Trigonometry is the study of relationships between angles and sides of triangles. The six trigonometric functions - sine, cosine, tangent, cosecant, secant, and cotangent - relate the angles and sides of a right triangle. These functions can be determined using the ratios of opposite, adjacent, and hypotenuse sides relative to a given angle. Trigonometric functions are commonly used in fields like navigation, surveying, and astronomy to solve problems involving triangles.

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Directions: Apply the Pythagorean theorem. Tell
whether each triangle satisfies the conditions of a
right triangle.

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1.The learner illustrates the six trigonometric
ratios: sine, cosine, tangent, secant, cosecant,
and cotangent. M9GE-IVa-1
2.finds the trigonometric ratios of special
angles. M9GE -IVb-c-1

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Trigonometry is thought to have
had its origin in ancient Egypt and
Mesopotamia. The ancient Egyptians,
Babylonians and Greeks developed
trigonometry to find the lengths of the
sides of triangles and measures of
their angles.
It was HIPPARCHUS, a Greek
mathematician, who introduced
trigonometry as gleaned from ancient
tablets and tables which reflected
work on the ratios of trigonometry.

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Trigonometry is derived from
the Greek words TRIGONON means
triangle and METRON means
measure.
Thus, trigonometry means
measurement of triangles. It was used
in ancient times in surveying,
navigation, and astronomy to find
relationships between the lengths of
the sides of a triangle and
measurement of angles.

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A trigonometric ratio is a ratio of two sides of a right
triangle. Trigonometric ratios are relations existing
between the sides and angles of a right triangle that are
expressed in the form of ratios.
The six trigonometric ratios are sine (sin), cosine
(cos), tangent (tan), secant (sec), cosecant (csc), and
cotangent (cot).

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• OPPOSITE refers to the side of the
triangle that is opposite of the
reference angle.
• ADJACENT refers to the side of the
triangle that is adjacent to the
reference angle (the adjacent side will
always form one side of the reference
angle).
• HYPOTENUSE is the side of the
triangle that is always opposite the
right angle.
reference angle.
The acute angle
of the right
triangle.

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In the right tringle ACB, the opposite
of the right angle C, is the Hypotenuse and
the side opposite the two acute angles are
the Legs.
• The Hypotenuse can be denoted by c,
corresponding lowercase of C,
• the side opposite angle A can be denoted
by a, and
• the side opposite angle B can be denoted
by b,
side a is said to be adjacent to angle B,
and side b is adjacent to angle A.

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Using the sides of the right triangle the definition of
trigonometric ratios are the following:
NAME ABBREV. RATIO
Sine Sin Sin𝑥=
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Cosine Cos Cosx=
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Tangent Tan Tanx=
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
Cosecant Csc Cscx=
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
or
1
Sin𝜃
Secant Sec Secx=
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
or
1
Cos𝜃
Cotangent Cot Cotx=
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
or
1
tan𝜃

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The trig functions can be summarized
using the following mnemonic device:
SOHCAHTOA
Sin =
opposite
hypotenuse
Cos =
adjacent
hypotenuse
Tan =
opposite
adjacent

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CHO SHA CAO
Csc =
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆
Sec =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
Cot=
𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕
𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆
The trig functions can be summarized
using the following mnemonic device:

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Find the values of the six trigonometric
functions for each of the following indicated
angle B.
The opposite of angle B = 6, the adjacent = 8,
hypotenuse?
Apply the Pythagorean theorem to get the value
of the hypotenuse.
c2 = a2 + b2 ; a2 = c2 - b2 ; b2 = c2 - a2
c2 = a2 + b2
c2 = 82 + 62
c2 = 64 + 36
c2 = 100
c = 10
10

