9 TRIGONOMETRIC RATIOS OF SPECIAL ANGLES - 4th QUARTER - Module 2: Department of Education - ZNNHS

Republic of the Philippines
 Department of Education
 Regional Office IX, Zamboanga Peninsula

 9 Zest for Progress
 Z Peal of artnership

 4th QUARTER – Module 2:
 TRIGONOMETRIC RATIOS OF SPECIAL
 ANGLES

Name of Learner: ___________________________
Grade & Section: ___________________________
Name of School: ___________________________

Mathematics – Grade 9
Alternative Delivery Mode
Quarter 4 - Module 2: Finding the Trigonometric Ratios of Special Angles
First Edition, 2020
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over them.

Published by the Department of Education
Secretary: Leonor Magtolis Briones
Undersecretary: Diosdado M. San Antonio

 Development Team of the Module
 Writer: Divine T. De Guzman
 Editors: Ma. Dolores J. Gregorio
 Shirly V. Gajilomo
 Reviewers: EPS, Mathematics Vilma A. Brown, Ed. D.
 Principal Mujim U. Abdurahim
 Management Team: SDS Roy C. Tuballa, EMD, JD, CESO VI
 ASDS Jay S. Montealto, CESO VI
 ASDS Norma T. Francisco, DM, CESE
 EPS Mathematics Vilma A. Brown, Ed. D.
 EPS LRMS Aida F. Coyme, Ed. D.

Printed in the Philippines
Department of Education – Region IX, Zamboanga Peninsula
Office Address: Tiguma, Airport Road, Pagadian City
Telefax: (062) – 215 – 3751; 991 – 5975
E-mail Address: region9@deped.gov.ph

 1

Introductory Message
This Self – Learning Module (SLM) is prepared so that you, our dear learners, can continue
your studies and learn while at home. Activities, questions, directions, exercises, and
discussions are carefully stated for you to understand each lesson.
Each SLM is composed of different parts. Each part shall guide you step-by-step as you
discover and understand the lesson prepared for you.
Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell
you if you can proceed on completing this module or if you need to ask your facilitator or your
teacher’s assistance for better understanding of the lesson. At the end of each module, you
need to answer the post-test to self-check your learning. Answer keys are provided for each
activity and test. We trust that you will be honest in using these.
In addition to the material in the main text, Notes to the Teacher are also provided to our
facilitators and parents for strategies and reminders on how they can best help you on your
home-based learning.
Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use
a separate sheet of paper in answering the exercises and tests. Read the instructions carefully
before performing each task.
If you have any questions in using this SLM or any difficulty in answering the tasks in this
module, do not hesitate to consult your teacher or facilitator.
Thank you.

 What I Need to Know

 This module is written as an aid in the basic trigonometric lesson of the fourth
quarter of grade 9 mathematics. The module follows a step – by – step approach to
computational trigonometry supported by examples and exercises. It covers the key
concepts of finding the trigonometric ratios of special angles.
 This module is designed to cater the academic needs of diverse learners in
achieving and improving the twin goals of mathematics in basic education which are
critical thinking and problem solving. The language used recognizes the vocabulary
level of grade 9 students. The lessons followed developmentally sequenced teaching
and learning processes to meet the curriculum requirement.
 After going through the module, you are expected to find the trigonometric
ratios of special angles. (M9GE-IVb-c-1)
 Believe that learning can continue amidst the health crisis. Good luck, stay safe,
and God bless.

 2

What I Know?

 Let us find out how much you already know about this module. Answer the
following questions as much as you can by writing on your answer sheet the letter that
you think is the correct answer. Take note of the items that you were not able to answer
correctly and then let us find out the correct answer as we go through this module.

