SENIOR HIGH SCHOOL

Pre-Calculus
Quarter 2 - Module 5
Trigonometric Identities

i
About the Module
This module was designed and written with you in mind. It is here to help you master
about Trigonometric Identities. The scope of this module permits it to be used in
many different learning situations. The language used recognizes the diverse
vocabulary level of students. The lessons are arranged based on the Most Essential
Learning Competencies (MELCs) released by the Department of Education (DepEd)
for this school year 2020 – 2021.

This module has two lessons:

Lesson 1 – Identity and Conditional Equations
Lesson 2 – Trigonometric Identities

After going through this module, you are expected to:

• determine whether an equation is an identity or a conditional equation;

and
• apply trigonometric identities to find other trigonometric identities.

ii
 What I Know (Pretest)

Instruction: Choose the letter of the correct answers to the following items. Write them
on a separate sheet of paper.

1. Which of the following equation is an identity?

𝑥 2 −4
A. 𝑥 2 − 1 = 0 C. 𝑥−2
=𝑥+2
B. 𝑥 − 1 = 𝑥 − 1
2
D. (𝑥 + 1)2 = 𝑥 2 + 𝑥 + 1

2. Which of the following equation is conditional?

𝑥 2 −1
A. (𝑥 + 7)2 = 𝑥 2 + 14𝑥 + 49 C. =𝑥−2
𝑥 2 +1
B. 3𝑥 + 1 = 2 D. 𝑥 − 4 = (𝑥 − 2)2
2

3. Which of the following is NOT an identity?

A. 𝑠𝑖𝑛2 𝛼+𝑐𝑜𝑠 2 𝛼 = 1 C. 1 + 𝑐𝑜𝑡 2 𝛼 = 𝑐𝑠𝑐 2 𝛼
B. 1 − 𝑠𝑒𝑐 2 𝛼 = 𝑡𝑎𝑛2 𝛼 D. 𝑠𝑖𝑛 𝛼 = 𝑡𝑎𝑛 𝛼 𝑐𝑜𝑠 𝛼

4. Which of the following is an identity?

sin 2𝛼
A. 𝑠𝑖𝑛(−𝛼) = 𝑠𝑖𝑛 𝛼 C. 𝑠𝑖𝑛 𝛼 𝑐𝑜𝑠 𝛼 = 2
𝑐𝑜𝑠 𝛼
B. 𝑠𝑖𝑛 𝛼 + 𝑐𝑜𝑠 𝛼 = 1 D. 𝑡𝑎𝑛 𝛼 =
𝑠𝑖𝑛 𝛼
5. What is the original trigonometric expression of the simplified expression
𝑠𝑖𝑛 𝑥?
cos 𝑥 cos 𝑥 sin 𝑥 sec 𝑥
A. cot 𝑥
B. sin 𝑥
C.cot 𝑥 D. cot 𝑥

6. What are the identities derived from the sum and difference identities?
A. Double-Angle C. Pythagorean
B. Half-Angle D. Reciprocal

7. Which of the following defines a conditional equation?

A. It satisfied by at least one real number.
B. It is true for all values of the variable in the domain of the equation.
C. It is true for all real numbers and whose both sides are defined and
identical.
D. An equation where values of the variable in the domain of the equation
satisfies the equation.

8. Which of the following DOES NOT constitute the Fundamental Trigonometric

Identity?
A. Double-Angle Identity C. Pythagorean Identity
B. Even-Odd Identity D. Quotient Identity

9. Which of the following is an equivalent of 𝑠𝑒𝑐 𝑥 − 𝑐𝑜𝑠 𝑥?

A. 𝑠𝑖𝑛 𝑥 𝑡𝑎𝑛 𝑥 B. 𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 𝑥 C. 𝑠𝑒𝑐 𝑥 𝑐𝑜𝑡 𝑥 D. 𝑐𝑜𝑡 𝑥 𝑡𝑎𝑛 𝑥

1
10. How do you express 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 𝜃 in terms of 𝑐𝑜𝑠 𝜃?
A. ±√1 + 𝑐𝑜𝑠 2 𝜃 B. ±√1 − 𝑐𝑜𝑠 2 𝜃 C. 𝑐𝑜𝑠 𝜃 + 1 D. 𝑐𝑜𝑠 𝜃 − 1

11. What is the exact value of 𝑠𝑖𝑛 105°?

√2+√6 √2−√6 √6 √2
A. B. C. D.
4 4 4 4

24 𝜋
12. Given that sin 𝑡 = 25 and 2
< 𝑡 < 𝜋, what is the exact value 𝑠𝑖𝑛 2𝑡?
225 336
A. − B. − C. 336 D. 225
224 225

24 𝜋
13. Given that sin 𝑡 = 25 and 2
< 𝑡 < 𝜋, what is the exact value 𝑐𝑜𝑠 2𝑡?
565 24 527 7
A. − B. − C. − D.
627 25 625 25

𝜋
14. What is the exact value of 𝑡𝑎𝑛 8
?
A. 2 + √3 B. √2 − 1 C. √3 +1 D.2
𝜃 4
15. What is the exact value of 𝑡𝑎𝑛 ( ) if 𝑡𝑎𝑛 𝜃 = and 𝑐𝑜𝑠 𝜃 < 0 ?
2 3
A. -2 B. 2 C. 0.2 D. √2

2
 Lesson Identity and Conditional Equations
1

What I Need to Know

At the end of this lesson, you are expected to:
o determine whether an equation is an identity or a conditional equation.

What’s In

An equation is a statement with two equal expressions.

