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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![### Locating the Angle θ on the Unit Circle
#### Problem Statement
Locate the angle \(\theta = -\frac{7\pi}{4}\) radians.
#### Diagram Description
The provided diagram is a unit circle centered at the origin of a coordinate plane. The coordinate axes are labeled as follows:
- The horizontal axis as \(x\).
- The vertical axis as \(y\).
#### Explanation
To find \(\theta = -\frac{7\pi}{4}\) radians on the unit circle, follow these steps:
1. Understanding Radian Measure:
- Radians measure angles via the length of the arc on the circle's circumference.
- A complete revolution around the circle is \(2\pi\) radians.
- Negative angles represent counterclockwise rotation.
2. Simplifying the Angle:
- Since \( -\frac{7\pi}{4} \) is a negative angle, it represents clockwise rotation.
- A full circle (clockwise) would correspond to \(-2\pi\) radians, since a clockwise movement goes in the negative direction.
- Simplify the given angle by adding \(2\pi\):
\[
-\frac{7\pi}{4} + 2\pi = -\frac{7\pi}{4} + \frac{8\pi}{4} = \frac{\pi}{4}
\]
- Therefore, locating \(\theta = -\frac{7\pi}{4}\) is equivalent to finding \(\frac{\pi}{4}\) radians on the unit circle.
3. Position on the Circle:
- The angle \(\theta = \frac{\pi}{4}\) radians represents an angle of 45 degrees.
- On the unit circle, this is in the first quadrant forming a line that makes a 45-degree angle with the positive \(x\)-axis.
- Precisely, it is counterclockwise from the positive \(x\)-axis.
#### Conclusion
The angle \(\theta = -\frac{7\pi}{4}\) radians locates the same point on the unit circle as the angle \(\theta = \frac{\pi}{4}\) radians, which is in the first quadrant making a 45-degree angle with the positive \(x\)-axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8aee7c7d-285e-49c9-ad3b-faae5963cfad%2F2c432fa1-a109-4274-bd57-98022d5b3f42%2Fu3msfd9_processed.png&w=3840&q=75)
Transcribed Image Text:### Locating the Angle θ on the Unit Circle
#### Problem Statement
Locate the angle \(\theta = -\frac{7\pi}{4}\) radians.
#### Diagram Description
The provided diagram is a unit circle centered at the origin of a coordinate plane. The coordinate axes are labeled as follows:
- The horizontal axis as \(x\).
- The vertical axis as \(y\).
#### Explanation
To find \(\theta = -\frac{7\pi}{4}\) radians on the unit circle, follow these steps:
1. Understanding Radian Measure:
- Radians measure angles via the length of the arc on the circle's circumference.
- A complete revolution around the circle is \(2\pi\) radians.
- Negative angles represent counterclockwise rotation.
2. Simplifying the Angle:
- Since \( -\frac{7\pi}{4} \) is a negative angle, it represents clockwise rotation.
- A full circle (clockwise) would correspond to \(-2\pi\) radians, since a clockwise movement goes in the negative direction.
- Simplify the given angle by adding \(2\pi\):
\[
-\frac{7\pi}{4} + 2\pi = -\frac{7\pi}{4} + \frac{8\pi}{4} = \frac{\pi}{4}
\]
- Therefore, locating \(\theta = -\frac{7\pi}{4}\) is equivalent to finding \(\frac{\pi}{4}\) radians on the unit circle.
3. Position on the Circle:
- The angle \(\theta = \frac{\pi}{4}\) radians represents an angle of 45 degrees.
- On the unit circle, this is in the first quadrant forming a line that makes a 45-degree angle with the positive \(x\)-axis.
- Precisely, it is counterclockwise from the positive \(x\)-axis.
#### Conclusion
The angle \(\theta = -\frac{7\pi}{4}\) radians locates the same point on the unit circle as the angle \(\theta = \frac{\pi}{4}\) radians, which is in the first quadrant making a 45-degree angle with the positive \(x\)-axis.
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