Lesson no. 6 (Graphs of Sine and Cosine Functions)

LESSON NO. 6
GRAPHS OF SINE AND
COSINE FUNCTIONS

Lesson No. 6| Graphs of Sine and Cosine Functions
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Topics:
• Graphs of y = sin x and y = cos x
• Graphs of y= a sin bx and y = cos bx
• Graphs of y = a sin b (x – c) + d and y = a cos b (x – c) + d

Lesson No. 6 |Graphs of Sine and Cosine Functions
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Introduction
• There are many things that occur periodically.
Phenomena like rotation of the planets and
comets, high and low tides, and yearly change of
the seasons follow a periodic pattern.
• In this lesson, we will graph circular functions
and we will see that they are periodic in nature.

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ENGAGE

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Engagement Activity 1 - ““Domain & Range Illustrator”
– Review on domain and range of a function
Author :Tim Brzezinski
Topic: Functions
Reference: https://www.geogebra.org/m/DUx2uB5f

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Engagement Activity 1
Questions:
1. What can you say about the domain of the given
function?
2. What can you say about the domain of the given
function?
3. How will you define (in your own words) the domain of
any function?
4. How will you define (in your own words) the range of
any function?

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Engagement Activity 2 -The Graph of Sine & Cosine Functions
Author:Tim Brzezinski
Topic:Cosine, Functions, Function Graph, Sine,Trigonometric Functions
Reference: http://www.geogebra.org

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Questions:
1) Consider the function f(x) = sin(x).
What are the values of a, b, c, and d for this parent
sine function? What is its period? How about
amplitude?
2)What do the parameters a, b, c, and d do to the
graph of the function f(x) = sin(x) under the
transformation y = a*sin(bx - c) + d?

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Questions:
3) Consider the function g(x) = cos(x). What are the values
of a, b, c, and d for this parent cosine function? What is its
period? How about amplitude?
4)What do the parameters a, b, c, and d do to the graph of
the function f(x) = cos(x) under the transformation y =
a*cos(bx - c) + d?
5) What are the domain and range of f(x) = sin(x)? How
about g(x) = cos(x)?

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Small-Group Interactive Discussion
Graphs of Sine & Cosine Functions

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Small-Group Interactive Discussion on
Inquiry Guide Questions:
• What can you say about the graphs of sine and cosine functions in terms of the
following:
– Domain;
– Range;
– Amplitude and;
– Period?
• What are the important properties of the graphs of sine and cosine functions?
• What are the domains of the sine and cosine functions?
• What are the ranges of the sine and cosine functions?

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Small-Group Interactive Discussion on
Inquiry Guide Questions:
-What are the ranges of the sine and cosine functions?
-What are the periods of the sine and cosine functions?
What does period mean?
-How does the amplitude affect the graph of the sine or
cosine functions?
-How do you graph sine and cosine functions? What are the
things to be considered in graphing the said functions?

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__

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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 – Correct Answer/well-Explained/with
Systematic Solution

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Explore
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 one cycle of y = 3 sin (x + Π/4 )
and y = 3 cos (x + Π/4 )
Sketch the graph of one cycle of y = 1/2 sin (-2x/3)
and
y = 1/2 cos (-2x/3)

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Explore
Sketch the graph of one cycle of y = −𝟑𝒔𝒊𝒏
𝒙
𝟐
and y = −𝟑𝒄𝒐𝒔
𝒙
𝟐
Sketch the graph of one cycle of y =
𝟐 𝒔𝒊𝒏 𝟒𝒙 and
y = 𝟐 𝒄𝒐𝒔 𝟒𝒙

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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
answering the given problem-based task ?
• What methods did you use in answering the
given problem-based task?

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Explain
Guide Questions:
-How did you answer the given problem-based
task using that method?
-Are there still other ways to answer the problem-
based task ? How did you do it?
-Are there any limitations to your answer to the
given problem-based task ?
-What particular mathematical concept in
trigonometry did you apply to answer 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
sine and cosine functions?
- What are the domain and range of sine and
cosine Functions?
- How do we determine the Amplitude,
Period, and Phase Shift of Sine and Cosine
Functions?

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Elaborate
Integration of Philosophical Views:
In this part, the teacher and the learners will relate
the terms/content/process learned in the lesson about
Graphs of Sine and Cosine 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
Questions
 What are the things/situations/instances that
you can relate with regards to the lesson about
Graphs of Sine and Cosine Functions?
 How will you connect the terms/content/process
of the lesson in real-life
situations/instances/scenario considering your
philosophical views?

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Elaborate
Philosophical Views Integration from the Teacher:
Graphs of Sine and Cosine
The graphs of sine and cosine can be found
everywhere. It is present in the radio waves, electrical
currents, tides, and musical tones. When we look at
seismic waves on a map of what is happening beneath us,
we can see this graph. The graphs of the sine and cosine
both have the hills and valleys in a repeating pattern. In life,
this pattern signifies the ups and downs that people face.

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Elaborate
Philosophical Views Integration from the
Teacher: Graphs of Sine and Cosine
We see the sine curves the way we react on things
naturally like the occurring phenomena. Take water
waves as an example; when waves have more energy,
the more vigorous they go up and down. The
amplitude - the distance from the resting position is an
indication of the amount of energy that the waves
contain.

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Elaborate
Philosophical Views Integration from the Teacher: Graphs of
Sine and Cosine
In the same manner, when people have low
amplitude, they have low energy to fight against the
challenges that they are facing. With them becoming less
energetic, the less vigorous the graphs go up or down. The
graph of the sine at the beginning shows the people when
they are at the top while the beginning of the cosine
shows the bottom. The movement depends on the energy
of the person.The graph may go down or may rise.

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EVALUATE

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Evaluate
Answer the following:
a) Sketch the graph of the function y = −2 cos (x −
𝛱
2
) + 3 over two
periods.
b) Graph the given sine and cosine functions with its amplitude,
period, and phase shift and determine its domain & range.
i) y = 3sin(x) and y = 3cos(x)
ii) y = −sin(x +
𝜋
3
) and y = −cos(x +
𝜋
3
)

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Evaluate
Answer the following:
c) Explain how to find the amplitude of y = −3sinx and
describe how the negative coefficient affects the graph.
d) How will you compare and contrast the graphs of y =
2sinx and y = sin 2x?

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Assignment:
Answer the following questions:
1.What is the difference between secant and
cosecant graphs?
2. How do we graph secant and cosecant functions?
3.What are the domain, range & period of sine &
cosine functions?
Reference: DepED Pre-Calculus Learner’s Material, pages 154 – 157
-GNDMJR-

Lesson no. 6 (Graphs of Sine and Cosine Functions)

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