1. With respect to an angle, what is the ratio of the adjacent side to the hypotenuse?
 A. sine B. cosine C. tangent D. cosecant

2. Which of the following is a special angle?
 A. 10°-20°-150° B. 40°-50°-90° C. 45°-45°-90° D. 50°-50°-80°

3. What is the value of sin 30°?
 1 1 3
 A. 4 B. 2 C. 4 D. 1

4. What is the value of x, if cot x = 1?
 A.30° B. 45° C. 60° D. 90°

5. Find the exact value of sin2 30° + tan2 45°.
 1 1 1 1
 A. 14 B. 24 C. 1 2 D. 2 2

 3

LESSON FINDING THE TRIGONOMETRIC
 1 RATIOS OF SPECIAL ANGLES

 What’s In?
 ACTIVITY 1 NAME ME!
Directions: Match the trigonometric identities and the ratio by drawing a line to connect them.

 ℎ sine Ꝋ 
 
 cosine Ꝋ

 tangent Ꝋ
 ℎ 
 ℎ 
 secant Ꝋ
 cosecant Ꝋ
 
 tangent Ꝋ ℎ 

 ACTIVITY FIND MY MEASURE!
Directions: Use a protractor to find the measure of the angles in each triangle.

1. 2. 3.
 a=4

 b = 14
 c=7
 a=4
 c=4 b = 4ξ2
 c = 2ξ3

 a = 7ξ3

 b=2

 ∠A = _______ ∠A = _______ ∠A = _______
 ∠B = _______ ∠B = _______ ∠B = _______
 ∠C = _______ ∠C = _______ ∠C = _______

QUESTIONS:

1. What have you noticed about the length of the sides of each triangle?

 4

2. What have you observed about the measure of the angles in each triangle?

3. What do you call these triangles?

4. Write the mathematical concepts that you learned from the activity.

 45° - 45° - 90° Triangle 30° - 60° - 90° Triangle
 _______________________________ _______________________________
 _______________________________ _______________________________
 _______________________________ _______________________________

 What’s New?
 ACTIVITY WHAT’S MY MEASURE?
A. Directions: Given the angles of the triangles below, find the values of the six
 trigonometric ratios

 Let a be the leg of a 45°-45°-90° Let a be the leg of a 30°-60°-90° Let a be the leg of a 30°-60°-90°
 Triangle. Triangle. Triangle.

 sin 45° = _____ sec 45° = _____ sin 30° = _____ sec 30° = _____ sin 60° = _____ sec 60° = _____
 cos 45° = _____ csc 45° = _____ cos 30° = _____ csc 30° = _____ cos 60° = _____ csc 60° = _____
 tan 45° = _____ cot 45° = _____ tan 30° = _____ cot 30° = _____ tan 60° = _____ cot 60° = _____

B. Directions: Summarize the Six Trigonometric Ratios for Special Angles using the table
 below.
 TRIGONOMETRIC RATIOS OF SPECIAL ANGLES
 Ꝋ sin cos tan csc sec cot

 30°

 45°

 60°

 5

What is It?

 In geometry, a branch of mathematics that deals with the sides and angles of a
right angled triangle is Trigonometry. Trigonometric ratios are thus measured with
regard to sides and angles.

 The trigonometry ratios for a specific angle ‘θ’ are given below:
 opposite side ℎ 
 sin Ꝋ = csc Ꝋ = 
 ℎ 
 ℎ 
 cos Ꝋ = sec Ꝋ = 
 ℎ 
 
 tan Ꝋ = cot Ꝋ = 

 The trigonometric ratios for some specific angles such as 30°, 45°, and 60° are
given below, which are commonly used in mathematical calculations.

 TRIGONOMETRIC RATIOS OF SPECIAL ANGLES
 Ꝋ sin cos tan csc sec cot
 1 ξ3 1 2
 30° 2 ξ3
 2 2 ξ3 ξ3
 1 1
 45° 1 ξ2 ξ2 1
 ξ2 ξ2
 ξ3 1 2 1
 60° ξ3 2
 2 2 ξ3 ξ3
 Using the table of values of trigonometric ratios of special angles, let us
consider the following examples:

1. Determine the exact value of the expression sin 30° + cos 60°.

 1 1
 SOLUTION: sin 30° + cos 60° = +
 2 2
 2
 = 2 or 1
2. Find the exact value of csc2 30° + cos2 45°.

