JHS
TRIGONOMETRY
Quarter 4
AIRs - LM
Grade 9_Trigonometry_Module2
ABM – Business Mathematics
Module 8: Quarter 2
Second Edition, 2021
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Region I
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Grade 9_Trigonometry_Module2
JHS
Illustrating the Laws of Sines and
Cosines
Module 5: Quarter 4
Grade 9_Trigonometry_Module2
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 need to 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. And 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.
Grade 9_Trigonometry_Module2
Target
Good day!
This module will assess your knowledge on the different mathematical
concepts and skills in performing mathematical operations that will help you
understand laws of sines and cosines.
Before we start, let us first consider the learning competency.
1. Illustrates laws of sines and cosines (M9GE-IVe-1)
After going through this module, you are expected to:
1. illustrates the law of sine and law of cosine
2. use the concept of the sum of the measures of the interior angles is 180° in
finding the missing one angle.
3. apply the formula of sine and cosine in finding the missing part of an
oblique triangle.
For you to understand the lesson well, do the following activities.
Good luck!
Pre-Assessment Test
1
Grade 9_Trigonometry_Module2
Directions: Choose the letter of the correct answer. Write your answer on a
separate sheet of paper
1. Which of the following is the formula used in the law of sine?
A.
𝑠𝑖𝑛𝐴
𝑎
=
𝑠𝑖𝑛𝐵
𝑏
=
𝑠𝑖𝑛𝐶
B. 𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐(cos 𝐴)
𝑐
C. 𝑏2 = 𝑎2 + 𝑐2 − 2𝑎𝑐(cos 𝐵)
D. 𝑐2 = 𝑎2 + 𝑏2 − 2𝑎𝑏(cos 𝐶)
2. Which of the following is NOT the formula used in the law of cosine?
A.
𝑠𝑖𝑛𝐴
𝑎
=
𝑠𝑖𝑛𝐵
𝑏
=
𝑠𝑖𝑛𝐶
B. 𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐(cos 𝐴)
𝑐
C. 𝑏2 = 𝑎2 + 𝑐2 − 2𝑎𝑐(cos 𝐵)
D. 𝑐2 = 𝑎2 + 𝑏2 − 2𝑎𝑏(cos 𝐶)
3. The Law of Cosines is appropriate to use in solving oblique triangles when
you know the following information.
A. SAA
B. ASA
C. SSA
D. SAS
4. Using the law of cosine, which of the following should be used to find the
measure of side c?
A. 𝑐2 = 𝑎2 + 𝑏2 − 2𝑎𝑏(cos 𝐶)
B. 𝑐2 = 𝑎2 + 𝑏2 − 2𝑎𝑏(cos 𝐴)
2
2
C. 𝑐 = 𝑎 + 𝑏 2 − 2𝑎𝑏(cos 𝐵)
D. 𝑐2 = 𝑎2 + 𝑏2 − 2𝑏𝑐(cos 𝐶)
5. Which one will NOT give you information about law of sine?
A. ASA Case
B. SAA Case
C. SSA Case
D. SSS Case
6. Which one will give you information about law of cosine?
A. ASA Case
B. SAA Case
C. SSA Case
D. SSS Case
For items 7 – 8, refer to the ΔABC below.
C
3
108°
B
9
A
7. Find side a.
A. 10.27
B. 10.33
C. 13.33
8. Find angle B.
A. 12°
B. 14°
C. 16°
For items 9 – 10, refer to the ∆ABC below.
B
D. 18°
a = 12
c
A
9. Find angle B.
A. 25°
10. Find side b.
A. 11.18
D. 15
b
C
B. 50°
C. 75°
D. 90°
B. 12.28
C. 13.38
D. 14.48
2
Grade 9_Trigonometry_Module2
MODULE
2
Illustrates the Laws of Sines
and Cosines
Jumpstart
Activity 1. Let’s Recall
Direction: Find the value of x.
3
Grade 9_Trigonometry_Module2
Discover
From your previous self-learning modules, you have learned about the
special right triangles, namely: the 30°-60°-90° triangle and the isosceles right
triangle (also known as 45°-45°-90° triangle). Your knowledge of that will help
you better understand the lesson in finding the trigonometric ratios of special
angles.
In this lesson, we will learn the trigonometric functions of the special
angles 30˚, 45˚, and 60˚ and how to use them to find exact values of
trigonometric expressions without a calculator.
45°-45°-90° Right Triangle Theorem
In a 45°-45°-90° Right triangle,
 The legs are congruent
 The length of the hypotenuse is √2 times the length of the leg
The ratio of the sides is x : x : x√2
45°
45°
x√2
x
√2
1
90°
45°
45°
x
1
An Isosceles Right Triangle
Example 1: Find the length of the indicated variable.
