This document is a learning activity sheet for Pre-Calculus students in Region VI – Western Visayas of the Philippines. It contains instructions and examples for using reference angles to find the exact values of circular functions. Students are asked to use reference angles and consider the appropriate quadrant and sign to calculate values like sin 60°, cos 150°, and tan (π/6). The activity sheet provides the key definitions and formulas needed to complete the exercises. It aims to help students learn independently and meet the standards of the K-12 Basic Education Curriculum.
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1. REGION VI – WESTERN VISAYAS
Pre-Calculus
Learning Activity Sheet
Quarter 2 – MELC 5
(STEM_PC11T-IIb-2)
REGION VI – WESTERN VISAYAS
2. 1
Development Team of Pre-Calculus Learning Kit
Writer: Josette C. Cajutol
Illustrator: Patrick T. Lomigo
Layout Artist: Krishea Mae P. Jaruda
Division Quality Assurance Team:
Dr. Julio J. Villalon
Dr. Kim Jay C. Encio
Rosanna A. Daitao
Division of Iloilo Management Team: Dr. Roel F. Bermejo, Dr. Nordy D. Siason, Jr.
Dr. Lilibeth T. Estoque, Dr. Azucena T. Falales
Ruben S. Libutaque, Lilibeth E. Larupay
Dr. Kim S. Arceña
Regional Management Team:
Ma. Gemma M. Ledesma,
Dr. Josilyn S. Solana,
Dr. Elena P. Gonzaga,
Mr. Ronald T. Genine,
Mathematics – SHS Grade 11 Pre-Calculus
Activity Sheet No. 5
First Edition, 2020
Published in the Philippines
by the Department of Education
Region 6 – Western Visayas
Republic Act 8293, section 176 states that: No copyright shall subsist in any
work of the Government of the Philippines. However, prior approval of the government
agency or office wherein the work is created shall be necessary for exploitation of such
work for profit. Such agency or office may, among other things, impose as a condition
the payment of royalties.
This Learning Activity Sheet in Pre-Calculus is developed by DepEd Region 6
– Western Visayas.
ALL RIGHTS RESERVED. No part of this learning resource may be reproduced
or transmitted in any form or by any means electronic or mechanical without written
permission from the DepEd Regional Office 6 – Western Visayas.
3. 2
Introductory Message
Welcome to Senior High School Grade 11 Mathematics(Pre-Calculus).
The Pre-Calculus Learning Kit is a product of the collaborative efforts of the
Schools Division of Iloilo and DepEd Regional Office VI – Western Visayas through the
Curriculum and Learning Management Division (CLMD). This is developed to guide the
learning facilitators (teachers, parents, and responsible adults) in helping our learners
meet the standards set by the K to 12 Basic Education Curriculum.
The Learning Activity Sheet is self-directed instructional materials aimed to
guide our learners in accomplishing activities at their own pace and time using the
contextualized resources in the community. This will also assist learners in acquiring the
lifelong learning skills, knowledge and attitudes for productivity and employment.
For the learning facilitator:
The Learning Activity Sheet will help you facilitate the teaching-learning
activities specified in each Most Essential Competency (MELC) with minimal or no face-
to-face encounter between you and learner. This will be made available to the learners
with the references/links to ease the independent learning.
For the learner:
The Learning Activity Sheet is developed to help you continue learning even if
you are not in school. This learning material provides you with meaningful and engaging
activities for independent learning. Being an active learner, carefully read and
understand the instructions then perform the activities and answer the assessments.
This will be returned to your facilitator on the agreed schedule.
4. 3
Pre-Calculus Learning Activity Sheets (LAS) (For SHS Mathematics)
School:_________________________________________________________
Name of Learner:_________________________________________________
Grade Level:_____________________________________________________
Section:_________________________________________________________
Date:___________________________________________________________
PRE-CALCULUS LEARNING ACTIVITY SHEET
Learning Competency
Use reference angles to find exact values of circular functions. (STEM_PC11T-IIb-2).