To solve an equation means to find all numbers that make Examples

the equation true. These numbers are called solutions or of an equation:
roots of the equation. 𝑥+2=9

A number that is a solution of an equation is said to satisfy 11𝑥 = 5𝑥 + 6𝑥

the equation, and the solutions of an equation make up its
𝑥 2 − 2𝑥 − 1 = 0
solution set. Equations with the same solution set are
equivalent equations.

What’s New
Activity 1.1: Find the Impostor!
Instruction: The following crews have expressions equivalent to each other. Your task
is to find the impostor who has an expression not equivalent to the rest of the crews.
After the impostor is identified, write its corresponding name in the box provided
below to reveal a phrase that describes you. Write the phrase you found on your
answer sheet. Good luck!

1 2 3
Cute Not me You Self Me Yup Are
Good

1 2
3(𝑥 + 2) 9𝑥 + 18 2
6(𝑥 + 2𝑥 + 1) 6𝑥 + 12
3𝑥 + 6 2𝑥 + 3 6(𝑥 + 1)2
3 6𝑥 2 + 12𝑥 + 6

Loved Strong Beautiful

Amazing

3
20𝑥 − 10𝑦 −5(−4𝑥 + 2𝑦)
−5(−4𝑥 − 2𝑦) 5(4𝑥 − 2𝑦)

3
 What Is It

Identity and Conditional Equations

Equations in mathematics can be classified as conditional equations or

identity equations. According to Albay (2016), a conditional equation is an equation
that is satisfied by at least one real number. On the other hand, an equation that is
true for all real numbers and whose both sides are defined and identical is called an
identity equation.

Example 1.1: Identify if each of the following equations is a conditional or an identity

equation.
a. 3𝑥 − 4 = 8 You can simplify first each side of the
1 1 1
b. + = equation by applying necessary
3 𝑝 3𝑝 algebraic manipulations. When the
c. 𝑥 2 + 2𝑥 + 1 = (𝑥 + 1)2 equation is in simplest form, you can
tan 𝑥
d. sin 𝑥 = now determine whether it is an identity
sec 𝑥
or a conditional equation.
Solution:
a. The equation 3𝑥 − 4 = 8 is a conditional equation because this is only
satisfied if 𝑥 = 4. To show the solution, we have the following:

3𝑥 − 4 = 8
3𝑥 − 4 + 𝟒 = 8 + 𝟒 Add both sides with positive 4
3𝑥 = 12
3𝑥 12
= Divide both sides with 3
3 3
𝑥=4
1 1 1
b. The equation + = is true only for 𝑝 = −2. Therefore, the equation is
3 𝑝 3𝑝
a conditional equation. To show the solution, we have the following:
1 1 1
+ =
3 𝑝 3𝑝
1 1 1 Multiply both sides with 3p
3𝑝 ( + = ) 3𝑝
3 𝑝 3𝑝
(3𝑝) (3𝑝) 3𝑝
+ =
3 𝑝 3𝑝
𝑝+3=1
𝑝+3−𝟑=1−𝟑 Subtract both sides with 3
𝑝 = −2
c. Note that the left side can be expressed in factored form as (𝑥 + 1)2 .
Similarly, the right side can be expressed as 𝑥 2 + 2𝑥 + 1. Hence, we can
write the equation as follows:

(𝑥 + 1)2 = (𝑥 + 1)2 or 𝑥 2 + 2𝑥 + 1 = 𝑥 2 + 2𝑥 + 1

4
 Therefore, the equation is true for all values of 𝑥. Thus, it is an identity
equation.

d. The right side can be simplified.

tan 𝑥
sin 𝑥 =
sec 𝑥
sin 𝑥
sin 𝑥 = cos 𝑥
sec 𝑥
sin 𝑥 1
sin 𝑥 = ∙
cos 𝑥 sec 𝑥
sin 𝑥
sin 𝑥 = ∙ cos 𝑥
cos 𝑥
𝐬𝐢𝐧 𝒙 = 𝐬𝐢𝐧 𝒙
The equation is true to all 𝑥, thus, it is an identity equation.

What’s More

Activity 1.2: NOW IT’S YOUR TURN!

Instruction: Determine whether the following equations is a conditional or an identity
equation. If it is Conditional Equation, write CE and determine the value or values
of the variable that will make the equation true. Otherwise, write IE if it is an Identity
Equation. Write your answer on your answer sheets.

1. 2𝑥 2 = 72
2. (𝑥 + 1)(𝑥 − 1) = 𝑥(𝑥 + 1) − (𝑥 + 1)
1 6
3. + 1 =
𝑛 𝑛
4. (𝑥 + 1)2 −42 = (𝑥 − 3)(𝑥 + 5)
5. sin 𝑥 csc 𝑥 = cos 𝑥 sec 𝑥

What I Need to Remember

• A conditional equation is an equation that is satisfied by at least one real

number while identity equation is an equation that is true for all real
numbers and whose both sides are defined and identical.

5
 Lesson Trigonometric Identities
2

What I Need to Know

At the end of this lesson, you are expected to:
o apply trigonometric identities to find other trigonometric value.

What’s In

Recall that the sine and cosine functions (and four others: tangent, cosecant,
secant and cotangent) of angles measuring between 0° and 90° were defined as ratios
of sides of a right angle.

