2. Determine whether the following function is even, odd or neither. Justify your answer algebraically, and clearly state your findings. S(t)= tan(t)· sin(t)· cos(t)
2. Determine whether the following function is even, odd or neither. Justify your answer algebraically, and clearly state your findings. S(t)= tan(t)· sin(t)· cos(t)
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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![Sure! Here is the transcription as it might appear on an educational website:
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**Problem 2: Function Symmetry Analysis**
Determine whether the following function is even, odd, or neither. Justify your answer algebraically and clearly state your findings.
\[ f(t) = \tan(t) \cdot \sin(t) \cdot \cos(t) \]
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To determine if the function is even, odd, or neither, we analyze the properties of the individual components:
1. **Tangent Function (\(\tan(t)\))**: This function is odd since \(\tan(-t) = -\tan(t)\).
2. **Sine Function (\(\sin(t)\))**: This function is odd since \(\sin(-t) = -\sin(t)\).
3. **Cosine Function (\(\cos(t)\))**: This function is even since \(\cos(-t) = \cos(t)\).
To determine the overall symmetry of \(f(t)\), we need to consider the product \(\tan(t) \cdot \sin(t) \cdot \cos(t)\).
- Both \(\tan(t)\) and \(\sin(t)\) being odd results in their product being even, since multiplying two odd functions results in an even function.
- Multiplying this even result by the even function \(\cos(t)\) maintains the symmetry, resulting in an even function.
Therefore, \(f(t)\) is an even function because the product of two odd functions and one even function results in an even function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22d34706-9464-4e7c-bfd8-ca130c481c17%2Fa0c9c3ed-42e7-4dfb-9947-588625736a9e%2F5oi0538_processed.png&w=3840&q=75)
Transcribed Image Text:Sure! Here is the transcription as it might appear on an educational website:
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**Problem 2: Function Symmetry Analysis**
Determine whether the following function is even, odd, or neither. Justify your answer algebraically and clearly state your findings.
\[ f(t) = \tan(t) \cdot \sin(t) \cdot \cos(t) \]
---
To determine if the function is even, odd, or neither, we analyze the properties of the individual components:
1. **Tangent Function (\(\tan(t)\))**: This function is odd since \(\tan(-t) = -\tan(t)\).
2. **Sine Function (\(\sin(t)\))**: This function is odd since \(\sin(-t) = -\sin(t)\).
3. **Cosine Function (\(\cos(t)\))**: This function is even since \(\cos(-t) = \cos(t)\).
To determine the overall symmetry of \(f(t)\), we need to consider the product \(\tan(t) \cdot \sin(t) \cdot \cos(t)\).
- Both \(\tan(t)\) and \(\sin(t)\) being odd results in their product being even, since multiplying two odd functions results in an even function.
- Multiplying this even result by the even function \(\cos(t)\) maintains the symmetry, resulting in an even function.
Therefore, \(f(t)\) is an even function because the product of two odd functions and one even function results in an even function.
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