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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![### Understanding Undefined Values of Cotangent Function in Trigonometry
In trigonometry, certain values of the angle \(\Theta\) (\(\Theta\) is pronounced as theta) can result in the cotangent (cot) function being undefined. The cotangent of an angle is defined as the ratio of the cosine to the sine of that angle:
\[ \cot \Theta = \frac{\cos \Theta}{\sin \Theta} \]
For the cotangent function to be undefined, the denominator of this ratio (i.e., \(\sin \Theta\)) must be zero. Therefore, cotangent is undefined for angles where the sine value is zero.
#### Problem Statement:
For what value of \(\Theta\) is \(\cot \Theta\) undefined?
Here are the options provided:
I. \(0^\circ\)
II. \(180^\circ\)
III. \(270^\circ\)
The choices are:
- [ ] I only
- [ ] I and III only
- [ ] II and III only
- [ ] I, II, and III
#### Explanation:
To solve this, we need to determine the angles at which the sine function is zero:
1. At \(\Theta = 0^\circ\): \(\sin(0^\circ) = 0\), thus \(\cot 0^\circ\) is undefined.
2. At \(\Theta = 180^\circ\): \(\sin(180^\circ) = 0\), thus \(\cot 180^\circ\) is undefined.
3. At \(\Theta = 270^\circ\): \(\sin(270^\circ) = -1\), thus \(\cot 270^\circ\) is defined.
Given the above observations, \(\cot\) is undefined at \(0^\circ\) and \(180^\circ\) but not at \(270^\circ\). Therefore, the correct answer is:
- [ ] I and II only
Selecting this answer ensures the correct understanding of when the cotangent function is undefined in trigonometry based on the properties of the sine function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e08c0ed-2478-4250-8ecc-80933702ca06%2F76f14ef7-981b-450d-9397-08f28a73e7ef%2F8txyq0o_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Undefined Values of Cotangent Function in Trigonometry
In trigonometry, certain values of the angle \(\Theta\) (\(\Theta\) is pronounced as theta) can result in the cotangent (cot) function being undefined. The cotangent of an angle is defined as the ratio of the cosine to the sine of that angle:
\[ \cot \Theta = \frac{\cos \Theta}{\sin \Theta} \]
For the cotangent function to be undefined, the denominator of this ratio (i.e., \(\sin \Theta\)) must be zero. Therefore, cotangent is undefined for angles where the sine value is zero.
#### Problem Statement:
For what value of \(\Theta\) is \(\cot \Theta\) undefined?
Here are the options provided:
I. \(0^\circ\)
II. \(180^\circ\)
III. \(270^\circ\)
The choices are:
- [ ] I only
- [ ] I and III only
- [ ] II and III only
- [ ] I, II, and III
#### Explanation:
To solve this, we need to determine the angles at which the sine function is zero:
1. At \(\Theta = 0^\circ\): \(\sin(0^\circ) = 0\), thus \(\cot 0^\circ\) is undefined.
2. At \(\Theta = 180^\circ\): \(\sin(180^\circ) = 0\), thus \(\cot 180^\circ\) is undefined.
3. At \(\Theta = 270^\circ\): \(\sin(270^\circ) = -1\), thus \(\cot 270^\circ\) is defined.
Given the above observations, \(\cot\) is undefined at \(0^\circ\) and \(180^\circ\) but not at \(270^\circ\). Therefore, the correct answer is:
- [ ] I and II only
Selecting this answer ensures the correct understanding of when the cotangent function is undefined in trigonometry based on the properties of the sine function.
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