Grade 9 Instructional Support Material 4th Q

Republic of the Philippines
Department of Education
REGION III
SCHOOLS DIVISION OF TARLAC PROVINCE
Address: Macabulos Drive, San Roque, Tarlac City

Telephone No.: (045) 982-0374
Email Address: tarlac@deped.gov.ph
REGION III
TABLE OF CONTENTS
TRIANGLE TRIGONOMETRY
NO. MELC DURATION K to 12 CG PAGE

Code NO.
1 Illustrates the six trigonometric ratios:
sine, cosine, tangent, secant, cosecant,
M9GE-IVa-1 3
and cotangent.
Week 1 to 2
2 Finds the trigonometric ratios of special
M9GE -IVb-c-1 5
angles.
3 Illustrates angles of elevation and angles
M9GE-IVd-1 7
of depression.
Week 3 to 5
4 Uses trigonometric ratios to solve real-life
M9GE-IVe-1 10
problems involving right triangles.
5 Illustrates laws of sines and cosines. M9GE-IVf-g-1 13
6 Solves problems involving oblique Week 6 to 9
M9GE-IVh-j-1 16
triangles.

Telephone No.: (045) 982-0374
REGION III
L E S S O N I
ILLUSTRATING THE SIX TRIGONOMETRIC RATIO (Week 1)
Activity 1: Word Hunt!

Look for the following mathematical terms on the crossword puzzle below.
1. trigonometry 6. cotangent 11. hypotenuse
2. degree 7. secant 12. cofunction
3. sine 8. cosecant 13. complementary
4. cosine 9. adjacent 14. angles
5. tangent 10. opposite 15. ratio
A A D J A C E N T C O M P L E
T T C C O T A N G E N T T O P
N C E Z A R E O S R E A A P O
A N G O I T A R Y S H Y P O M
C L O V N T N C O F U N C T I
E C O A A T A C M A T H A A G
S T R I G O N O M E T R Y T U
O O A U G U S S H Y P O R Y E
C O S E C A N I A N G L E S L
A D J A C E N N A G U A Y O O
P D D E G R E E L Y D I A B U
O C C O M P L E M E N T A R Y
P C O F U N C T I O N A R Y O
P I P Y H Y P O T E N U S E L
O P P O S I T E H Y P O T E N

Telephone No.: (045) 982-0374
REGION III
Activity 2: Match Me.

Match the name of each trigonometric ratio in column A with the correct ratio of the
side lengths of a right triangle in column B. Write the letter of the correct answer in the blank
provided.
Column A Column B
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
__________1. cosine a.
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
__________2. Sine b. 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
__________3. Secant c. 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
__________4. Tangent d. 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
__________5. Cotangent e. ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
__________6. Cosecant f. ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Activity 3: Find Me!

Use ∆𝑀𝐴𝑇 to find each trigonometric ratio.
1. sin 𝑀 = _________ 6. cot 𝑇 =_________

2. cos 𝑇 = _________ 7. sec 𝑇 =_________
3. tan 𝑀 = _________ 8. cos 𝑀 =_________
4. sec 𝑀 = _________ 9. cot 𝑀 =_________
5. csc 𝑇 = _________ 10. sin 𝑇 =_________

Telephone No.: (045) 982-0374
REGION III
L E S S O N 2
ILLUSTRATING THE SIX TRIGONOMETRIC RATIOS OF SPECIAL ANGLES (Week 2)
Activity 1: Quantitative Comparison

Without using a scientific calculator, write
A if the quantity under column I is greater than that under column II.
B if the quantity under column I is lesser than that under column II.
C if the quantities are equal.
D if the quantities cannot be compared.
Column I Column II
_________1. sin 30° cos 30°
_________2. cos 45° sin 60°
_________3. tan 45° cot 45°
_________4. csc 30° sec 60°
_________5. sec 30° cos 45°
_________6. tan 45° sin 30°
_________7. cos 45° sin 45°
_________8. csc 30° + cot 45 sin 30° + cos 60°
_________9. sec 60° − csc 30° tan 60°
tan 60°
_________10. sin 60°
csc 30°
Activity 2: What Makes You Special?

Refer to the given triangles below to complete the table.
Find the
measures/
values of
the
following:
𝐴𝐶
𝐵𝐶
sin 𝐴
cos 𝐴
5

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REGION III
tan 𝐴
csc 𝐴
sec 𝐴
cot 𝐴
Activity 3: Another Solving!

Solve for the measurement of the unknown sides of the following triangles.
1. 𝑏 = ________ 2. 𝑎 = ________
𝑐 = ________ 𝑏 = ________
3. 𝑠 =________ 4. 𝑝 =________ 5. 𝑥 = ________

𝑢 =________ 𝑟 =________ 𝑦 =________

Telephone No.: (045) 982-0374
REGION III
L E S S O N 3
ILLUSTRATING ANGLES OF ELEVATION AND DEPRESSION (Week 3-5)
Activity 1: Fill Me.

