Lesson no. 8 (Graphs of Tangent and Cotangent Functions)

One of the instructional materials (Slide Presentations) packaged out of the lessons as a result of the study entitled: "INQUIRY-BASED LESSONS IN PRE-CALCULUS FOR SENIOR HIGH SCHOOL"

1. Lesson No. 8 Graphs of Tangent and Cotangent Functions

2. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Topics: Graphs of y = tan x and y = cot x Graphs of y= a tan bx and y = a cot bx Graphs of y = a tan b (x – c) + d and y = a cot b (x – c) + d

3. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ ENGAGE

4. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Engagement Activity 1 Illustrating Tangent and Cotangent with the Unit Circle Author:afrewin Topic : Trigonometry References: https:// www.geogebra.org/m/YUJvBfxw#material/fbQWQGsg https ://www.geogebra.org/m/YUJvBfxw#material/ueUcqGNG

5. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Illustrating Tangent and Cotangent with the Unit Circle Questions : 1. What can you say about the relationship between the tangent and cotangent function with the unit circle? 2. How will you describe the relationship between the tangent and cotangent function with the unit circle?

6. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Tangent Cotangent Relationship Author:carpenter Reference: https:// www.geogebra.org/m/Y74C5aNz Move the parameters a (vertical dilation), b (period dilation), c ( part of phase shift), and d (vertical shift) to see how the graphs of tangent and cotangent are related.

7. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Tangent Cotangent Relationship Questions : 1. Based on the graph, what can you say about the domain and range of tangent function? How about cotangent function? 2 . How will you describe the relationship of tangent and cotangent function in terms if their domain, range and phase shift?

8. Engagement Activity 2 Small-Group Interactive Discussion on Graphs of Tangent & Cotangent Functions Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________

9. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Small-Group Interactive Discussion on Graphs of Tangent & Cotangent Functions Inquiry Guide Questions: What can you say about the graphs of tangent and cotangent functions in terms of the following: Domain; Range; Phase Shift and; Period? What is your guide in graphing tangent and cotangent functions? What are the important properties of the graphs of tangent and cotangent functions?

10. Small-Group Interactive Discussion on Graphs of Tangent & Cotangent Functions Inquiry Guide Questions : -Do tangent and cotangent functions have amplitude? Why? -What are the domains of the tangent and cotangent functions? -What are the ranges of the tangent and cotangent functions? -What are the periods of the tangent and cotangent? What does period mean? How do you find the period of a given tangent or cotangent functions? -How do you graph tangent and cotangent functions? What are the things to be considered in graphing the said functions? Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________

11. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________

12. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________

13. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Graphs of Tangent & Cotangent Functions In general, to sketch the graphs of y = a tan bx and y = a cot bx , a ≠ 0 and b > 0, we may proceed with the following steps: (1) Determine the period Π/b. Then we draw one cycle of the graph on (-Π/2b, Π/2b) for y = tan bx , and on (0 , Π/b ) for y = a cot bx. (2) Determine the two adjacent vertical asymptotes. For y = a tan bx , these vertical asymptotes are given by x = ±Π/2b. For y = a cot bx , vertical asymptotes are given by x = 0 and x= Π/b. (3) Divide the interval formed by the vertical asymptotes in Step 2 into four equal parts , and get three division points exclusively between the asymptotes. (4) Evaluate the function at each of these x-values identified in Step 3. The points will correspond to the sign and x-intercept of the graph. (5) Plot the points found in Step 3, and join them with a smooth curve approaching to the vertical asymptotes. Extend the graph to the right and to the left, as needed.

14. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ EXPLORE

15. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Explore The class will be divided into 8 groups (5-6 members). Each group will be given a problem-based task card to be explored, answered and presented to the class. Inquiry questions from the teacher and learners will be considered during the explore activity .

16. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Explore Rubric/Point System of the Task : 0 point – No Answer 1 point – Incorrect Answer/Explanation/Solutions 2 points – Correct Answer but No Explanation/Solutions 3 points – Correct Answer with Explanation/Solutions 4 points – Correct Answer/well-Explained/with Systematic Solution

17. Assigned Role : Leader – 1 student Secretary/Recorder – 1 student Time Keeper – 1 Peacekeeper/Speaker – 1 student Material Manager – 1-2 students Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________

18. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Explore   Task 1 (Group 1 & Group 2 ): Sketch the graph of y = tan 2x   Task 2 (Group 3 & Group 4 ): Sketch the graph of y = 2 cot .   Task 3 (Group 5 & Group 6 ): Sketch the graph of y = – tanx + 2 Task 4 (Group 7 & Group 8): Sketch the graph of y = – 2cotx – 1  

19. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ EXPLAIN

20. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Explain Group Leader/Representative will present the solutions and answer to the class by explaining the problem/concept explored considering the given guide questions.

21. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Explain Guide Questions: What is the problem-based task all about? What are the given in the problem-based task? What are the things did you consider in solving/answering the problem-based task? What methods did you use in solving/answering the given problem-based task ?

22. Explain Guide Questions: - How did you solve/answer the problem-based task using that method? - Are there still other ways to answer the problem/task? How did you do it? - Are there any limitations to your solution/answer? - What particular mathematical concept in trigonometry did you apply to solve the problem-based task? Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________

23. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ ELABORATE

24. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Elaborate Generalization of the Lesson: - What are the properties of the graphs of tangent and cotangent functions? - What are the domain and range of tangent and cotangent functions? - How do we determine the asymptotes, period and phase shift of tangent and cotangent functions?

25. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Elaborate Integration of Philosophical Views In this part, the teacher and learners will relate the term(s)/content/process learned in Lesson 5about Graphs of Tangent and Cotangent Functions in real life situations/scenario/instances considering the philosophical views that can be integrated/associated to term(s)/content/process/ skills of the lesson.

26. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Elaborate Integration of Philosophical Views Questions : What are the things/situations/instances that you can relate with regard to the lesson about graphs of tangent and cotangent? Considering your philosophical views, how will you relate the terms/content/process of the lesson in real-life situations/instances/scenario?

27. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Integration of Philosophical Views from the teacher The graph of tangent and cotangent functions extends to positive and negative infinity. There are specific points where the graph is undefined and where the graph is restricted. As the concepts and properties of tangent and cotangent functions in real life, there are specific instances in our life that our undertakings have a positive or a negative outcome.

28. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Integration of Philosophical Views from the teacher - There can also be instances where the result is limited. Based on Newton's third law of motion, for every action, there is an equal and opposite reaction . -The result of your action may have infinite advantages or endless disadvantages, but there may be a limited result. In most cases, you consider your comfort zone, whereas, you restrict yourself from learning and experiencing new things because you do not want to take the risk.

29. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Integration of Philosophical Views from the teacher You want a situation in which you feel comfortable by not testing your ability and determination. Your comfort zone is your behavioral space where your activities and behaviors fit the routine and pattern that minimizes stress and risk. There is a sense of familiarity, security, and certainty in your comfort zone. So, opening yourselves up to the possibility of stress and anxiety when you step outside of your comfort zone is a difficult thing for you to do.

30. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Integration of Philosophical Views from the Teacher However , you would never know what life has to bring to you if you would try. Remember that it is good to step out, and to challenge yourself to perform to the best of your ability. You never know the kind of life that you may have been. It can be like the graph of tangent and cotangent function that extends to positive infinity, instead of staying at the negative infinity.

31. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ EVALUATE

32. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Evaluate Answer the following : a) Sketch the graph of the function y = over three periods. Find the domain and range of the function . b) Graph the given tangent and cotangent functions with its period, and phase shift and determine its domain & range. i ) y = ii) y = c ) How does the graph of y = tan x + 1 is different from y = tan x ? d) Are the graphs of y = (x) 1 different from the graph of y = cot (x)? Justify your answer.  

33. Lesson No. 8 | Graphs of Tangent and Cotangent Functions _____________________________________________________________________ Assignment: Answer the following questions : 1 . What is meant by simple harmonic motion? 2 . What are the equations of simple harmonic motion? 3 . Give example of solved situational problems involving graphs of circular functions. Reference : DepED Pre-Calculus Learner’s Material, pages 160-165 -GNDMJR-