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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![### Determining Corresponding Ordered Pair on the Unit Circle
In this exercise, we will determine the corresponding ordered pair for the point on the unit circle for the following angle:
\[ - \frac{23\pi}{4} \]
The multiple-choice options are as follows:
1. \( \left( -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) \)
2. \( \left( -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \) (Highlighted as the selected option)
3. \( \left( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) \)
4. \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \)
#### Explanation:
1. **Option 1**: \( \left( -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) \)
- This point is in the third quadrant.
2. **Option 2**: \( \left( -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \)
- This point is in the second quadrant and is the selected option.
3. **Option 3**: \( \left( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) \)
- This point is in the fourth quadrant.
4. **Option 4**: \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \)
- This point is in the first quadrant.
### Understanding Angles on the Unit Circle:
The unit circle is a circle with a radius of one centered at the origin of the coordinate system. Angles in the unit circle can be measured in radians, and each angle corresponds to a specific point, with coordinates \((x, y)\).
To find the coordinates of a given angle, you can use the following trigonometric identities:
- \( x = \cos(\theta) \)
- \( y = \sin(\theta) \)
For the angle \( - \frac{23](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F846a5d83-ebd2-4e0d-9bf0-b0f470211899%2F3845bbbf-5998-463e-a43e-b6c6f67e4017%2Fnuhoguh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Determining Corresponding Ordered Pair on the Unit Circle
In this exercise, we will determine the corresponding ordered pair for the point on the unit circle for the following angle:
\[ - \frac{23\pi}{4} \]
The multiple-choice options are as follows:
1. \( \left( -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) \)
2. \( \left( -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \) (Highlighted as the selected option)
3. \( \left( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) \)
4. \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \)
#### Explanation:
1. **Option 1**: \( \left( -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) \)
- This point is in the third quadrant.
2. **Option 2**: \( \left( -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \)
- This point is in the second quadrant and is the selected option.
3. **Option 3**: \( \left( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) \)
- This point is in the fourth quadrant.
4. **Option 4**: \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \)
- This point is in the first quadrant.
### Understanding Angles on the Unit Circle:
The unit circle is a circle with a radius of one centered at the origin of the coordinate system. Angles in the unit circle can be measured in radians, and each angle corresponds to a specific point, with coordinates \((x, y)\).
To find the coordinates of a given angle, you can use the following trigonometric identities:
- \( x = \cos(\theta) \)
- \( y = \sin(\theta) \)
For the angle \( - \frac{23
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