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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Question
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**Question:**
The vertical asymptotes of the tangent function are located at the same values of \( x \) for which the cosine function is equal to 0.
- ○ True
- ○ False
---
**Explanation:**
This question is related to the properties of the trigonometric functions tangent and cosine.
In trigonometry, the tangent function, \(tan(x)\), has vertical asymptotes where the cosine function, \(cos(x)\), has zeroes. The reason for this is that the tangent function is defined as the ratio of the sine function to the cosine function, \(tan(x) = \frac{sin(x)}{cos(x)}\). When \(cos(x) = 0\), the function becomes undefined because division by zero is undefined. Therefore, vertical asymptotes occur at these values.
### Graph/Diagram:
Although there is no graph in the provided image, let's conceptualize the relationship.
If we were to graph the cosine function, we would identify the points where \(cos(x) = 0\). These occur at:
\[x = \frac{\pi}{2} + k\pi\]
where \(k\) is any integer.
In turn, on a graph of the tangent function, we would see vertical asymptotes at these same values of \(x\):
\[x = \frac{\pi}{2} + k\pi\]
Visually, a tangent graph repeats every \(\pi\) units and has vertical lines (asymptotes) where it shoots up to \(\infty\) and down to \(-\infty\) at these critical points.
By answering 'True' to this question, students affirm their understanding that the vertical asymptotes of the tangent function directly correspond to the zeros of the cosine function.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f130582-8262-45ba-b1cd-6680cc0034f3%2F7d1160c9-d2ab-4270-8d5f-d4ae5e8c52d7%2Fc40zker_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Question:**
The vertical asymptotes of the tangent function are located at the same values of \( x \) for which the cosine function is equal to 0.
- ○ True
- ○ False
---
**Explanation:**
This question is related to the properties of the trigonometric functions tangent and cosine.
In trigonometry, the tangent function, \(tan(x)\), has vertical asymptotes where the cosine function, \(cos(x)\), has zeroes. The reason for this is that the tangent function is defined as the ratio of the sine function to the cosine function, \(tan(x) = \frac{sin(x)}{cos(x)}\). When \(cos(x) = 0\), the function becomes undefined because division by zero is undefined. Therefore, vertical asymptotes occur at these values.
### Graph/Diagram:
Although there is no graph in the provided image, let's conceptualize the relationship.
If we were to graph the cosine function, we would identify the points where \(cos(x) = 0\). These occur at:
\[x = \frac{\pi}{2} + k\pi\]
where \(k\) is any integer.
In turn, on a graph of the tangent function, we would see vertical asymptotes at these same values of \(x\):
\[x = \frac{\pi}{2} + k\pi\]
Visually, a tangent graph repeats every \(\pi\) units and has vertical lines (asymptotes) where it shoots up to \(\infty\) and down to \(-\infty\) at these critical points.
By answering 'True' to this question, students affirm their understanding that the vertical asymptotes of the tangent function directly correspond to the zeros of the cosine function.
---
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