**Understanding the Unit Circle: Calculating Cosine** The graph below illustrates the angle \( A = \frac{5\pi}{3} \) inside the unit circle. [Image Description] The unit circle is displayed with a central angle marked as \( A = \frac{5\pi}{3} \). The angle is drawn on the circle starting from the positive x-axis going clockwise. A right triangle is formed inside the circle, where the hypotenuse lies along the radius at angle \( A \), reaching the circumference of the unit circle. The adjacent side of the triangle lies on the x-axis, and the opposite side is perpendicular to the x-axis. **Objective**: Find the exact value (without rounding) for the cosine of the given angle. \[ \cos\left(\frac{5\pi}{3}\right) = \] --- In this educational explanation, students are encouraged to use their understanding of the unit circle and trigonometric identities to derive the cosine value of the specified angle \( \frac{5\pi}{3} \). ### Key Concepts to Consider: 1. **Unit Circle**: A circle with a radius of 1 centered at the origin (0,0). 2. **Standard Position of Angles**: - Angles measured counterclockwise from the positive x-axis. - For angles greater than \( 2\pi \) or less than 0, subtract/add \( 2\pi \) until the angle is within [0, \( 2\pi \)]. 3. **Cosine and Sine Values on the Unit Circle**: - The x-coordinate of the point where the terminal side of the angle intersects the unit circle gives the cosine value. - The y-coordinate gives the sine value. By following these principles, students should be able to determine the exact value of \( \cos\left(\frac{5\pi}{3}\right) \).
**Understanding the Unit Circle: Calculating Cosine** The graph below illustrates the angle \( A = \frac{5\pi}{3} \) inside the unit circle. [Image Description] The unit circle is displayed with a central angle marked as \( A = \frac{5\pi}{3} \). The angle is drawn on the circle starting from the positive x-axis going clockwise. A right triangle is formed inside the circle, where the hypotenuse lies along the radius at angle \( A \), reaching the circumference of the unit circle. The adjacent side of the triangle lies on the x-axis, and the opposite side is perpendicular to the x-axis. **Objective**: Find the exact value (without rounding) for the cosine of the given angle. \[ \cos\left(\frac{5\pi}{3}\right) = \] --- In this educational explanation, students are encouraged to use their understanding of the unit circle and trigonometric identities to derive the cosine value of the specified angle \( \frac{5\pi}{3} \). ### Key Concepts to Consider: 1. **Unit Circle**: A circle with a radius of 1 centered at the origin (0,0). 2. **Standard Position of Angles**: - Angles measured counterclockwise from the positive x-axis. - For angles greater than \( 2\pi \) or less than 0, subtract/add \( 2\pi \) until the angle is within [0, \( 2\pi \)]. 3. **Cosine and Sine Values on the Unit Circle**: - The x-coordinate of the point where the terminal side of the angle intersects the unit circle gives the cosine value. - The y-coordinate gives the sine value. By following these principles, students should be able to determine the exact value of \( \cos\left(\frac{5\pi}{3}\right) \).
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
Related questions
Question
![**Understanding the Unit Circle: Calculating Cosine**
The graph below illustrates the angle \( A = \frac{5\pi}{3} \) inside the unit circle.
[Image Description]
The unit circle is displayed with a central angle marked as \( A = \frac{5\pi}{3} \). The angle is drawn on the circle starting from the positive x-axis going clockwise. A right triangle is formed inside the circle, where the hypotenuse lies along the radius at angle \( A \), reaching the circumference of the unit circle. The adjacent side of the triangle lies on the x-axis, and the opposite side is perpendicular to the x-axis.
**Objective**: Find the exact value (without rounding) for the cosine of the given angle.
\[ \cos\left(\frac{5\pi}{3}\right) = \]
---
In this educational explanation, students are encouraged to use their understanding of the unit circle and trigonometric identities to derive the cosine value of the specified angle \( \frac{5\pi}{3} \).
### Key Concepts to Consider:
1. **Unit Circle**: A circle with a radius of 1 centered at the origin (0,0).
2. **Standard Position of Angles**:
- Angles measured counterclockwise from the positive x-axis.
- For angles greater than \( 2\pi \) or less than 0, subtract/add \( 2\pi \) until the angle is within [0, \( 2\pi \)].
3. **Cosine and Sine Values on the Unit Circle**:
- The x-coordinate of the point where the terminal side of the angle intersects the unit circle gives the cosine value.
- The y-coordinate gives the sine value.
By following these principles, students should be able to determine the exact value of \( \cos\left(\frac{5\pi}{3}\right) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc1123ec-a0e0-4e35-afe0-df78f47a58b5%2F2361a562-5ae7-4834-a4fa-7b880712f843%2Fhsrfrcv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding the Unit Circle: Calculating Cosine**
The graph below illustrates the angle \( A = \frac{5\pi}{3} \) inside the unit circle.
[Image Description]
The unit circle is displayed with a central angle marked as \( A = \frac{5\pi}{3} \). The angle is drawn on the circle starting from the positive x-axis going clockwise. A right triangle is formed inside the circle, where the hypotenuse lies along the radius at angle \( A \), reaching the circumference of the unit circle. The adjacent side of the triangle lies on the x-axis, and the opposite side is perpendicular to the x-axis.
**Objective**: Find the exact value (without rounding) for the cosine of the given angle.
\[ \cos\left(\frac{5\pi}{3}\right) = \]
---
In this educational explanation, students are encouraged to use their understanding of the unit circle and trigonometric identities to derive the cosine value of the specified angle \( \frac{5\pi}{3} \).
### Key Concepts to Consider:
1. **Unit Circle**: A circle with a radius of 1 centered at the origin (0,0).
2. **Standard Position of Angles**:
- Angles measured counterclockwise from the positive x-axis.
- For angles greater than \( 2\pi \) or less than 0, subtract/add \( 2\pi \) until the angle is within [0, \( 2\pi \)].
3. **Cosine and Sine Values on the Unit Circle**:
- The x-coordinate of the point where the terminal side of the angle intersects the unit circle gives the cosine value.
- The y-coordinate gives the sine value.
By following these principles, students should be able to determine the exact value of \( \cos\left(\frac{5\pi}{3}\right) \).
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