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Opposite = 6, adjacent = 8, hypotenuse = 10
SinB =
CosB
TanB
Find the values of the six trigonometric
functions for each of the following indicated angle B.
10
𝑜𝑝𝑝
ℎ𝑦𝑝
=
6
10
=
𝟑
𝟓
=
𝑎𝑑𝑗
ℎ𝑦𝑝
=
8
10
=
𝟒
𝟓
=
𝑜𝑝𝑝
𝑎𝑑𝑗
=
6
8
=
𝟑
𝟒
CscB
SecB
CotB
=
ℎ𝑦𝑝
𝑜𝑝𝑝
=
10
6
=
𝟓
𝟑
=
ℎ𝑦𝑝
𝑎𝑑𝑗
=
10
8
=
𝟓
𝟒
=
𝑎𝑑𝑗
𝑜𝑝𝑝
=
8
6
=
𝟒
𝟑

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In a right triangle, the legs measure 5cm and 12cm.
Find the values of six trigonometric functions of the two
acute angles of the right triangle.
5cm
12cm
𝜶
𝜷
Apply the Pythagorean theorem to
get the value of the hypotenuse.
c2 = a2 + b2
c2 = 52 + 122
c2 = 25 + 144
c2 = 169
c = 13
13cm

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THE TRIGONOMETRIC OF 𝜶
Opposite = 5, adjacent = 12, hypotenuse = 13
Sin𝛼 =
Cos𝛼
Tan𝛼
𝑜𝑝𝑝
ℎ𝑦𝑝
=
𝟓
𝟏𝟑
=
𝑎𝑑𝑗
ℎ𝑦𝑝
=
𝟏𝟐
𝟏𝟑
=
𝑜𝑝𝑝
𝑎𝑑𝑗
=
𝟓
𝟏𝟐
Csc𝛼
Sec𝛼
Cot𝛼
=
ℎ𝑦𝑝
𝑜𝑝𝑝
=
𝟏𝟑
𝟓
=
ℎ𝑦𝑝
𝑎𝑑𝑗
=
𝟏𝟑
𝟏𝟐
=
𝑎𝑑𝑗
𝑜𝑝𝑝
=
𝟏𝟐
𝟓
5cm
12cm
𝜶
𝜷 13cm
In a right triangle, the legs measure 5cm and 12cm.
Find the values of six trigonometric functions of the two
acute angles of the right triangle.

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The Trigonometric Of 𝜷
Opposite = 12, Adjacent = 5, Hypotenuse = 13
Sin𝛽 =
Cos𝛽
Tan𝛽
𝑜𝑝𝑝
ℎ𝑦𝑝
=
𝟏𝟐
𝟏𝟑
=
𝑎𝑑𝑗
ℎ𝑦𝑝
=
𝟓
𝟏𝟑
=
𝑜𝑝𝑝
𝑎𝑑𝑗
=
𝟏𝟐
𝟓
Csc𝛽
Sec𝛽
Cot𝛽
=
ℎ𝑦𝑝
𝑜𝑝𝑝
=
𝟏𝟑
𝟏𝟐
=
ℎ𝑦𝑝
𝑎𝑑𝑗
=
𝟏𝟑
𝟓
=
𝑎𝑑𝑗
𝑜𝑝𝑝
=
𝟓
𝟏𝟐
5cm
12cm
𝜶
𝜷 13cm
In a right triangle, the legs measure 5cm and 12cm.
Find the values of six trigonometric functions of the two
acute angles of the right triangle.

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A leg and the hypotenuse of a right triangle
measure 2 cm and 5cm, respectively. Find the six
trigonometric functions of angle 𝜽.
5cm
2cm
𝒂
𝜽
First, compute for the length of the
unknown leg, a, then apply the
Pythagorean theorem to get the
value of the hypotenuse.
a2 = c2 - b2
a2 = ( 5cm)2 - (2cm)2
a2 = 5cm2 - 4cm2
a2 = 1cm2
a2 = 1cm2
a = 1cm
1cm