Let 𝜃 be an angle in standard position and 𝑃(𝜃) = 𝑃(𝑥, 𝑦) the point on its
terminal side on the unit circle. Then,

1
sin 𝜃 = 𝑦 csc 𝜃 = 𝑦 , 𝑦 ≠ 0
1
cos 𝜃 = 𝑥 sec 𝜃 = 𝑥 , 𝑥 ≠ 0
𝑦 𝑥
tan 𝜃 = ,𝑥 ≠ 0 cot 𝜃 = ,𝑦 ≠ 0
𝑥 𝑦
Note that all these holds whether we take the values of the variable to
be real numbers (𝑖. 𝑒. , "𝑥") or measures of angles (𝜃).

The sine, cosine and tangent functions are known as the basic circular
functions. The cosecant, secant and cotangent functions are called reciprocal
functions because they are simply the reciprocal of the basic circular functions.

What’s New

Activity 2.1: Finding the Values

Instruction: Read the given below and give the values of the six trigonometric
functions. Write your solution and simplified answer on a separate sheet of paper.

√3 1
Let 𝑃 ( , ) be the terminal point of an arc length 𝑠 on the unit circle. Give the
2 2
values of the six circular functions of 𝑠.

sin 𝑠 = csc 𝑠 =
cos 𝑠 = sec 𝑠 =
tan 𝑠 = cot 𝑠 =

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 What Is It

Fundamental Trigonometric Identities

From the definition of the six circular functions of angles measuring between
0° and 90° as defined ratios of the sides of a right angle, the reciprocal and quotient
identities follow. More so, by simplifying a trigonometric expression using the
reciprocal and quotient identities, we obtain the Pythagorean identities.

Reciprocal Identities
1 1 1
𝑠𝑖𝑛 𝜃 = 𝑐𝑠𝑐 𝜃 𝑐𝑜𝑠 𝜃 = 𝑠𝑒𝑐 𝜃 𝑡𝑎𝑛 𝜃 = 𝑐𝑜𝑡 𝜃
Remember!

All these hold

Quotient Identities whether we take
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 the values of the
𝑡𝑎𝑛 𝜃 = 𝑐𝑜𝑠 𝜃 𝑐𝑜𝑡 𝜃 = 𝑠𝑖𝑛 𝜃 variable to be
real numbers
(i.e., “x”) or
Pythagorean Identities measures of
angles (𝜃).
𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1 1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐 2 𝜃 1 + 𝑐𝑜𝑡 2 𝜃 = 𝑐𝑠𝑐 2 𝜃

In addition to the eight identities above, we also have the following identities:

Even-Odd Identities
𝑠𝑖𝑛 (−𝜃) = −𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 (−𝜃) = 𝑐𝑜𝑠 𝜃 𝑡𝑎𝑛 (−𝜃) = −𝑡𝑎𝑛 𝜃

Equations sometimes involve expressions with trigonometric functions. The

fundamental trigonometric identities are used to simplify trigonometric expressions.

Example 2.1: Express the following trigonometric expressions as a single function.

(a) tan 𝑥 csc 𝑥
(b) 𝑠𝑖𝑛2 𝜃 + 𝑠𝑖𝑛2 𝜃𝑡𝑎𝑛2 𝜃
(c) sin 𝑥 𝑐𝑠𝑐 𝑥 𝑡𝑎𝑛 𝑥 𝑐𝑜𝑡 𝑥
sin 𝑥
(d) cos (𝑥)
cos(−𝑥)

7
 Quotient Identity
Solution:
𝐬𝐢𝐧 𝒙 1 Manipulating
(a) tan 𝑥 csc 𝑥 = ∙
cos 𝑥 𝐬𝐢𝐧 𝒙 Reciprocal Identity
1
=
cos 𝑥
Reciprocal Identity
= sec 𝑥

= 𝑠𝑖𝑛2 𝜃 (1 + 𝑡𝑎𝑛2 𝜃) Factoring out

(b) 𝑠𝑖𝑛2 𝜃 + 𝑠𝑖𝑛2 𝜃𝑡𝑎𝑛2 𝜃
𝑠𝑖𝑛2 𝜃 𝜃

= 𝑠𝑖𝑛2 𝜃 (𝑠𝑒𝑐 2 𝜃) Pythagorean

Identity

1 Manipulating
= 𝑠𝑖𝑛2 𝜃 ( )
𝑐𝑜𝑠 2 𝜃 Reciprocal Identity
𝑠𝑖𝑛2 𝜃
=
𝑐𝑜𝑠 2 𝜃
= 𝑡𝑎𝑛2 𝜃 Quotient Identity

Manipulating Reciprocal Identity

1 𝒔𝒊𝒏 𝒙 𝒄𝒐𝒔 𝒙
(c) sin 𝑥 𝑐𝑠𝑐 𝑥 𝑡𝑎𝑛 𝑥 𝑐𝑜𝑡 𝑥 = 𝐬𝐢𝐧 𝒙 ∙ ∙ ∙
𝒔𝒊𝒏 𝒙 𝒄𝒐𝒔 𝒙 𝒔𝒊𝒏 𝒙

Quotient Identities

= 1

sin 𝑥 sin 𝑥
(d) cos (𝑥) = cos 𝑥 cos (𝑥)
cos(−𝑥)

Even-Odd Identity

= sin 𝑥

8
Other Trigonometric Identities

The Fundamental Trigonometric Identities dealt only with one angle. Other
Trigonometric Identities involve two angles, say 𝐴 and 𝐵. The following identities are
used to determine the exact values of the sine and cosine of the sum and difference
of two given angles, 𝛼 and 𝛽.