 1 2
 SOLUTION: csc2 30° + cos2 45° = 22 + ( 2)
 ξ
 1
 =4+2
 1
 =42

 6

3. Determine the value of angle x when cot x = 1.

 SOLUTION: Referring to the table, cot 45° = 1. Therefore, x = 45°.

4. If x = 30°, show that cot2 x + csc x = 5.

 SOLUTION: cot2 x + csc x = 5
 (cot 30°)2 + csc 30°= 5
 (ξ3 )2 + 2 = 5
 3 +2=5
 5=5

5. What is (csc 30°)(sin 60°) equals to?

 ξ3
 SOLUTION: (csc 30°)(sin 60°) = 2 ( 2 )
 ξ3
 =2(2)
 = ξ3

6. Find the length of the legs of the 45-45°-90° triangle if the hypotenuse is 6cm.

 SOLUTION: ℎ = ξ2
 6 ξ2
 =
 ξ2 ξ2
 ξ 
 ( ) ( ) = 
 ξ ξ 
 6ξ 
 ( ) = 
 
 ( ξ ) = 

 What’s More?
A. Directions: Solve each problem and answer the riddle by writing the letter on the
 blank.
 HINT: Use the table of trigonometric ratios of special angles to answer the problem
 easily)

 H - 10 sin 30° Y - 3 cot 45°
 I - 2 tan 60° N - 18 sec 30°
 D - 7 cos 45° A - 50 sin 60°
 L - 11 csc 30° Z - 13 cot 45°

 7

MATH TRIVIA
 “ Who is the first-ever Filipina Mindanaoan who won gold for weightlifting
 during the 30th SEA games?

 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___
 5 22 3 13

B. Problem Solving.
 Directions: Given the triangle JOY with 30-60-90 angle measures, find the
 measure of side OY.

 O

 ?

 J Y

 What I Have Learned
Directions: Supply the trigonometric ratios of the given special angles.

 TRIGONOMETRIC RATIOS

 30° 45° 60°

 8

What I Can Do?

 Talon Talon National High School celebrates its Foundation Day every 10th of
November. One of the highlights are activities for Recreational Games which include
the “Palo Sebo”. Usually, they build the Pole with a height of 8 ft. This year, they wish
to put a 10-foot high of pole and decorate a bantings from the top of a pole to a point
on the ground. The assigned member is having difficulty to estimate the number of
wires to buy for the bantings. As a student of the school, at least, how many ft of wire
will you suggest connecting the point on the ground to the top of the pole? Show your
illustration and identify the figures.

 Illustration:

 Solution:

 Assessment
Directions: Choose the letter that corresponds to your answer.

1. With respect to an angle, what is the ratio of the opposite side to the hypotenuse?
 A. sine B. cosine C. tangent D. cosecant

2. Which of the following is a special angle?
 A. 10°-20°-150° B. 30°-60°-90° C. 35°-55°-90° D. 50°-50°-80°

3. What is the value of cos 60°?
 1 1 3
 A. 4 B. 2 C. 4 D. 1

4. What is the value of x, if sec x = 2?
 A.30° B. 45° C. 60° D. 90°

5. Find the exact value of sec 60° + sin 30°.
 1 1 1 1
 A. 14 B. 24 C. 1 2 D. 2 2

 9

6. What is (sin 45°) (sec 60°) equals to?
 1 2 3ξ2
 A. B. C. ξ2 D.
 ξ2 ξ2 2

7. Given 45°-45°-90° what is the measure of the hypotenuse if the leg measures 5cm?
 5 5 ξ2
 A. 5ξ2 B. C. D.
 2 ξ2 5

8. What is the measure of the shorter leg of a 30°-60°-90° triangle, given that its
 hypotenuse is equal to 24cm?
 A. 6 B. 8 C. 10 D. 12 

9. Which of the following situations can trigonometric ratios of special angles be
 applied?
 A. Finding the area of rectangular field.
 B. Finding the amount of space of a cylindrical container.
 C. Finding the number of students who passed the exam.
 D. Finding the height of an electric post.

