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

LESSON NO. 8
GRAPHS OF TANGENT
AND
COTANGENT
FUNCTIONS

Lesson No. 8 | Graphs of Tangent and Cotangent Functions
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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

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ENGAGE

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Engagement Activity 1
IllustratingTangent 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

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IllustratingTangent 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?

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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.

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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?

Engagement Activity 2
Small-Group Interactive Discussion on
Graphs ofTangent & Cotangent Functions
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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?

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
-How do you graph tangent and cotangent functions? What are the
things to be considered in graphing the said functions?
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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.

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EXPLORE

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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.

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Explore
Rubric/Point System of theTask:
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 – CorrectAnswer/well-Explained/with
Systematic Solution

Assigned Role:
Leader – 1 student
Secretary/Recorder – 1 student
Time Keeper – 1
Peacekeeper/Speaker – 1 student
Material Manager – 1-2 students
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Explore
Task 1 (Group 1 & Group 2): Sketch the graph of y =
1
2
tan 2x
Task 2 (Group 3 & Group 4): Sketch the graph of y = 2 cot
1
2
𝑥.
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

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EXPLAIN

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Explain
• Group Leader/Representative will present
the solutions and answer to the class by
explaining the problem/concept explored
considering the given guide questions.

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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?

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?
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ELABORATE

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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
- How do we determine the asymptotes, period and
phase shift of tangent and cotangent functions?

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Elaborate
Integration of PhilosophicalViews
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.

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Elaborate
Integration of PhilosophicalViews
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?

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Integration of PhilosophicalViews 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.

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-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.

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• 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.

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Integration of PhilosophicalViews from theTeacher
• 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.

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EVALUATE

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Evaluate
Answer the following:
a) Sketch the graph of the function y =
1
4
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.
i) y =
1
2
cot
1
3
𝑥 + 2 ii) y = −4 tan 𝑥 −
𝛱
4
− 1
c) How does the graph of y =
1
2
tan x + 1 is different from y = tan x?
d) Are the graphs of y = 𝑐𝑜𝑡 (x) − 1 different from the graph of y = cot (x)? Justify
your answer.

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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-

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

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