Determine a coterminal angle of 690°.
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter3: Additional Topics In Trigonometry
Section: Chapter Questions
Problem 40CT: To determine the angle of elevation of a star in the sky, you align the star and the top of the...
Related questions
Question
![**Determining a Coterminal Angle of 690°**
Coterminal angles are angles that share the same initial and terminal sides, but may have different measures. You can find coterminal angles by adding or subtracting full rotations (360°) from the given angle.
To find a coterminal angle of 690°:
1. Subtract 360° from 690° until you obtain an angle between 0° and 360°.
\[
690° - 360° = 330°
\]
Thus, 330° is a coterminal angle of 690°.
**Explanation with Diagram:**
Imagine the coordinate plane with the standard position of angles. The initial side of the angle is along the positive x-axis. Each full rotation (360°) brings you back to the starting point.
- Start at 0° along the positive x-axis.
- Rotate counterclockwise by 690° (this includes one full rotation of 360° plus an additional 330°).
On subtracting 360° from 690°:
- The first subtraction gives 330°, which is within the range from 0° to 360°. Therefore, the angle 330° is coterminal with the given angle of 690°.
A simple way to visualize this is by drawing a circle and marking the angles. Consequently, both 690° and 330° will have their terminal side coinciding.
This method can easily be used to find coterminal angles for any given angle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c9db27c-d267-4a61-b179-a00be574a092%2F57c3f931-14b6-4c42-ae1a-cf4b9ea1351e%2Fxniupd7_processed.png&w=3840&q=75)
Transcribed Image Text:**Determining a Coterminal Angle of 690°**
Coterminal angles are angles that share the same initial and terminal sides, but may have different measures. You can find coterminal angles by adding or subtracting full rotations (360°) from the given angle.
To find a coterminal angle of 690°:
1. Subtract 360° from 690° until you obtain an angle between 0° and 360°.
\[
690° - 360° = 330°
\]
Thus, 330° is a coterminal angle of 690°.
**Explanation with Diagram:**
Imagine the coordinate plane with the standard position of angles. The initial side of the angle is along the positive x-axis. Each full rotation (360°) brings you back to the starting point.
- Start at 0° along the positive x-axis.
- Rotate counterclockwise by 690° (this includes one full rotation of 360° plus an additional 330°).
On subtracting 360° from 690°:
- The first subtraction gives 330°, which is within the range from 0° to 360°. Therefore, the angle 330° is coterminal with the given angle of 690°.
A simple way to visualize this is by drawing a circle and marking the angles. Consequently, both 690° and 330° will have their terminal side coinciding.
This method can easily be used to find coterminal angles for any given angle.
![### Trigonometric Values of Angles
The following are the given angles in radians:
- **a.** \(\frac{3\pi}{2}\)
- **b.** \(\frac{5\pi}{3}\)
- **c.** \(\frac{7\pi}{4}\)
- **d.** \(\frac{11\pi}{6}\)
#### Explanation of the Angles:
1. **\(\frac{3\pi}{2}\)**:
- This angle is equivalent to 270 degrees.
- It is a straight angle directed downwards on the unit circle.
2. **\(\frac{5\pi}{3}\)**:
- This angle converts to 300 degrees.
- It is an obtuse angle located in the fourth quadrant of the unit circle.
3. **\(\frac{7\pi}{4}\)**:
- Converting this, the angle is 315 degrees.
- It is also positioned in the fourth quadrant of the unit circle, just 45 degrees away from completing a full circle.
4. **\(\frac{11\pi}{6}\)**:
- This angle translates to 330 degrees.
- Positioned in the fourth quadrant, it is 30 degrees away from completing a full circle.
These angles are significant within trigonometry and calculus, as they are used to determine various trigonometric values and identities. Understanding these angles in radians is fundamental for higher-level mathematics and physics applications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c9db27c-d267-4a61-b179-a00be574a092%2F57c3f931-14b6-4c42-ae1a-cf4b9ea1351e%2Fhugr027_processed.png&w=3840&q=75)
Transcribed Image Text:### Trigonometric Values of Angles
The following are the given angles in radians:
- **a.** \(\frac{3\pi}{2}\)
- **b.** \(\frac{5\pi}{3}\)
- **c.** \(\frac{7\pi}{4}\)
- **d.** \(\frac{11\pi}{6}\)
#### Explanation of the Angles:
1. **\(\frac{3\pi}{2}\)**:
- This angle is equivalent to 270 degrees.
- It is a straight angle directed downwards on the unit circle.
2. **\(\frac{5\pi}{3}\)**:
- This angle converts to 300 degrees.
- It is an obtuse angle located in the fourth quadrant of the unit circle.
3. **\(\frac{7\pi}{4}\)**:
- Converting this, the angle is 315 degrees.
- It is also positioned in the fourth quadrant of the unit circle, just 45 degrees away from completing a full circle.
4. **\(\frac{11\pi}{6}\)**:
- This angle translates to 330 degrees.
- Positioned in the fourth quadrant, it is 30 degrees away from completing a full circle.
These angles are significant within trigonometry and calculus, as they are used to determine various trigonometric values and identities. Understanding these angles in radians is fundamental for higher-level mathematics and physics applications.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Trigonometry (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781337278461/9781337278461_smallCoverImage.gif)
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Holt Mcdougal Larson Pre-algebra: Student Edition…](https://www.bartleby.com/isbn_cover_images/9780547587776/9780547587776_smallCoverImage.jpg)
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
![Trigonometry (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781337278461/9781337278461_smallCoverImage.gif)
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Holt Mcdougal Larson Pre-algebra: Student Edition…](https://www.bartleby.com/isbn_cover_images/9780547587776/9780547587776_smallCoverImage.jpg)
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
![Trigonometry (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305652224/9781305652224_smallCoverImage.gif)
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Elementary Geometry For College Students, 7e](https://www.bartleby.com/isbn_cover_images/9781337614085/9781337614085_smallCoverImage.jpg)
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,