Fill in the blanks to complete the definitions.
1. An angle of __________ is the angle formed by a horizontal line and a line of sight to a
point above the line.
2. An angle of __________ is the angle formed by a horizontal line and a line of sight to a
point below the line.
3. The line which is drawn from the eyes of the observer to the point being viewed on the
object is known as the __________.
4. __________is a straight line on the coordinate flat surface where all points on the line
have the same y-coordinate.
5. If two rays or two line segments meet at a common endpoint, then the point is known
as the __________.
Activity 2: Classify Me.

Marco breeds and trains homing pigeons on the roof of his building. Classify each
angle as an angle of elevation or an angle of depression.
1. ∠1: __________________
2. ∠2: __________________
3. ∠3: __________________
4. ∠4: __________________

Telephone No.: (045) 982-0374
REGION III
Activity 3: Illustrate and Solve!

Illustrate and solve the following problems. Use the table below.
1. A 12-meter-high post casts a 19-meter shadow. Find the angle of elevation of the sun.
Draw the diagram
What is/are the given?

What is asked?
What trigonometric ratio
must be used to solve the
problem?
Solution
2. The angle of elevation from a boat to the top of a 92-meter hill is 12o. How far is the
boat from the base of the hill?
Draw the diagram

What is asked?
problem?
Solution

Telephone No.: (045) 982-0374
REGION III
3. From the top of a tower which is 175 ft tall, the angle of depression to a house is 13°.
How far is the house from base of the tower?
Draw the diagram

What is asked?
problem?
Solution
4. When the kite is 120 ft high, it makes an angle of 40° with the level ground. How long
is the string?
Draw the diagram

What is asked?
problem?
Solution

Telephone No.: (045) 982-0374
REGION III
5. From the top of a cliff 350 ft high, the angle of depression of a boat on the sea is 15°.
How far is the boat from the base of the cliff?
Draw the diagram

What is asked?
problem?
Solution
L E S S O N 4
USES TRIGONOMETRIC RATIOS TO SOLVE REAL-LIFE PROBLEMS INVOLVING RIGHT
TRIANGLES (Week 3-5)
Activity 1: Problem Solved!
Read each of the following problems carefully and solve.
1. Opposite corners of a rectangular garden are joined by a

wire 72 m long and it forms an angle of 50° with the
length. Find the dimensions of the garden.
10

Telephone No.: (045) 982-0374
REGION III
2. A flagpole stands in the middle of a flat, level

field. Fifty feet away from its base, a surveyor
measures the angle to the top of the flagpole as
48°. How tall is the flagpole?
3. A statue is put on the top of a 1.8 m

high stone platform. The angle of
elevation of the top of the statue
from point 𝐴 on the ground is 28°. If
point 𝐴 is 8 m away from the statue,
find the height of the statue.
4. A ladder 5.7 m long rests against a vertical wall at an

angle of 34° with the vertical wall. How far is the foot of
the ladder from the wall?
5. A piece of land triangular in shape measures 140 m on

two sides which intersect at an angle of 80°. How long
is the frontage of the land in nearest meter?
Activity 2: Challenge Accepted!

Answer the following problems. If the answers are not exact, round off to the nearest
tenth.
1. Two men are on the opposite sides of a pole. From their positions, they measure the
angles of elevation of the top of the pole as 25° and 20° respectively. What is the distance
between the two men if the height of the pole is 22 m?
11

Telephone No.: (045) 982-0374
REGION III
2. John’s house is 16 m from a television tower. From his window, he figured that the angle
of depression to the base of the tower is 24°, and the angle of elevation to the top of the
tower is 42°. Find the height of the tower.
3. From a height 42 m above sea level, two ships are sighted due east. The angles of
depression are 52° and 27°. How far apart are the ships?
4. A Jose Rizal statue stands on a 2.8 m high podium. At a point 10 m away from the base
of the podium, the angle of elevation to the top of the statue is 43°. How high is the
statue?
5. A ray of sunlight casts a shadow of a flagpole on the ground at an angle of depression of
58°. If the length of the shadow is 3 m shorter than the height of the flagpole, find the
height of the flagpole.
Activity 3: Let’s Make It Right!

