9

MATHEMATICS
Quarter 4 - Module 2
Trigonometric Ratios of Special
Angles

NegOr_Q4_Mathematics9_Module2_v2
NegOr_Q4_Mathematics9_Module2_v2
Mathematics – Grade 9
Alternative Delivery Mode
Quarter 4 – Module 2: Trigonometric Ratios of Special Angles
Second Edition, 2021

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NegOr_Q4_Mathematics9_Module2_v2
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
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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
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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.

i NegOr_Q4_Mathematics9_Module2_v2
 I

This module was designed to provide you with fun and meaningful opportunities for
guided and independent learning at our own pace and time. You will be enabled to process the
contents of the learning resource while being an active learner.

The module is intended for you to find the trigonometric ratios of special angles.

MULTIPLE CHOICE. Write the letter of the correct answer.

1. Determine the exact value of sin 30° tan 45° + tan 30° sin 60°.
a. -1 b. 0 c. 1 d. 2

2. What is the exact value of cos 30° sin 45° + sin 30° tan 30°?
3√6 +2√3 3√6 +3√2
a. b.
12 12

6√3 +2√3 6√3 +3√2

c. d.
12 12

3. Find the difference of csc230° and cot245°.

a. 2 b. 3 c. 4 d. 5

4. Find the exact value of sin 30 cos 30 sin 60 cos 60.

√2 √3 √2 √3
a. b. − c. − d.
3 2 3 2

5. What is the exact value of 2sin30°cos30°.

1 1 1 1
a. b. c. d.
8 2 3 6

1 NegOr_Q4_Mathematics9_Module2_v2
 ’s In

What Makes Us Special?

Given the angles of the triangles below, find the values of the six trigonometric
ratios.

1. Let a be the leg of a 45º-45º-90º Triangle.

45°
Sin 45° = _______________ csc 45° = __________
a c = a√2
Cos 45° = __________ sec 45° = __________

Tan 45° = __________ cot 45° = __________

45°
b=a

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

60° Sin 30° = __________ csc 30° = __________

a c = 2a
Cos 30° = __________ sec 30° = __________

30° Tan 30° = __________ cot 30° = __________

b = a√3

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

60° Sin 60° = __________ csc 60° = __________

a c = 2a
Cos 60° = __________ sec 60° = __________

30° Tan 60° = __________ cot 60° = __________

b = a√3

2 NegOr_Q4_Mathematics9_Module2_v2
 ’s New

Look at the Table of Trigonometric Values below and analyze.

𝑎𝑛𝑔𝑙𝑒 𝜃
𝜃° 30° 45° 60° 90°
ratio
½ √2 √3
sin 𝜃 0 2 2 1

√3 √2 ½
cos 𝜃 1 2 2 0

tan 𝜃 √3
0 3 1 √3 Undefined

2√3
csc 𝜃 undefined 2 √2 3 1

2√3
sec 𝜃 1 3 √2 2 Undefined

cot 𝜃 √3
undefined √3 1 3 0

The table above shows the corresponding values of trigonometric ratios

from each special angles( 𝜃°, 30°, 45°, 60°, and 90°).

3 NegOr_Q4_Mathematics9_Module2_v2
 is It

Finding Trigonometric Ratios of Special Angles

In trigonometry, 0°, 30°, 45°, 60° and 90° are called as special angles. These
angles lie in the first quadrant. They are frequently seen in applications and we
can use geometry to determine the trigonometric ratios of these angles.

Examples: Determine the exact values of each of the following:

a) sin30°tan45° + tan30°sin60°
b) 2sin30°cos30°
c ) sin²45° + sec²60°

Solutions:
a.) sin30°tan45° + tan30°sin60°
• locate first their corresponding ratios from the table on
trigonometric ratios:
sin30° = ½
tan45° = 1
√3
tan30° =
3
√3
sin60° = 2

• solve:
1
sin30°tan45° = (½) (1) =
2

√3 √3 √9 3 1
tan30°sin60° = ( )( ) = = =
3 2 6 6 2
1 1
sin30°tan45° + tan30°sin60° = + =1
2 2

• therefore, sin30°tan45° + tan30°sin60° = 1

4 NegOr_Q4_Mathematics9_Module2_v2
 b.) 2sin30°cos30°
• locate first their corresponding ratios from the table on
trigonometric ratios:
sin30° = ½
√3
cos30° = 2