Sum and Difference Identities for Sine, Cosine and Tangent

𝑠𝑖𝑛 (𝛼 + 𝛽) = 𝑠𝑖𝑛 𝛼 𝑐𝑜𝑠 𝛽 + 𝑐𝑜𝑠 𝛼 𝑠𝑖𝑛 𝛽 𝑠𝑖𝑛 (𝛼 − 𝛽) = 𝑠𝑖𝑛 𝛼 𝑐𝑜𝑠 𝛽 − 𝑐𝑜𝑠 𝛼 𝑠𝑖𝑛 𝛽

𝑐𝑜𝑠 (𝛼 + 𝛽) = 𝑐𝑜𝑠 𝛼 𝑐𝑜𝑠 𝛽 − 𝑠𝑖𝑛 𝛼 𝑠𝑖𝑛 𝛽 𝑐𝑜𝑠 (𝛼 − 𝛽) = 𝑐𝑜𝑠 𝛼 𝑐𝑜𝑠 𝛽 + 𝑠𝑖𝑛 𝛼 𝑠𝑖𝑛 𝛽
𝑡𝑎𝑛 𝛼 + 𝑡𝑎𝑛 𝛽 𝑡𝑎𝑛 𝛼 − 𝑡𝑎𝑛 𝛽
𝑡𝑎𝑛 (𝛼 + 𝛽) = 𝑡𝑎𝑛 (𝛼 − 𝛽) =
1 − 𝑡𝑎𝑛 𝛼 𝑡𝑎𝑛 𝛽 1 + 𝑡𝑎𝑛 𝛼 𝑡𝑎𝑛 𝛽

These formulas or identities can be utilized to find the exact values of

trigonometric functions involving the sum and difference of special angles.

Example 2.2:

Find the exact value of the following using Sum and Difference Identities.
(a) 𝑠𝑖𝑛 15°
𝜋
(b) 𝑐𝑜𝑠 12
3 12
(c) 𝑡𝑎𝑛 (𝛼 + 𝛽) if 𝑠𝑖𝑛 𝛼 = and 𝑐𝑜𝑠 𝛽 = − , and 𝛼 is in Quadrant I while 𝛽 is in
5 13
Quadrant II.

Solution:

(a) The given angle can be expressed in terms of two special angles. That is,
15° = 45° − 30°.
Use the formula for the sine of the difference of the angles to obtain the
exact value of 𝑠𝑖𝑛 15°.
Difference
𝑠𝑖𝑛 15° = 𝑠𝑖𝑛 (45° − 30°)
Identity for Sine
= sin 45°𝑐𝑜𝑠 30° − 𝑐𝑜𝑠 45° 𝑠𝑖𝑛 30°
Apply unit
triangle
√2 √3 √2 1 property or
=( )( ) − ( )( )
2 2 2 2 use a
scientific
√6 √2
= − calculator.
4 4

√6 − √2
=
4

9
 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋
(b) Observe that = − . Thus, 𝑐𝑜𝑠 = 𝑐𝑜𝑠 ( − ).
12 3 4 12 3 4

𝜋 𝜋 𝜋 Difference Identity
𝑐𝑜𝑠 = 𝑐𝑜𝑠 ( − )
12 3 4 for Cosine
𝜋 𝜋 𝜋 𝜋
= 𝑐𝑜𝑠 𝑐𝑜𝑠 + 𝑠𝑖𝑛 𝑠𝑖𝑛
3 4 3 4 Apply unit
triangle
1 √2 √3 √2 property or
= ( )( ) + ( )( )
2 2 2 2 use a
√2 √6 scientific
= + calculator.
4 4

√2 + √6
=
4

(c) To find 𝑡𝑎𝑛 𝛼, use the Pythagorean Identity 𝑠𝑖𝑛2 𝛼 + 𝑐𝑜𝑠 2 𝛼 = 1 and solve for
the value of 𝑐𝑜𝑠 𝛼.

𝑠𝑖𝑛2 𝛼 + 𝑐𝑜𝑠 2 𝛼 = 1
3 2
Given ( ) + 𝑐𝑜𝑠 2 𝛼 = 1
5
9
+ 𝑐𝑜𝑠 2 𝛼 = 1
25
9
𝑐𝑜𝑠 2 𝛼 = 1 −
25
25 − 9
𝑐𝑜𝑠 2 𝛼 =
25
16
𝑐𝑜𝑠 2 𝛼 =
25

16
√𝑐𝑜𝑠 2 𝛼 = √
25

4
𝑐𝑜𝑠 𝛼 = ±
5
𝟒
From the given, 𝛼 is in quadrant I, thus 𝒄𝒐𝒔 𝜶 = 𝟓. To find the value of 𝑡𝑎𝑛 𝛼,
𝑠𝑖𝑛 𝛼
use the equation 𝑡𝑎𝑛 𝛼 = 𝑐𝑜𝑠 𝛼.
3
5
𝑡𝑎𝑛 𝛼 = 4
5
3 𝟓
=∙
𝟓 4
𝟑
𝒕𝒂𝒏 𝜶 =
𝟒

10
 Do the same procedure as presented above to find the value of 𝑡𝑎𝑛 𝛽.
To find 𝑡𝑎𝑛 𝛽, use the Pythagorean Identity 𝑠𝑖𝑛2 𝛽 + 𝑐𝑜𝑠 2 𝛽 = 1 and solve
for the value of 𝑠𝑖𝑛 𝛽.

