Lesson no. 4 (Circular functions on Real Numbers)

LESSON NO. 4
CIRCULAR FUNCTIONS
ON REAL NUMBERS

Lesson No. 4|Circular Functions on Real Numbers
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Introduction
• ATrigo function song in the tune of “One Call
Away” downloaded from youtube
https://www.youtube.com/watch?v=_M6WdLP2Qq
o will be played.
• We define the six trigonometric function in such a
way that the domain of each function is the set of
angles in standard position.The angles are
measured either in degrees or radians.

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Introduction
In this lesson, we will modify
these trigonometric functions so
that the domain will be real numbers
rather than set of angles.

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ENGAGE

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Engagement Activity 1
• Author: Nick Kochis
• Topic:Circle, Unit Circle
• Reference: https://www.geogebra.org/m/G7xgNRxm

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“Unit Circle - ExactValues”
Task:
Investigate the exact trigonometric values using Geogebra
applet by dragging the green dot around the unit circle.

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Questions:
1. Do you observe any patterns with the sine
and cosine functions relating
to the coordinates?
2. How will you relate the coordinates with
the sine and cosine functions?

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Small-Group Interactive Discussion
Circular Functions on Real Numbers

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on Circular Functions on Real Numbers
Inquiry Guide Questions:
- What are the values of the circular functions on
real numbers considering θ as the given angle and
P(θ) = P(x, y) be the point on the unit circle?
- How do we define the six functions on real
numbers if we let s be any real number and θ be
the angle in standard position with measure s rad?

on Circular Functions on Real Numbers
Inquiry Guide Questions:
- How do we find the exact values of trigonometric
functions considering the coordinates of the terminal
point on the unit circle of the given angle?
- How do we define the six circular functions if θ be an
angle in standard position, Q(x, y) any point on the
terminal side of θ, and
r = 𝑥2 + 𝑦2 > 0?
___________________________________________________________________

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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

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Explore
Assigned Role:
Leader – 1 student
Peacekeeper/Speaker – 1 student
Secretary/Recorder – 1 student
Time Keeper – 1
Material Manager – 1-2 students

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Explore
• Problem 1 (Group 1 & Group 2): Find the
values of cos 135°, tan 135°, sin(-60°), and
sec (-60°).
• Problem 2 (Group 3 & Group4): Find the
exact values of sin
3𝜋
2
, cos
3𝜋
2
, and tan
3𝜋
2
.

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Explore
• Problem 3 (Group 5 & Group 6): Suppose s is a
real number such that sin
s = −
3
4
and cos s > 0. Find cos s.
• Problem 4 (Group 7 & Group 8): Suppose s is a
real number such that
cos s =
1
2
and sin s > 0. Find sin s.

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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 all about?
• What are the given in the problem?
• What are the things did you consider in
solving the given problem?
• What is/are the unknown in the given
problem?

______________________________________________________________________________
Explain
Guide Questions:
 What method(s) did you use in solving the given
problem?
 How did you solve the given problem using that
method(s)?
 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:
- Considering θ as the given angle and
P(θ) = P(x, y) be the point on the unit
circle, what are the values of the six
circular functions on real numbers?

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Elaborate
Integration of PhilosophicalViews
In this part, the teacher and learners will relate
the terms/content/process learned in the lesson
about circular functions on real numbers in real life
situations/scenario/instances considering the
philosophical views that can be
integrated/associated to the
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 circular functions
on real numbers in real-life?
 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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Elaborate
Sample Philosophical Views Integration from the Teacher: Circular
Functions on Real Numbers
Circular Functions on Real Numbers deal
with finding the exact values of a given angle.
In real-life, the exact values of a given angle
can be connected or associated to our value
or worth in dealing life’s function as a person
to others.

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Elaborate
Sample PhilosophicalViews Integration from theTeacher:
Circular Functions on Real Numbers
True wisdom is looking at things from a
wide (obtuse) and positive angle. This way, we
would be able to examine the situations, look
for fresh dots, explore the things we thought
we knew, and trigger new insights.

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EVALUATE

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Evaluate
Solve the following problems:
a. Find the values of:
i) sin 30° ii) cos 150°
iii) tan (-150°) iv) sec (-30°)

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Evaluate
Solve the following problems:
b. Find the exact values of:
i) sin
11𝜋
6
ii) cos
11𝜋
` 6
iii) tan
11𝜋
6
iv) cot (−
4𝜋
3
)
c. Determine whether 3 sin 60° = sin 180° is true or false.
Explain your answer.

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Assignment:
Answer the following questions:
1.What is a reference angle?
2. How do we find the value of a circular
function at a number θ?
Reference: DepED Pre-Calculus Learner’s Material, pages 139-141.
-GNDMJR-

Lesson no. 4 (Circular functions on Real Numbers)

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