Read each problem carefully and analyze what is wrong with the suggested solution.
Then, write the correct solution in the space provided.
Problem 1: An airplane cruising at an altitude of 9 km wants to land at NAIA begins to descent at

an angle of depression of 7 °. What is the horizontal distance between the plane and the airport
at that time?
Suggested Solution My Solution

Let A represent the airplane and B the airport.
Let 𝑥 be the horizontal distance.
𝑥
tan 7° =
9
𝑥 = 9 (tan 7°)
𝑥 ≈ 1.11 km
12

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REGION III
The horizontal distance between the plane and

the airport is approximately 1.11 km
Problem 2: An observer stands 12 m from a tree and finds that the line of sight to the top of the
tree is 32° above the horizontal. Find the height of the tree above eye level.
Suggested Solution My Solution
Let 𝑥 be the height of the tree above eye level.
𝑥
sin 32° =
12
𝑥 = 12 (sin 32°)
𝑥 ≈ 6.36 m
The height of the tree above eye level is

approximately 6.36 m.
L E S S O N 5
ILLUSTRATES LAWS OF SINES AND COSINES (Week 6-9)
Activity 1: Seek the truth!

Write T if the statement is true and F if the statement is false.
_______1. An oblique triangle is always acute.
_______2. The Law of Sines may be applied to solve the missing parts of a triangle given two of
its sides and the angle between them.
_______3. The Law of Cosines may be applied to determine the measurements of the angles of
a triangle given its sides.
13

Telephone No.: (045) 982-0374
REGION III
𝑎 𝑏 𝑐
_______4. The Law of Sines is represented by the equation = = .
sin 𝐴 sin 𝐵 sin 𝐶
_______5. The sine value of an acute angle is less than 1.
Activity 2: Law of Sines or Law of Cosines?
Determine what law is applicable to use in solving the triangle given the following
information.
_____________ 1. 𝐴 = 50° 𝐵 = 65° 𝑎 = 10

_____________ 2. 𝐴 = 50° 𝐵 = 60° 𝑐 = 10
_____________ 3. 𝐵 = 60° 𝑎 = 12 𝑐 = 15
_____________ 4. 𝐶 = 60° 𝐴 = 12° 𝑐 = 15
_____________ 5. 𝐵 = 30° 𝑏 = 15 𝑐 = 20
_____________ 6. 𝐴 = 30° 𝑏 = 20 𝑎 = 10
_____________ 7. 𝑎 = 4 𝑏=9 𝑐=6
_____________ 8. 𝑎 = 10 𝑐=7 𝐵 = 24°
_____________ 9. 𝑎 = 39 𝑏 = 52 𝐶 = 122°
_____________ 10. 𝑐 = 20 𝑏 = 30 𝐶 = 25°
Activity 3: Looking for 𝒙!
Use law of sine or cosine to find the value of 𝑥 in each of the following triangles.
14

Telephone No.: (045) 982-0374
REGION III
Activity 4: Draw me!

Draw your own triangles based on the given information then solve for the other parts.
Round off all decimals to the nearest tenth.
1. ∠𝐴 = 73°, 𝑎 = 18, 𝑏 = 11 2. 𝑎 = 20, 𝑏 = 16, 𝑐 = 23
∠𝐵 = ____ ∠𝐶 = _____ 𝑐 = _____ ∠𝐴 = ____ ∠𝐵 = _____ ∠𝐶 = _____
15

Telephone No.: (045) 982-0374
REGION III
L E S S O N 6
SOLVING PROBLEMS INVOLVING OBLIQUE TRIANGLES (Week 6-9)
Activity 1: I Can Do This!

Solve the following problems. Round your answers to the nearest tenth.
1. An isosceles triangle has sides that measures 36 cm, 36 cm, and 27 cm. Find the measure
of each angle.
2. In a town, the church (𝐶) is located 3 km from a public school (𝐴) and 2 km from a
private school (𝐵). If the line connecting 𝐴 to 𝐶 makes an angle of 40° with the line
connecting 𝐴 to 𝐵, how far is 𝐴 from 𝐵?
3. The measure of two of the angles of a triangle are 50° and 55°. If its longest side
measures 17 cm. Find the perimeter of the triangle.
4. To support a tent, two pieces of 2.5 m long are tied to the top of a post forming an
angle of 110° between them. What is the distance between the pegs to which the
other ends of each rope are tied?
5. Bryan’s (𝐵) and Carl’s (𝐶) houses are along the riverbanks. On the opposite side is Angel’s
(𝐴) house, which is 275 m away from Carl’s. The angles ∠𝐶𝐴𝐵 𝑎𝑛𝑑 ∠𝐴𝐶𝐵 are measured
and they are found to be 125° and 49° respectively. Find the distance between Angel’s
and Bryan’s houses.
Activity 2: More on Problem Solving…

Read each problem carefully, then solve. Round your answers to the nearest tenth.
1. To find the distance between two points 𝐴 and 𝐵 that lie on opposite banks of a river, a
surveyor lays off a line segment 𝐴𝐶 of the length 250 m along one bank and determines
that the measure of ∠𝐵𝐴𝐶 and ∠𝐴𝐶𝐵 are 64° and 43°, respectively. Find the distance
from 𝐴 to 𝐵.
2. The Leaning Tower of Pisa leans at an angle of about 84.7°. A point 52.12 m away from
the base of the tower, the angle of elevation to the top is 50°. Find the distance from the
base to the top of the tower.
3. A developer has a triangular lot at the intersection of two streets. The streets meet at an
angle of 72°, and the lot has 300 ft frontage along one street and 416 ft frontage along
the other street. Find the length of the third side of the lot.
16

Telephone No.: (045) 982-0374
REGION III
Activity 3: What is your Problem?