𝑠𝑖𝑛2 𝛽 + 𝑐𝑜𝑠 2 𝛽 = 1
2
12 2
𝑠𝑖𝑛 𝛽+ (− ) = 1
13

Given

144
𝑠𝑖𝑛2 𝛽 + =1
169
144
𝑠𝑖𝑛2 𝛽 = 1 −
169
169 − 144
𝑠𝑖𝑛2 𝛽 =
169
25
𝑠𝑖𝑛2 𝛽 =
169

25
√ 𝑠𝑖𝑛2 𝛽 = √
169

5
𝑠𝑖𝑛 𝛽 = ±
13
𝟓
From the given, 𝛽 is in quadrant II, thus 𝒔𝒊𝒏 𝛽 = . To find the value of 𝑡𝑎𝑛 𝛽,
𝟏𝟑
𝑠𝑖𝑛 𝛽
use the equation 𝑡𝑎𝑛 𝛽 = .
𝑐𝑜𝑠 𝛽
5
13
𝑡𝑎𝑛 𝛽 = 12
(− )
13
5 𝟏𝟑
= ∙ (− )
𝟏𝟑 12
𝟓
𝒕𝒂𝒏 𝜷 = −
𝟏𝟐

The exact value of 𝑡𝑎𝑛 (𝛼 + 𝛽) is:

𝑡𝑎𝑛 𝛼 + 𝑡𝑎𝑛 𝛽
𝑡𝑎𝑛 (𝛼 + 𝛽) =
1 − 𝑡𝑎𝑛 𝛼 𝑡𝑎𝑛 𝛽
3 + (− 5 )
=
4 12
1 − ( ) (− 5 )
3
4 12
3− 5
= 4 12
1 + 15
48

11
 9−5 You can
= 12 reduce the
48 + 15 numerator
48 and
4 denominat
= 12 or by
63
48 removing
4 𝟒𝟖 its
= ∙ common
𝟏𝟐 63
value.
16
𝑡𝑎𝑛 (𝛼 + 𝛽) =
63

Through the sum and difference identities, the following double-angle

identities can be derived.

Double-Angle Identities

𝑠𝑖𝑛 2𝛼 = 2𝑠𝑖𝑛 𝛼 𝑐𝑜𝑠 𝛼

2𝑡𝑎𝑛 𝛼
2 2 𝑡𝑎𝑛 2𝛼 =
𝑐𝑜𝑠 2𝛼 = 𝑐𝑜𝑠 𝛼 − 𝑠𝑖𝑛 𝛼 1−𝑡𝑎𝑛2 𝛼

𝑐𝑜𝑠 2𝛼 = 2𝑐𝑜𝑠 2 𝛼 − 1

𝑐𝑜𝑠 2𝛼 = 1 − 𝑠𝑖𝑛2 𝛼

Example 2.3:

4 𝜋
If 𝑠𝑖𝑛 𝛼 = 5 and 2
< 𝛼 < 𝜋, find the exact values of the following:
(a) 𝑠𝑖𝑛 2𝛼
(b) 𝑐𝑜𝑠 2𝛼
(c) 𝑡𝑎𝑛 2𝛼

Solution:

First, solve for the value of 𝑐𝑜𝑠 𝛼 and 𝑡𝑎𝑛 𝛼.

𝑠𝑖𝑛2 𝑎 + 𝑐𝑜𝑠 2 𝑎 = 1 Pythagorean
Identity
4 2
Given ( ) + 𝑐𝑜𝑠 2 𝑎 = 1
5
16
+ 𝑐𝑜𝑠 2 𝑎 = 1
25
16
𝑐𝑜𝑠 2 𝑎 = 1 −
25
25 − 16
𝑐𝑜𝑠 2 𝑎 =
25
9
𝑐𝑜𝑠 2 𝑎 =
25
9
√𝑐𝑜𝑠 2 𝑎 =12√
25
 3
𝑐𝑜𝑠 𝑎 = ±
5

𝜋 𝟑
Because 2
< 𝛼 < 𝜋, then 𝒄𝒐𝒔 𝜶 = − 𝟓. Therefore,
4
5
𝑡𝑎𝑛 𝛼 = 3
−
5
4 𝟓
= ∙ (− )
𝟓 3
𝟒
=−
𝟑
Then,
(a) 𝑠𝑖𝑛 2𝛼 = 2𝑠𝑖𝑛 𝛼 𝑐𝑜𝑠 𝛼

4 3
𝑠𝑖𝑛 2𝛼 = 2 ( ) (− )
5 5
24
=−
25

(b) 𝑐𝑜𝑠 2 α = 𝑐𝑜𝑠 2 α − 𝑠𝑖𝑛2 α

3 2 4 2
𝑐𝑜𝑠 2 α = (− ) − ( )
5 5
9 16
= −
25 25
7
=−
25

2𝑡𝑎𝑛 𝛼
(c) 𝑡𝑎𝑛 2𝛼 =
1−𝑡𝑎𝑛2 𝛼
4
2(− )
3
𝑡𝑎𝑛 2𝛼 =
4 2
1−(−3)
8
−3
= 16
1− 9
8
−3
= 9−16
9
8
−3
= −7
9
8 9
= − ∙−
3 7
24
=
7
13
 𝛼
The half-angle identities can be derived by replacing 𝛼 with in the double-
2
angle identity for cosine.

Half-Angle Identities

𝛼 1 − 𝑐𝑜𝑠 𝛼 𝛼 1−𝑐𝑜𝑠 𝛼
𝑠𝑖𝑛 = ±√ 𝑡𝑎𝑛 2
= ±√ ; 𝑐𝑜𝑠 𝛼 ≠ 1
2 2 1+𝑐𝑜𝑠 𝛼

𝛼 1 + 𝑐𝑜𝑠 𝛼 𝛼 1−𝑐𝑜𝑠 𝛼
𝑐𝑜𝑠
2
= ±√
2
𝑡𝑎𝑛
2
= ; 𝑠𝑖𝑛 𝛼 ≠ 0
𝑠𝑖𝑛 𝛼

𝛼 𝑠𝑖𝑛 𝛼
𝑡𝑎𝑛
2
= ; 𝑐𝑜𝑠 𝛼 ≠ −1
1−𝑐𝑜𝑠 𝛼

Example 2.4:

Find the exact value of the following using half-angle identities.