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The Trigonometric Of 𝜽
Opposite = 2, Adjacent = 1, Hypotenuse = 5
Sin𝜽=
Cos𝜽
Tan𝜽
𝑜𝑝𝑝
ℎ𝑦𝑝
=
2
5
=
𝑎𝑑𝑗
ℎ𝑦𝑝
=
1
5
=
𝑜𝑝𝑝
𝑎𝑑𝑗
=
2
1
Csc𝜽
Sec𝜽
Cot𝜽
=
ℎ𝑦𝑝
𝑜𝑝𝑝 =
𝟓
𝟐
=
ℎ𝑦𝑝
𝑎𝑑𝑗
=
𝟓
𝟏
=
𝑎𝑑𝑗
𝑜𝑝𝑝
=
𝟏
𝟐
A leg and the hypotenuse of a right triangle measure
2 cm and 5cm, respectively. Find the six trigonometric
functions of angle 𝜽.
5cm
2cm
𝒂
𝜽
1cm
=
2
5
*
5
5
=
𝟐 𝟓
𝟓
=
2
5
*
5
5
=
𝟓
𝟓
=2

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Sec𝜽 =
ℎ𝑦𝑝
𝑎𝑑𝑗
=
𝟏𝟕
𝟏𝟓
Sec𝜽 =
ℎ𝑦𝑝
𝑎𝑑𝑗
=
𝟏𝟑
𝟏𝟐
Cot𝜽 =
𝑎𝑑𝑗
𝑜𝑝𝑝
=
𝟒
𝟑
Csc𝜽 =
ℎ𝑦𝑝
𝑜𝑝𝑝
=
𝟏𝟕
𝟏𝟓

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1. Find Cscθ, Secθ, Cotθ if TanΘ =
3
4
TOA = Tangent =
𝑜𝑝𝑝𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
=
3
4
Opposite = 3, Adjacent = 4,
C2 = 32 + 42
C2 = 9 + 16
C2 = 25
C = 5
Hypotenuse = 5
Find the value of the trig function
indicated.
Csc𝜽
Sec𝜽
Cot𝜽
=
ℎ𝑦𝑝
𝑜𝑝𝑝
=
𝟓
𝟑
=
ℎ𝑦𝑝
𝑎𝑑𝑗
=
𝟓
𝟒
=
𝑎𝑑𝑗
𝑜𝑝𝑝
=
𝟒
𝟑

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1. Find tanθ, cosθ if sin θ =
4
5
2. Find cotθ and cscθ if sin θ =
12
13
Find the value of the trig function
indicated.

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1.The learner illustrates the six trigonometric
ratios: sine, cosine, tangent, secant, cosecant,
and cotangent. M9GE-IVa-1
2.finds the trigonometric ratios of special
angles. M9GE -IVb-c-1

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28. Basic Education Department
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ISOCELES RIGHT TRIANGLE THEOREM
In an isosceles right triangle, the length of the hypotenuse is equal to 2
times the length of a leg.
An isosceles triangle is defined as a
triangle that has two sides of equal measure and
also known as 45°-45°-90° triangle.
45°
45°
h

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Find the six trigonometric ratios of the given angle.
𝑜𝑝𝑝
ℎ𝑦𝑝
=
𝒂
𝒂 𝟐
=
𝑎𝑑𝑗
ℎ𝑦𝑝
=
𝑜𝑝𝑝
𝑎𝑑𝑗
=
𝒂
𝒂
Csc45°
Sec45°
Cot45°
=
ℎ𝑦𝑝
𝑜𝑝𝑝
=
ℎ𝑦𝑝
𝑎𝑑𝑗
=
𝒂 𝟐
𝒂
=
𝑎𝑑𝑗
𝑜𝑝𝑝
Sin45° =
Cos45°
Tan45°
=
𝟐
𝟐
=
𝟐
𝟏
= 𝟐
=𝟏
=
𝟏
𝟐
=
𝟏
𝟐
∗
𝟐
𝟐
=
𝒂
𝒂 𝟐 =
𝟐
𝟐
=
𝟏
𝟐
=
𝟏
𝟐
∗
𝟐
𝟐
=
𝒂 𝟐
𝒂
=
𝟐
𝟏
= 𝟐
=
𝒂
𝒂
=𝟏