Generalizations
A reference angle is an acute angle formed by the terminal side and the x-axis.
Below is a guide in finding reference angles:
Quadrant of 𝜽 Reference Angle
I 𝜃
II 180° - 𝜃
III 𝜃 - 180°
IV 360° - 𝜃
The signs of the coordinates of P(𝜃) depends on the quadrant or axis where it
terminates. It is important to know the sign of each circular function in each quadrant.
Quadrant II Quadrant I
(-,+) (+,+)
Only sine and cosecant All circular functions
functions are positive. are positive.
Quadrant III Quadrant IV
(-,-) (+,-)
Only tangent and cotangent Only cosine and secant function
functions are positive. are positive.
5. 4
Using the fact that the unit circle is symmetric with respect to the x-axis, the y-axis, and
the origin, we can identify the coordinates of all the points using the coordinates of the
corresponding points in the Quadrant I for the special angles. Refer to the figure below.
Example 1:
Use reference angle and appropriate sign to find the exact value of each expression.
1. sin 45° 3. sin
5𝜋
3
2. cos (— 135 °) 4. tan 150°
Solution:
To find the exact value of the given function, consider the quadrant where the given angle lies
and its sign and then solve the reference angle using the formula in the table above.
Given Angle Quadrant Reference Angle Sign Exact Value
1. sin 45° 45° I 45° + sin 45° =
√2
2
2. cos(−135°) −135° III 180° − 135° = 45° − cos(−135 °) = −
√2
2
3. sin
5𝜋
3
5𝜋
3
IV 2𝜋 −
5𝜋
3
=
6𝜋
3
−
5𝜋
3
=
𝜋
3
− sin
5𝜋
3
= −
√3
3
4. tan 150° 150° II 180° − 150° = 30° −
tan 150° =
𝑦
𝑥
= −
1
2
√3
2
tan 150° = −
1
2
∙
2
√3
tan 150° = −
1
√3
= −
√3
3
Exercises 1
Use reference angle and appropriate sign to find the exact value of each expression.
1. sin 60° 4. csc ( - 300°)
2. cos 150° 5. sec ( -
3𝜋
4
)
3. tan
𝜋
6
6. 5
Example 2:
Find the values of the six circular functions of 𝜃 whose terminal side is at
5𝜋
6
.
Solution:
The coordinates of the terminal point are (-
√3
2
,
1
2
). Thus, the values of the six circular
functions of 𝜃 are:
sin
5𝜋
6
= y =
1
2
csc
5𝜋
6
=
1
𝑦
=
1
1
2
= 2
cos
5𝜋
6
= x = -
√3
2
sec
5𝜋
6
=
1
𝑥
=
1
−
√3
2
= 1 ∙ -
2
√3
= -
2√3
3
tan
5𝜋
6
=
𝑦
𝑥
=
1
2
−
√3
2
=
1
2
∙ −
2
√3
= -
1
√3
= -
√3
3
cot
5𝜋
6
=
𝑥
𝑦
=
− √3
2
1
2
= -
√3
2
∙
2
1
= - √3
Exercises 2
Determine the values of the six circular functions of 𝜃 whose terminal side contains the
terminal point of an arc given the following lengths.
a. 45° b.
7𝜋
6
c.
4𝜋
3
d. - 135°
Enrichment Activity
Circular functions have many practical applications in real life. Using any online
references, research on how the circular functions are applied in the following fields:
a. Health
b. Meteorology
c. Music
d. Amusement parks
Present your output in the form of power point presentation that will show how the
circular functions are applied in the indicated fields.
Reflection
1. What challenges did you encounter in this module? How did you overcome these
challenges?
2. What concepts, skills, or values have you learned in this module which you think you can
use in your future endeavors in mathematics or in your everyday life?
References
1. Diwa Senior High School Series: Precalculus Book, 2016, pp.147- 154.
2. Pre-Calculus Learner’s Material.1st
Edition. (2016). Pasig City: Department of Education-
Bureau of Learning Resources (DepEd-BLR). pp. 139-142.