5𝜋
(a) 𝑠𝑖𝑛 8
𝜃 3
(b) 𝑐𝑜𝑠 2
if 𝑐𝑜𝑠 𝜃 = 5 and 𝜃 is in quadrant I
𝜃
(c) 𝑡𝑎𝑛 if 𝑡𝑎𝑛 𝜃 = 2 and 𝜃 is in quadrant III
2

Solution:
𝛼 5𝜋 5𝜋
(a) Let 2
= 8
. Then 8
is in quadrant II where the sine is positive.
5𝜋 5𝜋 𝛼 1−𝑐𝑜𝑠 𝛼
Also, 𝛼 = 2 ( 8 ) = 4 . Since sine is positive, 𝑠𝑖𝑛 2 = √ 2 .
5𝜋
5𝜋 √1 − 𝑐𝑜𝑠 4
𝑠𝑖𝑛 =
8 2

√2
1 − (− )
√ 2
=
2

√2
√1 + 2
=
2

2 + √2
√ 2
=
2

2 + √2 1
=√ ∙
2 2

5𝜋 2 + √2
𝑠𝑖𝑛 =√
8 4

14
 𝜃
(b) Since 𝜃 is in quadrant I, 𝑐𝑜𝑠 is positive.
2

𝜃 1 + 𝑐𝑜𝑠 𝛼
𝑐𝑜𝑠 =√
2 2

3
𝜃 √1 + 5
𝑐𝑜𝑠 =
2 2

5+3
√ 5
=
2
8
√5
=
2
8 1
=√ ∙
5 2

4
=√
5

𝜃 2√5
𝑐𝑜𝑠 =
2 5

𝜃
(c) Note that 180° < 𝜃 < 270°, then 90° < < 135° which is in quadrant II.
2
𝜃
Therefore, 𝑡𝑎𝑛 2
must be negative.
First, solve the values of 𝑠𝑖𝑛 𝜃 and 𝑐𝑜𝑠 𝜃. Tha value of 𝑐𝑜𝑠 𝜃 can
be solved using the formula 1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐 2 𝜃.

1 + (2)2 = 𝑠𝑒𝑐 2 𝜃
1 + 4 = 𝑠𝑒𝑐 2 𝜃
𝑠𝑒𝑐 2 𝜃 = 5
√𝑠𝑒𝑐 2 𝜃 = √5
𝑠𝑒𝑐 𝜃 = ±√5

Because 𝜃 is in quadrant III, then 𝑠𝑒𝑐 𝜃 = √5. It follows that

√5
𝑐𝑜𝑠 𝜃 = − 5
. From the equation 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1 and using
√5 2√5
𝑐𝑜𝑠 𝜃 = − , you will find that 𝑠𝑖𝑛 𝜃 =− .
5 5

𝜃
Solving for the value of 𝑡𝑎𝑛 2 ,

𝜃 𝑠𝑖𝑛 𝜃
𝑡𝑎𝑛 =
2 1 − 𝑐𝑜𝑠 𝜃

15
 2√5
−
= 5
√5
1 − (− )
5

2√5
−
= 5
√5
1+
5
2√5
−
= 5
5 + √5
5
2√5 𝟓
=− ∙
𝟓 5 + √5
2√5
=−
5 + √5

Rationalizing the denominator by multiplying the numerator and

denominator by the conjugate of the denominator.
𝜃 2√5
𝑡𝑎𝑛 = −
2 5 + √5
2√5 5 − √5
=− ∙
5 + √5 5 − √5
−10√5 + 10
=
20
𝜃 −√5 + 1
𝑡𝑎𝑛 =
2 2

What’s More

Activity 2.2: NOW IT’S YOUR TURN!

Instruction: Answer the following. Write your answers on your answer sheet.
A. Simplify the following trigonometric expressions using the fundamental
trigonometric identities.
1. 𝑠𝑖𝑛 𝑥 𝑐𝑜𝑡 𝑥
𝑐𝑜𝑠 𝑥
2. cot 𝑥

