graphs of tangent and cotangent function

Graphing Tangent and Cotangent Function

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 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

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

5. 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.

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

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

8. 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?

9. 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

10. 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 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. asymptotes are given by x = (2) Determine the two adjacent vertical asymptotes. For y = a tan bx, these vertical ±Π/2b. For y = a cot bx, vertical asymptotes are given by x = and x= Π/b. Divide the interval formed by the vertical asymptotes in Step 2 into four equal parts, and get three division points exclusively between the asymptotes. 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. Plot the points found in Step 3, and join them with a smooth curve approaching to the vertical asymptotes. Exten the graph to the right and to the left, as needed.

13. Lesson No. 8 | Graphs of Tangent and Cotangent Functions Explore The class will be divided into 10 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.

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

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

16. Lesson No. 8 | Graphs of Tangent and Cotangent Functions Explore 1 2 Task 1 (Group 1 & Group 2): Sketch the graph of y = tan 2x 1 2 Task 2 (Group 3 & Group 4): Sketch the graph of y = 2 cot 𝑥 . Task 3 (Group 5 , Group 6 , &Group7 ): Sketch the graph of y = –tanx + 2 Task 4 (Group 8, Group 9, & Group 10 ): Sketch the graph of y = –2cotx – 1

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

18. 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?

19. 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

20. Lesson No. 8 | Graphs of Tangent and Cotangent Functions ELABOR A TE

21. 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?

22. 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?

23. Lesson No. 8 | Graphs of Tangent and Cotangent Functions Evaluate Answer the following: 1 4 a) Sketch the graph of the function y = tan 𝑥 − 𝛱 4 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. 1 2 i) y = cot 1 3 𝑥 + 2 ii) y = −4 tan 𝑥 − 𝛱 4 − 1 1 c) How does the graph of y = tan x + 1 is different from y = tan x? 2 d) Are the graphs of y = 𝑐𝑜𝑡 (x) − 1 different from the graph of y = cot (x)? Justify your answer.

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