 10. Evaluate the measures of each side of the 30°-60°-90° triangle, which of the side
 has a correct or exact measure of the triangle?
 A. only shorter leg is correct
 Y
 60° B. only longer leg is correct
 4 cm
2 cm C. only hypotenuse is correct
 D. all three sides are correct
 90° 30°
 X Z
 2ξ3 cm

 Additional Activities
Directions: Use the choices in the box to complete the equations below. Use the
Table for trigonometric ratios of special angles to answer the following:

 3 7
 sin 30° tan 30° tan 45°
 4 4

 1 ξ2
 sin 60° cot 30° 6ξ2
 2 4

 1
 1 sec 30° csc 60° 0
 4

 10

1. (sin 45°)2 = __________

2. tan 60° ( __________) = 3

3. (sin 60°)(cos 30°) + tan 45° = _____________

4. 2 sin 30° + ___________ = 2

5. (sin 30°)(cot 30°)(________) = 1

 11

12
What’s New:
A.
 1 1
sin 45° = cos 45° = tan 45° = 1 csc 45° = ξ2 sec 45° =ξ2 cot 45° = 1
 ξ2 ξ2
 1 ξ3 1 2
sin 30° = cos 30° = tan 30° = csc 30° = 2 sec 30° = cot 30° = ξ3
 2 2 ξ3 ξ3
 ξ3 1 2 1
sin 60° = cos 60° = tan 60° = ξ3 csc 60° = sec 60° = 2 cot 60° =
 2 2 ξ3 ξ3
 TRIGONOMETRIC RATIOS OF SPECIAL ANGLES
 Ꝋ sin cos tan csc sec cot
 1 ξ3 1 2
 30° 2 ξ3
 2 2 ξ3 ξ3
 1 1
 45° 1 ξ2 ξ2 1
 ξ2 ξ2
 ξ3 1 2 1
 60° ξ3 2
 2 2 ξ3 ξ3
What’s In:
ACTIVITY 1:
 opposite hypotenuse
sin Ꝋ = csc Ꝋ =
 hypotenuse opposite
 adjacent hypotenuse
cos Ꝋ = sec Ꝋ =
 hypotenuse adjacent
 opposite adjacent
tan Ꝋ = cot Ꝋ =
 adjacent opposite
ACTIVITY 2:
1. ∠A = 45° 2. ∠A = 90° 3. ∠A = 60°
 ∠C = 45° ∠B = 30° ∠B = 90°
 ∠B = 90° ∠C = 60° ∠C = 30°
What I Know:
1. B 2. C 3. B 4. B 5. A
 Answer Key

13
Additional Activities: (Optional)
 1
1. 2
 4. tan 45° or cot 45°
2. cot 30° 5. sec 30°
 7
3. 4
Assessment:
1. A 2. B 3. B 4. C 5. D
6. C 7. A 8. D 9. D 10. D
What I Can Do:
Hypotenuse ≈ 14.14 
What I Have Learned:
What’s More:
Answer: A) HIDILYN DIAZ B) 5

References:

Dr. Yvette F. Lim et. al., Math for Engaged Learning (Grade 9) Quezon: Sibs
Publishing House, Inc., 2014, 4 – 24.

Arvie D. Ubarro et. al., Soaring 21st Century Mathematics (Second Edition) Quezon :
Phoenix Publishing House, Inc., 2018, 121 – 148.

Rigor B. Ponsones, Shirlee R. Ocampo, and Regina M. Tresvalles, Math Ideas and
Life Applications (Second Edition) Quezon: Abiva Publishing House, Inc., 2017, 189-
215

Von Anthony G. Torio et. al., Smart in Math 9 Quezon: Isa-Jecho Publishing, Inc.,
2017, 81 – 117

“Applications of Quadratic Functions”, CK-12, 2019, ck12.org

“Applications of Quadratic Functions”, Monterey Institute for Technology and
Education, 2011, montereyinstitute.org.