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30°-60°-90° TRIANGLE THEOREM
In 30°-60°-90° triangle, the length of the hypotenuse is twice the length of
the shorter leg and the length of the longer leg is 3 times the length of the
shorter leg.
2a
a
𝑎 3

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𝑜𝑝𝑝
ℎ𝑦𝑝
=
𝒂
𝟐𝒂
=
𝑎𝑑𝑗
ℎ𝑦𝑝
=
𝒂 𝟑
𝟐𝒂
=
𝑜𝑝𝑝
𝑎𝑑𝑗
=
𝒂
𝒂 𝟑
Csc30°
Sec30°
Cot30°
=
ℎ𝑦𝑝
𝑜𝑝𝑝
=
𝟐𝒂
𝒂
=
ℎ𝑦𝑝
𝑎𝑑𝑗
=
𝟐𝒂
𝒂 𝟑
=
𝑎𝑑𝑗
𝑜𝑝𝑝
=
𝒂 𝟑
𝒂
Sin30° =
Cos30°
Tan30°
=
𝟏
𝟐
=
𝟑
𝟐
=
𝟏
𝟑
=
𝟏
𝟑
∗
𝟑
𝟑
=
𝟑
𝟑
=
𝟐
𝟏
=2
=
𝟐
𝟑
=
𝟐
𝟑
∗
𝟑
𝟑
=
𝟐 𝟑
𝟑
=
𝟑
𝟏
= 𝟑

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𝑜𝑝𝑝
ℎ𝑦𝑝 =
𝒂 𝟑
𝟐𝒂
=
𝑎𝑑𝑗
ℎ𝑦𝑝
=
𝒂
𝟐𝒂
=
𝑜𝑝𝑝
𝑎𝑑𝑗 =
𝒂 𝟑
𝒂
Csc60°
Sec60°
Cot60°
=
ℎ𝑦𝑝
𝑜𝑝𝑝
=
ℎ𝑦𝑝
𝑎𝑑𝑗
=
𝟐𝒂
𝒂 𝟑
=
𝑎𝑑𝑗
𝑜𝑝𝑝
Sin60° =
Cos60°
Tan60°
=
𝟑
𝟐
=
𝟏
𝟐
=
𝟑
𝟏
=
𝟐
𝟑
=
𝟐
𝟑
∗
𝟑
𝟑
=
𝟐 𝟑
𝟑
= 𝟑
=
𝟐𝒂
𝒂
=
𝟐
𝟏
=2
=
𝒂
𝒂 𝟑
=
𝟏
𝟑
=
𝟏
𝟑
∗
𝟑
𝟑
=
𝟑
𝟑

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𝟏
𝟐
𝟑
𝟐
𝟑
𝟑
𝟐
𝟐 𝟑
𝟑
𝟑
𝟐
𝟐
𝟐
𝟐
𝟏 𝟐 𝟏
𝟐
𝟑
𝟐
𝟏
𝟐
𝟑
𝟐 𝟑
𝟑
𝟐 𝟑
𝟑

34. Basic Education Department
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1. Determine the exact value of the
expression sin 30° + cos 60°.
Solution:
sin 30° + cos 60°
=
1
2
+
1
2
= 1
2. Determine the exact value of the
expression 𝑐𝑜𝑠245° + 𝑠𝑖𝑛245° .
Solution:
𝑐𝑜𝑠245° + 𝑠𝑖𝑛245°
= (
𝟐
𝟐
)2
+ (
𝟐
𝟐
)2
=
4
4
=
2
4
+
2
4
= 1
Using the table of values of trigonometric ratios of special
angles, let us consider the following examples:

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1. Find the exact value of
𝑐𝑠𝑐2
30° + 𝑐𝑜𝑠2
45°.
Solution:
𝑐𝑠𝑐2 30° + 𝑐𝑜𝑠2 45°
=22
+(
𝟐
𝟐
)2
= 4 +
2
4
2. Determine the exact value of the
expression
𝑠𝑖𝑛30° 𝑡𝑎𝑛30°
𝑠𝑒𝑐30° −𝑐𝑜𝑠30°
.
Solution:
=
𝑠𝑖𝑛30° 𝑡𝑎𝑛30°
𝑠𝑒𝑐30° −𝑐𝑜𝑠30°
=
𝟑
𝟔
𝟒 𝟑 − 𝟑 𝟑
𝟔
=
1
2
(
𝟑
𝟑
)
𝟐 𝟑
𝟑
−
𝟑
𝟐
=
𝟑
𝟔
*
𝟔
𝟑
Using the table of values of trigonometric ratios of special
angles, let us consider the following examples:
=
9
2
or 4.5 = 1

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38. Basic Education Department
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The basic unit of angle measure is DEGREE,
consider a right angle is 90° angle, and
if this angle is divided equally to 90 parts,
then one part is 1°
Thus 1° = 60’ or 60 minutes
1’ = 60” or 60 seconds
12°25’ means 12 degrees and 25 minutes.
19°6’1’’ means 19 degrees, 6 minutes and 1 second
5°2’’ means 5 degrees, 0 minute and 2 seconds.

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Use the Table of trigonometric Ratios to
find the following: refer to page 414-428
a. sin12°45′
= 0.22070
b. tan35°20′
= 0.70891
c. cot89°40′
= 0.00582
a. cos9°30′
= 6.0589
b. cot14°25′
= 3.8900
c. sin17°55′
= 0.30763

40. Basic Education Department
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Use a scientific a calculator to
find the value of the following
in 5-decimal palces:
a. sin10°10′10’’
Solution:
= 0.17656 3RD PRESS THE DEGREE
SYMBOL
4TH THEN PRESS THE
NUMBER EX. 10
6TH THEN PRESS THE
NUMBER EX. 10

41. Basic Education Department
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Use a scientific a calculator to
find the value of the following
in 5-decimal palces:
b. Cos 200°
2ND THEN PRESS THE
NUMBER EX. 200
3RD PRESS THE DEGREE
SYMBOL
= -0.93970

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0.98481 𝟑
𝟐
𝟏
𝟐
𝟎. 𝟖𝟑𝟗𝟏𝟎

43. Basic Education Department
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Using the sides of the right triangle the definition of
trigonometric ratios are the following:
NAME ABBREV. RATIO
Sine Sin Sin𝑥=
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Cosine Cos Cosx=
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Tangent Tan Tanx=
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
Cosecant Csc Cscx=
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
or
1
Sin𝜃
Secant Sec Secx=
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
or
1
Cos𝜃
Cotangent Cot Cotx=
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
or
1
tan𝜃

44. Basic Education Department
Junior High School
TRIGONOMETRIC RATIOS USING THE
CALCULATOR!
COSECANT reciprocal of SINE
first press the FRACTION SYMBOL
then, for the numerator PRESS 1,
for the denominator, press sin symbol.
SECANT reciprocal of COSINE
COTANGENT reciprocal of TANGENT
1ST PRESS THE fraction
symbol
Press 1 for the
numerator
The sin symbol

45. Basic Education Department
Junior High School
2.79042
1.16663

46. Basic Education Department
Junior High School
Use a scientific a calculator to
find the value of the following
in 5-decimal places:
a. Cot7’’
Solution:
= 29466.40088
Then PRESS number 1
3rd, press the
Tan
4th, PRESS number 0
5th PRESS degree symbol
6th PRESS 0
7th PRESS degree symbol
8th PRESS 7
9th PRESS degree symbol

47. Basic Education Department
Junior High School
Find the Approximate principal value of
angle 𝜃
a. sin𝜃=
𝟑
𝟐
= 𝜃 = 60°
b. Tan 𝜃 =
𝟑
𝟑
𝜃 = 30°
c. Tan 𝜃 = 0.11246°
refer to the book on page 414
𝜃 = 6°25’

48. Basic Education Department
Junior High School