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
![### Question
Given \( x = \frac{\pi}{3} \), what is the exact value of \( \sin(2\pi - x) \)?
### Options
1. \( \boxed{\frac{1}{2}} \)
2. \( \boxed{-\frac{1}{2}} \)
3. \( \boxed{\frac{\sqrt{3}}{2}} \)
4. \( \boxed{-\frac{\sqrt{3}}{2}} \)
---
### Explanation
In this question, you need to determine the value of \( \sin(2\pi - x) \) given that \( x = \frac{\pi}{3} \).
### Strategy to Solve
1. **Understanding the Sine Function**
- The sine function is periodic with a period of \( 2\pi \).
- The sine of an angle \( 2\pi - x \) is the same as the sine of the angle \( -x \), and \(\sin(-x) = -\sin(x)\).
2. **Substituting the Given Value**
- Given \( x = \frac{\pi}{3} \), substitute \( x \) into the expression to get \( \sin(2\pi - \frac{\pi}{3}) \).
3. **Simplifying the Argument**
- Simplify \( 2\pi - \frac{\pi}{3} \):
\[
2\pi - \frac{\pi}{3} = \frac{6\pi}{3} - \frac{\pi}{3} = \frac{5\pi}{3}
\]
4. **Using Sine Function Properties**
- \(\sin(2\pi - x) = \sin(-x)\), and \(\sin(-x) = -\sin(x)\).
5. **Finding \(\sin(\frac{5\pi}{3})\)**
- The angle \( \frac{5\pi}{3} \) in the unit circle is equivalent to \( -\frac{\pi}{3} \) because it completes one full cycle (i.e., \( 2\pi = \frac{6\pi}{3} \), so \( \frac{5\pi}{3} = 2\pi - \frac{\pi}{3}\)).
6. **Calculating the Sine](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e08c0ed-2478-4250-8ecc-80933702ca06%2F9223480f-2b62-430f-b362-a9d996b816d0%2F7nui18r_processed.png&w=3840&q=75)
Transcribed Image Text:### Question
Given \( x = \frac{\pi}{3} \), what is the exact value of \( \sin(2\pi - x) \)?
### Options
1. \( \boxed{\frac{1}{2}} \)
2. \( \boxed{-\frac{1}{2}} \)
3. \( \boxed{\frac{\sqrt{3}}{2}} \)
4. \( \boxed{-\frac{\sqrt{3}}{2}} \)
---
### Explanation
In this question, you need to determine the value of \( \sin(2\pi - x) \) given that \( x = \frac{\pi}{3} \).
### Strategy to Solve
1. **Understanding the Sine Function**
- The sine function is periodic with a period of \( 2\pi \).
- The sine of an angle \( 2\pi - x \) is the same as the sine of the angle \( -x \), and \(\sin(-x) = -\sin(x)\).
2. **Substituting the Given Value**
- Given \( x = \frac{\pi}{3} \), substitute \( x \) into the expression to get \( \sin(2\pi - \frac{\pi}{3}) \).
3. **Simplifying the Argument**
- Simplify \( 2\pi - \frac{\pi}{3} \):
\[
2\pi - \frac{\pi}{3} = \frac{6\pi}{3} - \frac{\pi}{3} = \frac{5\pi}{3}
\]
4. **Using Sine Function Properties**
- \(\sin(2\pi - x) = \sin(-x)\), and \(\sin(-x) = -\sin(x)\).
5. **Finding \(\sin(\frac{5\pi}{3})\)**
- The angle \( \frac{5\pi}{3} \) in the unit circle is equivalent to \( -\frac{\pi}{3} \) because it completes one full cycle (i.e., \( 2\pi = \frac{6\pi}{3} \), so \( \frac{5\pi}{3} = 2\pi - \frac{\pi}{3}\)).
6. **Calculating the Sine
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