Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Related questions
Question
Can I get help with this trig problem? Can You show me how you got the answer with steps, please?
![**Graphing Trigonometric Functions: Step-by-Step**
**Objective:**
Sketch one complete cycle of the following function by first graphing the appropriate sine or cosine curve and then using the reciprocal relationships.
**Function:**
\[ y = 2 \csc \left( 2x - \frac{\pi}{3} \right) \]
**Steps Involved:**
1. **Identify the Related Sine Function:**
- The cosecant (csc) function is the reciprocal of the sine function (sin).
- Therefore, start by sketching the sine function that corresponds to this cosecant function.
2. **Rewrite the Function in Terms of Sine:**
- Recognize that \( \csc \theta = \frac{1}{\sin \theta} \).
- Rewrite the given function as \( y = \frac{2}{\sin \left( 2x - \frac{\pi}{3} \right)} \).
3. **Graph the Sine Function:**
- Consider \( y = \sin \left( 2x - \frac{\pi}{3} \right) \), noting the amplitude and period adjustments.
- Period: The period of \( \sin(2x) \) is \(\frac{\pi}{2}\) (since \( \frac{2\pi}{2} = \pi \)).
- Phase Shift: The phase shift is \( \frac{\pi}{3} \), shifted to the right.
- Amplitude: This sine function will have an amplitude of 1, but the csc function will be more complex due to the reciprocal relationship.
4. **Sketch the Sine Curve:**
- Draw the sine curve indicating key points such as maxima, minima, and intercepts, shifted right by \( \frac{\pi}{3} \).
5. **Graph the Cosecant Function:**
- Identify the vertical asymptotes where the sine function intercepts the x-axis (these are the points where csc is undefined).
- Draw curves going towards these asymptotes, keeping in mind that csc(x) = 1/sin(x) will stretch the graph.
6. **Adjust for Amplitude:**
- Given the amplitude of the csc function is multiplied by 2, exaggerate the peaks and troughs accordingly, making them twice the distance from the midline as they](https://content.bartleby.com/qna-images/question/87f6c903-7c82-4f0e-8549-b569dd14b65f/ea5080fd-c7a6-4846-9c8b-a47d3132a503/zr5y1w.png)
Transcribed Image Text:**Graphing Trigonometric Functions: Step-by-Step**
**Objective:**
Sketch one complete cycle of the following function by first graphing the appropriate sine or cosine curve and then using the reciprocal relationships.
**Function:**
\[ y = 2 \csc \left( 2x - \frac{\pi}{3} \right) \]
**Steps Involved:**
1. **Identify the Related Sine Function:**
- The cosecant (csc) function is the reciprocal of the sine function (sin).
- Therefore, start by sketching the sine function that corresponds to this cosecant function.
2. **Rewrite the Function in Terms of Sine:**
- Recognize that \( \csc \theta = \frac{1}{\sin \theta} \).
- Rewrite the given function as \( y = \frac{2}{\sin \left( 2x - \frac{\pi}{3} \right)} \).
3. **Graph the Sine Function:**
- Consider \( y = \sin \left( 2x - \frac{\pi}{3} \right) \), noting the amplitude and period adjustments.
- Period: The period of \( \sin(2x) \) is \(\frac{\pi}{2}\) (since \( \frac{2\pi}{2} = \pi \)).
- Phase Shift: The phase shift is \( \frac{\pi}{3} \), shifted to the right.
- Amplitude: This sine function will have an amplitude of 1, but the csc function will be more complex due to the reciprocal relationship.
4. **Sketch the Sine Curve:**
- Draw the sine curve indicating key points such as maxima, minima, and intercepts, shifted right by \( \frac{\pi}{3} \).
5. **Graph the Cosecant Function:**
- Identify the vertical asymptotes where the sine function intercepts the x-axis (these are the points where csc is undefined).
- Draw curves going towards these asymptotes, keeping in mind that csc(x) = 1/sin(x) will stretch the graph.
6. **Adjust for Amplitude:**
- Given the amplitude of the csc function is multiplied by 2, exaggerate the peaks and troughs accordingly, making them twice the distance from the midline as they
Expert Solution
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 nowThis is a popular solution!
Step by stepSolved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, trigonometry and related others by exploring similar questions and additional content below.Similar questions
- following up for the solutions to part b! thank you!arrow_forwardI have attached a picture below. I am unsure of how to complete this problem. I believe A is "no", but I do not know how to do the other 2 questions.arrow_forwardI need to explain how to get the answer and i dont know how. See the pictures for the problem and what I need to explain. Thank youarrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
- Trigonometry (11th Edition)TrigonometryISBN:9780134217437Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie DanielsPublisher:PEARSON
Trigonometry (MindTap Course List)TrigonometryISBN:9781305652224Author:Charles P. McKeague, Mark D. TurnerPublisher:Cengage Learning
Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning
Trigonometry (11th Edition)
Trigonometry
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:9781337278461
Author:Ron Larson
Publisher:Cengage Learning