https://byjus.com/maths/trigonometric-ratios/

 14

I AM A FILIPINO
 by Carlos P. Romulo
I am a Filipino – inheritor of a glorious past, hostage to the It is the mark of my manhood, the symbol of my dignity as
uncertain future. As such, I must prove equal to a two-fold a human being. Like the seeds that were once buried in the
task – the task of meeting my responsibility to the past, and tomb of Tutankhamen many thousands of years ago, it shall
the task of performing my obligation to the future. grow and flower and bear fruit again. It is the insigne of my
I am sprung from a hardy race – child many generations race, and my generation is but a stage in the unending
removed of ancient Malayan pioneers. Across the centuries, search of my people for freedom and happiness.
the memory comes rushing back to me: of brown-skinned I am a Filipino, child of the marriage of the East and the
men putting out to sea in ships that were as frail as their hearts West. The East, with its languor and mysticism, its passivity
were stout. Over the sea I see them come, borne upon the and endurance, was my mother, and my sire was the West
billowing wave and the whistling wind, carried upon the that came thundering across the seas with the Cross and
mighty swell of hope – hope in the free abundance of the new Sword and the Machine. I am of the East, an eager
land that was to be their home and their children’s forever. participant in its struggles for liberation from the imperialist
This is the land they sought and found. Every inch of shore yoke. But I know also that the East must awake from its
that their eyes first set upon, every hill and mountain that centuried sleep, shake off the lethargy that has bound its
beckoned to them with a green and purple invitation, every limbs, and start moving where destiny awaits.
mile of rolling plain that their view encompassed, every river For I, too, am of the West, and the vigorous peoples of the
and lake that promised a plentiful living and the fruitfulness West have destroyed forever the peace and quiet that once
of commerce, is a hollowed spot to me. were ours. I can no longer live, a being apart from those
By the strength of their hearts and hands, by every right of whose world now trembles to the roar of bomb and cannon
law, human and divine, this land and all the appurtenances shot. For no man and no nation is an island, but a part of the
thereof – the black and fertile soil, the seas and lakes and main, and there is no longer any East and West – only
rivers teeming with fish, the forests with their inexhaustible individuals and nations making those momentous choices
wealth in wild and timber, the mountains with their bowels that are the hinges upon which history revolves. At the
swollen with minerals – the whole of this rich and happy land vanguard of progress in this part of the world I stand – a
has been for centuries without number, the land of my forlorn figure in the eyes of some, but not one defeated and
fathers. This land I received in trust from them, and in trust lost. For through the thick, interlacing branches of habit and
will pass it to my children, and so on until the world is no custom above me I have seen the light of the sun, and I
more. know that it is good. I have seen the light of justice and
I am a Filipino. In my blood runs the immortal seed of heroes equality and freedom, my heart has been lifted by the vision
– seed that flowered down the centuries in deeds of courage of democracy, and I shall not rest until my land and my
and defiance. In my veins yet pulses the same hot blood that people shall have been blessed by these, beyond the power
sent Lapulapu to battle against the alien foe, that drove Diego of any man or nation to subvert or destroy.
Silang and Dagohoy into rebellion against the foreign I am a Filipino, and this is my inheritance. What pledge
oppressor. shall I give that I may prove worthy of my inheritance? I
That seed is immortal. It is the self-same seed that flowered shall give the pledge that has come ringing down the
in the heart of Jose Rizal that morning in Bagumbayan when corridors of the centuries, and it shall be compounded of the
a volley of shots put an end to all that was mortal of him and joyous cries of my Malayan forebears when first they saw
made his spirit deathless forever; the same that flowered in the contours of this land loom before their eyes, of the battle
the hearts of Bonifacio in Balintawak, of Gregorio del Pilar cries that have resounded in every field of combat from
at Tirad Pass, of Antonio Luna at Calumpit, that bloomed in Mactan to Tirad Pass, of the voices of my people when they
flowers of frustration in the sad heart of Emilio Aguinaldo at sing:
Palanan, and yet burst forth royally again in the proud heart “I am a Filipino born to freedom, and I shall not rest until
of Manuel L. Quezon when he stood at last on the threshold freedom shall have been added unto my inheritance—for
of ancient Malacanang Palace, in the symbolic act of myself and my children and my children’s children—
possession and racial vindication. The seed I bear within me forever.”
is an immortal seed.

 15