = \cos x \) from \(-3\pi/2\) to \(3\pi/2\). The cosine curve will have intercepts at \( -3\pi/2, -\pi/2, \pi/2, \) and \( 3\pi/2 \) where \( y = 0 \), peaks at \( -\pi \) and \( \pi \) where \( y = 1 \), and troughs at \( 0 \) where \( y = -1 \). Now, graph \( y = \sec x \), noting that it will have vertical asymptotes at \( -\pi/2, \pi/2, \) and \( 3\pi/2 \) where \( \cos x = 0 \). \( \sec x \) takes positive values when \( \cos x > 0 \), negative values when \( \cos x < 0 \), and \( \sec x \rightarrow \infty \) or \( -\infty \) near asymptotes.
03
Commenting on \( \sec x \) in relation to \( \cos x \)
The secant curve \( y = \sec x \) is defined where the cosine is not zero and extends to infinity at points where \( \cos x = 0 \). When \( \cos x > 0 \), \( \sec x \) is positive and mirrors \( \cos x \) growing towards infinity near \( \cos x = 0 \). Similarly, when \( \cos x < 0 \), \( \sec x \) is negative.
04
Understanding the Sine and Cosecant Functions
The function \( y = \sin x \) is the sine function, which oscillates between -1 and 1 with a period of \( 2\pi \). The function \( y = \csc x \) is the cosecant function, which is the reciprocal of the sine function. For \( y = \csc x \), it is undefined where \( \sin x = 0 \), occurring at multiples of \( \pi \).
05
Graphing \( y = \sin x \) and \( y = \csc x \)
Graph \( y = \sin x \) from \(-\pi\) to \(2\pi\), finding intercepts at \( -\pi, 0, \pi, \) and \( 2\pi \), peaks at \( \pi/2, 5\pi/2, \) and troughs at \( 3\pi/2 \). Now, graph \( y = \csc x \), noting vertical asymptotes at \( -\pi, 0, \pi, \) and \( 2\pi \). \( \csc x \) takes positive values when \( \sin x > 0 \), negative values when \( \sin x < 0 \), and tends towards infinity as it approaches these asymptotes.
06
Commenting on \( \csc x \) in relation to \( \sin x \)
The cosecant curve \( y = \csc x \) is defined where the sine is not zero and heads to infinity at points where \( \sin x = 0 \). \( \csc x \) follows the behavior of \( \sin x \) but inversely extends towards infinity or negative infinity where \( \sin x \) is small and non-zero.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cosine Function
The cosine function, represented by the equation \( y = \cos x \), is a fundamental trigonometric function that exhibits periodic behavior. It oscillates between -1 and 1 as \( x \) varies. This predictable wave-like structure has a period of \( 2\pi \), meaning it completes one full cycle every \( 2\pi \) units of \( x \).
Some key points to understand about the cosine graph are:
Some key points to understand about the cosine graph are:
- The maximum value occurs at \( \cos(0), \cos(2\pi), \cos(-2\pi), ...\) where \( y = 1 \).
- The minimum value of -1 is reached at \( x = \pi, 3\pi, -\pi, -3\pi, ... \).
- Critical points, where \( \cos x = 0 \), appear at odd multiples of \( \pi/2 \), such as \( \pm\pi/2, \pm3\pi/2, ... \).
Secant Function
The secant function \( y = \sec x \) is the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). It shares some traits with cosine but diverges at points where the cosine equals zero. At those points, the secant function becomes undefined, creating vertical asymptotes in its graph. These asymptotes occur at odd multiples of \( \pi/2 \), just where \( \cos x = 0 \).
Important properties of the secant function include:
Important properties of the secant function include:
- When \( \cos x > 0 \) (for example, from \(-\pi\) to \(0\)), \( \sec x \) is positive.
- It takes negative values wherever \( \cos x < 0 \) (such as from \( \pi \) to \( 2\pi \)).
- \( \sec x \) approaches infinity as it gets closer to \( \cos x = 0 \).
Sine Function
The sine function \( y = \sin x \) is another primary trigonometric function. Like the cosine function, it oscillates between -1 and 1. However, its graph initiates at the origin and has a period of \( 2\pi \), completing a wave every \( 2\pi \) units of \( x \).
Notable features of the sine wave include:
Notable features of the sine wave include:
- The function hits its peak value of 1 at \( \pi/2, 5\pi/2, ... \).
- It reaches its lowest value of -1 at \( 3\pi/2, 7\pi/2, ... \).
- \( \sin x = 0 \) at all integer multiples of \( \pi \), such as \( 0, \pi, 2\pi, ...\).
Cosecant Function
The cosecant function \( y = \csc x \) is defined as the reciprocal of the sine function: \( \csc x = \frac{1}{\sin x} \). This function is significant in trigonometry due to its relationship with sine. It's undefined wherever \( \sin x = 0 \), resulting in vertical asymptotes at those points: \( 0, \pi, 2\pi, ... \).
Key aspects of the cosecant graph include:
Key aspects of the cosecant graph include:
- \( \csc x \) attains positive values where \( \sin x > 0 \), for instance, from \( 0 \) to \( \pi \).
- It takes negative values when \( \sin x < 0 \), such as from \( \pi \) to \( 2\pi \).
- \( \csc x \) trends towards either infinity or negative infinity as \( \sin x \) approaches zero.
