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:**
Determine the corresponding ordered pair of the point on the unit circle for the following angle: \(\frac{16\pi}{3}\).
**Options:**
1. \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)
2. ⬅️ \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)
3. \(\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)
4. \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)
**Explanation:**
To determine the corresponding ordered pair, we need to identify the point on the unit circle for the specified angle \( \frac{16\pi}{3} \).
By reducing the angle:
\[ \frac{16\pi}{3} = \frac{16\pi}{3} - 2\pi \times 2 = \frac{16\pi}{3} - \frac{12\pi}{3} = \frac{4\pi}{3} \]
The angle \(\frac{4\pi}{3}\) is in the third quadrant of the unit circle. The coordinates for \(\frac{4\pi}{3}\) are \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\).
Therefore, the corresponding ordered pair for the given angle is:
\(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\).
However, based on the provided options, the person has preselected the second option \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\), indicating this may be based on understanding or subsequently correcting the choice. The correct option should actually be \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\) if cross-referenced accurately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F136e7ace-7487-4e46-aa8b-2e1ffd8bd45d%2Fca6bc042-7a57-4a8b-b79e-9fb1bf985009%2Fa0g5s6r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
Determine the corresponding ordered pair of the point on the unit circle for the following angle: \(\frac{16\pi}{3}\).
**Options:**
1. \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)
2. ⬅️ \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)
3. \(\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)
4. \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\)
**Explanation:**
To determine the corresponding ordered pair, we need to identify the point on the unit circle for the specified angle \( \frac{16\pi}{3} \).
By reducing the angle:
\[ \frac{16\pi}{3} = \frac{16\pi}{3} - 2\pi \times 2 = \frac{16\pi}{3} - \frac{12\pi}{3} = \frac{4\pi}{3} \]
The angle \(\frac{4\pi}{3}\) is in the third quadrant of the unit circle. The coordinates for \(\frac{4\pi}{3}\) are \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\).
Therefore, the corresponding ordered pair for the given angle is:
\(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\).
However, based on the provided options, the person has preselected the second option \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\), indicating this may be based on understanding or subsequently correcting the choice. The correct option should actually be \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\) if cross-referenced accurately.
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