3. 𝑐𝑜𝑠 2 𝜃+𝑐𝑜𝑠 2 𝜃𝑡𝑎𝑛2 𝜃

B. Find the exact value of the following:
1. 𝑐𝑜𝑠 105°
3 𝜋
2. 𝑠𝑖𝑛 2𝛼 if 𝑠𝑖𝑛 𝛼 = and 0 < 𝛼 <
5 2

𝛼 12
3. 𝑐𝑜𝑠 2 if 𝑐𝑜𝑠 𝛼 = − 13 and 180° < 𝛼 < 270°

16
 What I Need to Remember

• An identity is an equation that is true for all real numbers and whose two sides
are both defined. A conditional equation, on the other hand, is an equation that
is true only for at least one real number.
• When simplifying equations that involve trigonometric expressions, it is
necessary to consider the following fundamental trigonometric identities:
A. Reciprocal Identities
1 1 1
𝑠𝑖𝑛 𝜃 = 𝑐𝑠𝑐 𝜃 𝑐𝑜𝑠 𝜃 = 𝑠𝑒𝑐 𝜃 𝑡𝑎𝑛 𝜃 = 𝑐𝑜𝑡 𝜃
B. Quotient Identities
𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃
𝑡𝑎𝑛 𝜃 = 𝑐𝑜𝑠 𝜃 𝑐𝑜𝑡 𝜃 = 𝑠𝑖𝑛 𝜃
C. Pythagorean Identities
𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1 1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐 2 𝜃 1 + 𝑐𝑜𝑡 2 𝜃 = 𝑐𝑠𝑐 2 𝜃
D. Even-Odd Identities
s𝑖𝑛 (−𝜃) = −𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 (−𝜃) = 𝑐𝑜𝑠 𝜃 𝑡𝑎𝑛 (−𝜃) = −𝑡𝑎𝑛 𝜃
• Other trigonometric identities involve functions of the sum or difference of
two angles, double angle, or half angle. The following are the sum and
difference identities for sine, cosine and tangent functions.
𝑠𝑖𝑛 (𝛼 + 𝛽) = 𝑠𝑖𝑛 𝛼 𝑐𝑜𝑠 𝛽 + 𝑐𝑜𝑠 𝛼 𝑠𝑖𝑛 𝛽
𝑠𝑖𝑛 (𝛼 − 𝛽) = 𝑠𝑖𝑛 𝛼 𝑐𝑜𝑠 𝛽 − 𝑐𝑜𝑠 𝛼 𝑠𝑖𝑛 𝛽
𝑐𝑜𝑠 (𝛼 − 𝛽) = 𝑐𝑜𝑠 𝛼 𝑐𝑜𝑠 𝛽 + 𝑠𝑖𝑛 𝛼 𝑠𝑖𝑛 𝛽 𝑐𝑜𝑠 (𝛼 + 𝛽) = 𝑐𝑜𝑠 𝛼 𝑐𝑜𝑠 𝛽 − 𝑠𝑖𝑛 𝛼 𝑠𝑖𝑛 𝛽

𝑡𝑎𝑛 𝛼 − 𝑡𝑎𝑛 𝛽 𝑡𝑎𝑛 𝛼 + 𝑡𝑎𝑛 𝛽

𝑡𝑎𝑛 (𝛼 − 𝛽) = 𝑡𝑎𝑛 (𝛼 + 𝛽) =
1 + 𝑡𝑎𝑛 𝛼 𝑡𝑎𝑛 𝛽 1 − 𝑡𝑎𝑛 𝛼 𝑡𝑎𝑛 𝛽

• From these identities, the double-angle identities which are shown below,
can be derived.
𝑠𝑖𝑛 2𝛼 = 2𝑠𝑖𝑛 𝛼 𝑐𝑜𝑠 𝛼 2𝑡𝑎𝑛 𝛼
𝑡𝑎𝑛 2𝛼 =
1−𝑡𝑎𝑛2 𝛼
𝑐𝑜𝑠 2𝛼 = 𝑐𝑜𝑠 2 𝛼 − 𝑠𝑖𝑛2 𝛼

𝑐𝑜𝑠 2𝛼 = 2𝑐𝑜𝑠 2 𝛼 − 1

𝑐𝑜𝑠 2𝛼 = 1 − 𝑠𝑖𝑛2 𝛼

• The double angle identities can be utilized to derive the half-angle

identities. Here are the half-angle identities:

𝛼 1 − 𝑐𝑜𝑠 𝛼 𝛼 1−𝑐𝑜𝑠 𝛼
𝑠𝑖𝑛 = ±√ 𝑡𝑎𝑛
2
= ±√ ; 𝑐𝑜𝑠 𝛼 ≠ 1
2 2 1+𝑐𝑜𝑠 𝛼

𝛼 1 + 𝑐𝑜𝑠 𝛼 𝛼 1−𝑐𝑜𝑠 𝛼
𝑐𝑜𝑠
2
= ±√
2
𝑡𝑎𝑛 2
= ; 𝑠𝑖𝑛 𝛼 ≠ 0
𝑠𝑖𝑛 𝛼

𝛼 𝑠𝑖𝑛 𝛼
𝑡𝑎𝑛 2
= ; 𝑐𝑜𝑠 𝛼 ≠ −1
1−𝑐𝑜 𝑠 𝛼

17
 What I Can Do

Activity 2.3: LET’S GET REAL

Instruction: Answer the following. Write your answer on your answer sheet.
A. Group the equations inside the box into Conditional Equation and Identity
Equation.
\

𝑥2 − 4 13 1 1 11
𝑥2 − 1 = 0 = 2𝑥 − 1 𝑥+ = 𝑥−
𝑥−2 6 3 2 2
(𝑥 + 7)2 = 𝑥 2 + 14𝑥 + 49 (𝑥 + 7)2 = 𝑥 2 + 49 1 1 1
( 𝑥 − √3) ( 𝑥 + √3) = 𝑥 2 − 3
𝑥 2 − 1 = (𝑥 − 1)(𝑥 + 1) 2 2 4
𝑥2 − 4
=𝑥+2
𝑥−2

B. Simplify the following trigonometric equations.

1. 𝑐𝑜𝑠 𝑥𝑠𝑖𝑛2 𝑥 − 𝑐𝑜𝑠 𝑥
2. (𝑐𝑠𝑐 2 𝜃 − 1)𝑠𝑖𝑛2 𝜃
1 1
3. 𝑠𝑖𝑛2 𝜃 − 𝑡𝑎𝑛2 𝜃

C. Find the exact values of the following.

5 7
1. 𝑡𝑎𝑛 (𝛼 − 𝛽) if 𝑠𝑖𝑛 𝛼 = and 𝑐𝑜𝑠 𝛽 = −
13 25
, and 𝛼 is in quadrant II while 𝛽 is in
quadrant III
5𝜋
2. 𝑠𝑖𝑛 12
and 𝑡𝑎𝑛 165°

18
 Assessment (Posttest)

Instruction: Choose the letter of the correct answer. Write them on

a separate sheet of paper.