• solve:
1 2
2sin30° = 2 (2) = =1
2

√ 3
2sin30°cos30° = (1) ( )
2

√3
=
2

√𝟑
• therefore, 2sin30°cos30° =
𝟐

c ) sin²45° + sec²60°

• locate first their corresponding ratios from the table on

trigonometric ratios:
√2
sin45° = 2

sec60° = 2

• solve:
2
√2 √4 2
sin²45° = ( 2 ) = = =1
2 2

sec²60° = (2)2 = 4
sin²45° + sec²60° = 1 + 4 = 5

• therefore, sin²45° + sec²60° = 5

5 NegOr_Q4_Mathematics9_Module2_v2
 ’s More

Determine the six trigonometric ratios for angle D in the right triangle below.

5
4

D 3 F

I Have Learned

Fill-in the chart below. Write your answers on your notebook.

3 things I learned

2 things that interest me

1 application of what I learned

I Can Do

In your notebook, make your own problem using the trigonometric ratios and the four
fundamental operations (addition subtraction, multiplication and division). Show your
solutions. Your output will be rated by your teacher using the scale of 1 – 10; 1 being the lowest
and 10 being the highest.

6 NegOr_Q4_Mathematics9_Module2_v2
Connect Me!

Read, analyze, solve and write the letter of the correct answer in each problem until
you could form a phrase. The choices are on the table below its corresponding letter.

Find the exact value of the following:

1.) sin30° + csc60°

2.) 5sec30°tan60°

3.) 3sin45° + 4cos45°

4.) sec45°csc45° - 1

sin45°
5.) 1 -
cos45°

𝑡𝑎𝑛 45°
6.)
(𝑡𝑎𝑛 30° + 𝑡𝑎𝑛 60°)

7.) tan2 60° - 2tan2 45° - cot2 30° + 2sin2 30°

8.) 4 (sin4 30° + cos460°) - 3 (cos245° - sin290°)

9.) 6 cos290° + 3 sin290° + 4 tan245°

10.) 4 cot245 - sec260 + sin260 + cos260

A C E I L

3
√3 7 −
3 0 2
4 2

O P S U Y

5
7 10 2 1 2

7 NegOr_Q4_Mathematics9_Module2_v2
Find the exact value of the variable/s. If your answer is not an integer, leave it in simplest
radical form.

1.)
6 x

60°
12

1
a. 2 b. 12√3 c. 2 d.6√3

2.)
x y

30°
20√3

a. x =10√3, 𝑦 = 30 b. x =10, y = 30√3

c. x = 30√3, 𝑦 = 10 d. x =30, y = 10√3

8 NegOr_Q4_Mathematics9_Module2_v2
What I What’s In What’s Assessment Additional
Know More Activities
1. C √2 4 1. S 1. D
1. sin 45° = sin D =
2 5
2. A 3 2. P 2. D
√2 cos D =
3. B cos 45°= 5 3. E
2
4
4. D tan 45°= 1 tan D = 4. C
3
5. A csc 45° = √2 5 5. I
csc D =
4 6. A
sec 45°= √2 5
sec D = 7. L
cot 45°= 1 3
3 8. Y
2. sin 30° =
1 cot D =
4
2 9. O
√3
cos 30° = 10. U
2
√3
tan 30° =
3
csc 30° = 2
2√3
sec 30°=
3
cot 30°= √3
√3
3. sin 60° =
2
1
cos 60° =
2
tan 60° = √3
2√3
csc 60° =
3
sec 60°= 2
√3
cot 60°=
3

9 NegOr_Q4_Mathematics9_Module2_v2
References
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2021. May 26. https://bit.ly/3K1mQm0.

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2021. July 7. https://bit.ly/3Fa5DTL.

n.d. https://www.coursehero.com/file/107301052/Math-9-Q4-Module-2pdf/.

n.d. https://bit.ly/3JVDtj4.

n.d. https://www.sarthaks.com/176804/evaluate-4-sin-4-30-cos-4-60-3-cos-2-45-sin-2-90.

2019. February 25. https://brainly.com/question/12194259.

n.d. https://brainly.ph/question/9914540.

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Corporation.

Bryant, M. et al. 2014. Mathematics Grade 9 Learner's Module. DepEd-IM.

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