Create a problem based on the illustration shown below. Then, solve for what is asked.
APPROVAL SHEET
CONTENT WRITERS:
CESAR B. AGUAYO SERWIN A. TOLENADA LUCITA E. REYES
MA. KRISTINE T. TIGLAO LYDIA L. BUSTOS RENELYN A. LLAPITAN
Reviewed by:
AUGUSTO L. BALLESTEROS DR. BOBBY P. CAOAGDAN

EPSvr - Math Secondary EPSvr – LRMDS
Recommended by:
DR. PAULINO D. DE PANO DR. MARIA CELINA L. VEGA

Chief-Education Supervisor – CID Asst. Schools Division Superintendent
APPROVED:
RONALDO A. POZON, Ph. D., CESO V

Schools Division Superintendent
17

Telephone No.: (045) 982-0374

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Republic of the Philippines

Address: Macabulos Drive, San Roque, Tarlac City

NO. MELC DURATION K to 12 CG PAGE

Address: Macabulos Drive, San Roque, Tarlac City

Activity 1: Word Hunt!

Address: Macabulos Drive, San Roque, Tarlac City

Activity 2: Match Me.

Activity 3: Find Me!

1. sin 𝑀 = _________ 6. cot 𝑇 =_________

Address: Macabulos Drive, San Roque, Tarlac City

Activity 1: Quantitative Comparison

Activity 2: What Makes You Special?

Address: Macabulos Drive, San Roque, Tarlac City

Activity 3: Another Solving!

3. 𝑠 =________ 4. 𝑝 =________ 5. 𝑥 = ________

Address: Macabulos Drive, San Roque, Tarlac City

Activity 1: Fill Me.

Activity 2: Classify Me.

Address: Macabulos Drive, San Roque, Tarlac City

Activity 3: Illustrate and Solve!

Draw the diagram

What is/are the given?

Draw the diagram

What is/are the given?

Address: Macabulos Drive, San Roque, Tarlac City

Draw the diagram

What is/are the given?

Draw the diagram

What is/are the given?

Address: Macabulos Drive, San Roque, Tarlac City

Draw the diagram

What is/are the given?

1. Opposite corners of a rectangular garden are joined by a

Address: Macabulos Drive, San Roque, Tarlac City

2. A flagpole stands in the middle of a flat, level

3. A statue is put on the top of a 1.8 m

4. A ladder 5.7 m long rests against a vertical wall at an

5. A piece of land triangular in shape measures 140 m on

Activity 2: Challenge Accepted!

Address: Macabulos Drive, San Roque, Tarlac City

Activity 3: Let’s Make It Right!

Problem 1: An airplane cruising at an altitude of 9 km wants to land at NAIA begins to descent at

Suggested Solution My Solution

Address: Macabulos Drive, San Roque, Tarlac City

The horizontal distance between the plane and

The height of the tree above eye level is

Activity 1: Seek the truth!

_______1. An oblique triangle is always acute.

Address: Macabulos Drive, San Roque, Tarlac City

_______5. The sine value of an acute angle is less than 1.

Activity 2: Law of Sines or Law of Cosines?

_____________ 1. 𝐴 = 50° 𝐵 = 65° 𝑎 = 10

Activity 3: Looking for 𝒙!

Address: Macabulos Drive, San Roque, Tarlac City

Activity 4: Draw me!

1. ∠𝐴 = 73°, 𝑎 = 18, 𝑏 = 11 2. 𝑎 = 20, 𝑏 = 16, 𝑐 = 23

∠𝐵 = ____ ∠𝐶 = _____ 𝑐 = _____ ∠𝐴 = ____ ∠𝐵 = _____ ∠𝐶 = _____

Address: Macabulos Drive, San Roque, Tarlac City

1. sin 𝑀 = _ 6. cot 𝑇 =_

3. 𝑠 = 4. 𝑝 = 5. 𝑥 =

∠𝐵 = __ ∠𝐶 = _ 𝑐 = _ ∠𝐴 = ∠𝐵 = _ ∠𝐶 = ___