45°
x√2
5
Using the ratio:
Using the definition:
x : x : x√2
The length of the
hypotenuse
is
√2
times the length of the
leg
5 : 5 : 5√2, then x = 5
5√2 is the hypotenuse
45°
hypotenuse = leg • √2
5
5 • √2
4
Grade 9_Trigonometry_Module2
hypotenuse = 5√2
Now let us apply the six trigonometric ratios for 45° angle.
sin 45° =
𝑜𝑝𝑝
ℎ𝑦𝑝
𝑜𝑝𝑝
cos 45° =
tan 45° =
ℎ𝑦𝑝
𝑜𝑝𝑝
𝑎𝑑𝑗
=
5
5 √2
=
=
5
5 √2
5
=
=
5
5√2
5
5√2
=
=
1
√2
1
√2
•
•
√2
√2
√2
= =
2
√2
√4
√2
√2
√2
√2
=
√4
=
sec 45° =
sec 45° =
2
=1
5
cot 45° =
5√2
5
5√2
5
5
5
=
5√2
=
5√2
5
5
=
√2
=
√2
=1
30°-60°-90° Right Triangle Theorem
In a 30°-60°-90° Right triangle,
 The length of the hypotenuse is twice the length of the shorter leg
Hypotenuse = 2 shorter leg

The length of the longer leg is √3 times the length of the shorter leg
Longer leg = √3 shorter leg
The ratio of the sides is
x : 2x : x√𝟑
30°
2x
30°
60°
90°
60°
x
2
Example 2: Find the length of the indicated side
Using the ratio x : 2x : x√3
x = shorter leg
=7
60°
7
2√3
4
x√3
y
2x = hypotenuse
= 2 (7)
y = 14
30°
7√3
x√3= longer leg
= 7√3
5
Grade 9_Trigonometry_Module2
Using the definiton:
 The length of the hypotenuse is twice the length of the shorter leg
Hypotenuse = 2 shorter leg
= 2 (7)
y = 14

The length of the longer leg is √3 times the length of the shorter leg
Longer leg = √3 shorter leg
= √3 • 7
= 7√3
Now let us apply the six trigonometric ratios for the 30° angle.
sin 30° =
cos 30° =
tan 30° =
csc 30° =
sec 30° =
cot 30° =
𝑜𝑝𝑝
=
ℎ𝑦𝑝
𝑜𝑝𝑝
ℎ𝑦𝑝
𝑜𝑝𝑝
𝑎𝑑𝑗
14
7
14
=
=
7
14
1
=
2
1
7 √3
7√3 √3
=2
=
14
14
2
7
1
√3
√3
=
•
=
7√3 √3
√3
√9
=
√3
3
=2
=1
7√3
7√3
2
14
=
7 √3
2
√3
•
√3
√3
=
2√3
√9
=
2√3
=
2√3
3
= √3
7
Let us also solve for the 60° angle.
sin 60° =
cos 60° =
tan 60° =
sec 60° =
csc 60° =
cot 60° =
𝑜𝑝𝑝
=
ℎ𝑦𝑝
𝑜𝑝𝑝
ℎ𝑦𝑝
𝑜𝑝𝑝
𝑎𝑑𝑗
14
7√3
14
7
7
7 √3
14
7
=
=
14
=
7 √3
=1
7
214
7 √3
√3
2
=
1
2
=
√3
=
2
√3
•
√3
√3
=
2√3
√9
=2
7√3
=
7
7 √3
=
1
√3
•
√3
√3
=
√3
√9
=
6
Grade 9_Trigonometry_Module2
√3
3
3
Explore
Here are some enrichment activities for you to work on to master and strengthen the basic
concepts you learned from this lesson.
ACTIVITY 1: LET’S PRACTICE!
Direction: Match Column A with the corresponding values in Column B.
Column A
Column B
1. cos 60°
a. √2
√3
2. csc 45°
b.
2
3. sin 45°
c. √3
√2
4. sin 60°
d.
2
1
5. tan 60°
e.
2
ACTIVITY 2: KEEP PRACTICING!
Direction: Solve the following to decode the mystery word
G. sec 30°
O. cos 45°
L. tan 30°
Hint: It is the large number 10100. In decimal notation, it is written as the digit
1 followed by one hundred zeroes.
Deepen
ACTIVITY 3: TEST YOURSELF!
Direction: Complete the table.
7
Grade 9_Trigonometry_Module2
Gauge
Directions: Read each of the following carefully. Choose the letter that
corresponds to the correct answer.
1. What is the value of csc 45?
A. −√3
B. √2
C. −√2
2. Which of the following is the value of sec 60°?
A. 2
B. -2
C. 1
3. What is the value of tan 45?
A. 1
B. √3
D. √3
D. -1
C. √2
D.
√2
2
√3
2
D.
√2
2
4. What is the value of sin 60?
A.
√3
3
B.
2√3
C.
3
5. Which among the angle measures is the correct value of θ if sin 𝜃 =
A. 120°
B. 90°
6. What is the value of sin 30?
A.
1
2
B.
C. 60°
√3
2
C.
1
C.
D. 30°
√2
2
D.
√2
2
D.
√3
3
7. Which among the following is the value of sec 30°?
2√3
1
2√3
1
A.
B. − 3
C.
D. − 2
3
2
8. What is the value of cos 30?
A.
2√3
3
B.
2
9. What is the value of θ if tan θ =
A. 30
√3
3
√3
2
?
B. 45
C. 60
D. 90
10. What is the value of tan 45𝑜?
A. 1
B. −1
1
C. − 2
8
Grade 9_Trigonometry_Module2
D.
1
2
√3
?
2
Answer Key
References
Printed Materials:
Website:
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Grade 9_Trigonometry_Module2