Graph the equation over a one-period interval. 18. y 2-cosx 19. y= sin 2x
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°.
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Question
![### Graph the equation over a one-period interval.
#### 18. \( y = 2 - \cos x \)
Graph Description:
- The diagram shows an empty Cartesian plane with the x-axis and y-axis both extending infinitely in positive and negative directions with arrows at the ends indicating this.
- To graph this equation (\( y = 2 - \cos x \)), plot the cosine curve starting from \( x = 0 \) to \( x = 2\pi \), shifted upwards by 2 units and reflected about the x-axis.
Key Points for Graphing:
1. **Cosine Function Basics**:
- One period of \( \cos x \) spans from \( x = 0 \) to \( x = 2\pi \).
- \( \cos x \) at \( x = 0 \) is 1, \( \cos x \) at \( x = \pi \) is -1, and \( \cos x \) at \( x = 2\pi \) is 1.
2. **Transformations Applied**:
- Vertical shift upwards by 2 units: \( \cos x \rightarrow y \).
- Reflected about the x-axis and then shifted upwards by 2 units.
3. **New Points**:
- At \( x = 0 \), \( y = 2 - \cos(0) = 1 \), so the graph starts at point \( (0, 1) \).
- At \( x = \pi \), \( y = 2 - \cos(\pi) = 3 \), so the graph peaks at \( (\pi, 3) \).
- At \( x = 2\pi \), \( y = 2 - \cos(2\pi) = 1 \), so the graph ends at point \( (2\pi, 1) \).
#### 19. \( y = \sin 2x \)
Graph Description:
- The diagram shows an empty Cartesian plane with the x-axis and y-axis both extending infinitely in positive and negative directions with arrows at the ends indicating this.
- To graph this equation (\( y = \sin 2x \)), plot the sine curve starting from \( x = 0 \) to \( x = \pi \), horizontally compressed by a factor of 2.
Key Points for Graphing:
1. **Sine](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce0f29cc-98be-45bb-9fb8-5b9efd247a01%2Ff9b52ad9-4b5b-41ba-b9ec-6987f8e5ac4d%2Fpjg96t3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Graph the equation over a one-period interval.
#### 18. \( y = 2 - \cos x \)
Graph Description:
- The diagram shows an empty Cartesian plane with the x-axis and y-axis both extending infinitely in positive and negative directions with arrows at the ends indicating this.
- To graph this equation (\( y = 2 - \cos x \)), plot the cosine curve starting from \( x = 0 \) to \( x = 2\pi \), shifted upwards by 2 units and reflected about the x-axis.
Key Points for Graphing:
1. **Cosine Function Basics**:
- One period of \( \cos x \) spans from \( x = 0 \) to \( x = 2\pi \).
- \( \cos x \) at \( x = 0 \) is 1, \( \cos x \) at \( x = \pi \) is -1, and \( \cos x \) at \( x = 2\pi \) is 1.
2. **Transformations Applied**:
- Vertical shift upwards by 2 units: \( \cos x \rightarrow y \).
- Reflected about the x-axis and then shifted upwards by 2 units.
3. **New Points**:
- At \( x = 0 \), \( y = 2 - \cos(0) = 1 \), so the graph starts at point \( (0, 1) \).
- At \( x = \pi \), \( y = 2 - \cos(\pi) = 3 \), so the graph peaks at \( (\pi, 3) \).
- At \( x = 2\pi \), \( y = 2 - \cos(2\pi) = 1 \), so the graph ends at point \( (2\pi, 1) \).
#### 19. \( y = \sin 2x \)
Graph Description:
- The diagram shows an empty Cartesian plane with the x-axis and y-axis both extending infinitely in positive and negative directions with arrows at the ends indicating this.
- To graph this equation (\( y = \sin 2x \)), plot the sine curve starting from \( x = 0 \) to \( x = \pi \), horizontally compressed by a factor of 2.
Key Points for Graphing:
1. **Sine
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