1. Which of the following equation is conditional?

𝑥 2 −4
𝐴. 𝑥 2 − 1 = 0 C. 𝑥−2
=𝑥+2
2
𝐵. 𝑥 − 1 = (𝑥 − 1)(𝑥 + 1) D. (𝑥 + 1)2 = 𝑥 2 + 2𝑥 + 1

2. Which of the following equation is an identity?

𝑥 2 −1
𝐴. (𝑥 + 7)2 = 𝑥 2 + 14𝑥 + 49 C. =𝑥−2
𝑥 2 +1
𝐵. 3𝑥 + 1 = 2 D. 𝑥 + 2 = 3𝑥
2

3. Which of the following is NOT an identity?

𝐴. 𝑠𝑖𝑛2 𝛼+𝑐𝑜𝑠 2 𝛼 = 1 C. 1 + 𝑐𝑜𝑡 2 𝛼 = 𝑐𝑠𝑐 2 𝛼
𝐵. 𝑠𝑖𝑛 𝛼 = 𝑡𝑎𝑛 𝛼 𝑐𝑜𝑠 𝛼 D. 1 − 𝑠𝑒𝑐 2 𝛼 = 𝑡𝑎𝑛2 𝛼

4. Which of the following is an identity?

sin 2𝛼
𝐴. 𝑠𝑖𝑛 𝛼 𝑐𝑜𝑠 𝛼 = 2
C. 𝑠𝑖𝑛(−𝛼) = 𝑠𝑖𝑛 𝛼
𝑐𝑜𝑠 𝛼
𝐵. 𝑠𝑖𝑛 𝛼 + 𝑐𝑜𝑠 𝛼 = 1 D. 𝑡𝑎𝑛 𝛼 =
𝑠𝑖𝑛 𝛼
cos 𝑥
5. What is the equivalent expression of ?
cot 𝑥
𝐴. 𝑐𝑜𝑠 𝑥 B. 𝑠𝑖𝑛 𝑥 C. 𝑡𝑎𝑛 𝑥 D. 𝑠𝑒𝑐 𝑥
6. What are the identities derived from the sum and difference identities?
A. Double-Angle C. Pythagorean
B. Half-Angle D. Reciprocal

7. Which of the following defines an identity equation?

A. It is satisfied by at least one real number.
B. It is not true for all values of the variable in the domain of the equation.
C. It is true for all real numbers and whose both sides are defined and
identical.
D. An equation where values of the variable in the domain of the equation
do not satisfy the equation.

8. Which of the following DOES NOT constitute the Fundamental Trigonometric

Identity?
A. Quotient Identity C. Pythagorean Identity
B. Even-Odd Identity D. Double-Angle Identity
3
9. If 𝑠𝑖𝑛 𝜃 = − 4 and 𝑐𝑜𝑠 𝜃 > 0. What is 𝑐𝑜𝑠 𝜃?
4 √7 √7 5
𝐴. 7 B. 5
C. 4
D. 7

19
10. How do you express 𝑠𝑒𝑐 𝜃 in terms of 𝑐𝑜𝑠 𝜃?
1
𝐴. 𝑐𝑜𝑠 𝜃
B. 𝑐𝑜𝑠 𝜃 − 1 C. 𝑐𝑜𝑠 𝜃 + 1 D. 𝑐𝑜𝑠 𝜃

11. What is the exact value of 𝑐𝑜𝑠 105°?

√2+√6 √2−√6 √6 √2
𝐴. 4
B. 4
C. 4
D. 4

3 𝜋
12. Given that sin 𝑡 = 5 and 2
< 𝑡 < 𝜋, what is the exact value 𝑠𝑖𝑛 2𝑡?
25 24 7
𝐴. − 24 B. − 25 C. 25 D. 25

3 𝜋
13. Given that sin 𝑡 = 5 and 2
< 𝑡 < 𝜋, what is the exact value 𝑐𝑜𝑠 2𝑡?
25 24 7
𝐴. − 24 B. − 25 C. 25 D. 25

14. What is the exact value of 𝑡𝑎𝑛 75° using the Half-Angle Identity?
𝐴. 2 + √3 B. 2 − √3 C. √3 D.2
4 𝜃
15. If 𝑡𝑎𝑛 𝜃 = and 𝑐𝑜𝑠 𝜃 < 0. What is the exact value of 𝑡𝑎𝑛 ( ) ?
3 2
A. 2 B. -2 C. 0.2 D. √2

20
References
Book
Cabral, E.A. et.al, Application of Trigonometric Functions in Periodic Phenomena,
Precalculus (Ateneo de Manila Press,2010), 270-291

Albay, E. M.,Senior High Schoo Series: Pre Calculus (DIWA Learning System, Inc.,
2016), 169-194

PDF File
Garces, I. J.,Mathematical Induction, Precalculus: Learner’s Material for Senior High
School (Quezon City © 2016), 161-170

Images
Impostor Crew Images, retrieved from tinyurl.com/3d733wx5 on January 27, 2021

Congratulations!
You are now ready for the next module. Always remember the following:

1. Make sure every answer sheet has your

▪ Name
▪ Grade and Section
▪ Title of the Activity or Activity No.
2. Follow the date of submission of answer sheets as agreed with your
teacher.
3. Keep the modules with you AND return them at the end of the school year
or whenever face-to-face interaction is permitted.

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