Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Cotangent Function Asymptotes and Sine Function Zeros
**Question:**
The vertical asymptotes of the cotangent function are located at the same values of \(x\) for which the sine function is equal to \(0\).
- True
- False
**Explanation:**
The cotangent function, \(\cot(x)\), is defined as the quotient of the cosine function and the sine function:
\[ \cot(x) = \frac{\cos(x)}{\sin(x)} \]
Vertical asymptotes of the cotangent function occur where its denominator is zero. Therefore, vertical asymptotes are found where \(\sin(x) = 0\).
The sine function, \(\sin(x)\), is zero at integer multiples of \(\pi\) (i.e., \(0, \pm\pi, \pm2\pi, \ldots\)).
Hence, the statement "The vertical asymptotes of the cotangent function are located at the same values of x for which the sine function is equal to 0" is:
- True](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f130582-8262-45ba-b1cd-6680cc0034f3%2F6b9ca550-8e81-425d-bab6-6f0989dde565%2Fqaoquc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Cotangent Function Asymptotes and Sine Function Zeros
**Question:**
The vertical asymptotes of the cotangent function are located at the same values of \(x\) for which the sine function is equal to \(0\).
- True
- False
**Explanation:**
The cotangent function, \(\cot(x)\), is defined as the quotient of the cosine function and the sine function:
\[ \cot(x) = \frac{\cos(x)}{\sin(x)} \]
Vertical asymptotes of the cotangent function occur where its denominator is zero. Therefore, vertical asymptotes are found where \(\sin(x) = 0\).
The sine function, \(\sin(x)\), is zero at integer multiples of \(\pi\) (i.e., \(0, \pm\pi, \pm2\pi, \ldots\)).
Hence, the statement "The vertical asymptotes of the cotangent function are located at the same values of x for which the sine function is equal to 0" is:
- True
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