SENIOR HIGH SCHOOL

Pre-Calculus
Quarter 2- Module 1
Angles in a Unit Circle, Standard
Position of Angles, and Its
Coterminal Angles

i
Introductory Message
Welcome to the Pre-Calculus on Angles in a Unit Circle, Standard Position of Angles, and Its
Coterminal Angles!

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

This module has the following parts and corresponding icons:

This part includes an activity that aims
What I Know to check what you already know about
(Pre-Test)
the lesson to take.
This will give you an idea of the skills or
What I Need to Know competencies you are expected to learn
(Objectives)
in the module.
This is a brief drill or review to help you
What’s In
link the current lesson with the previous
(Review/Springboard)
one.
In this portion, the new lesson will be
What’s New introduced to you in various ways; a
(Presentation of the Lesson) story, a song, a poem, a problem opener,
an activity, or a situation.
This section provides a brief discussion
What is It of the lesson. This aims to help you
(Discussion) discover and understand new concepts
and skills.
This section provides activities which will
What’s More help you transfer your new knowledge or
(Application)
skill into real life situations or concerns.

What I Need To Remember This includes key points that you need
(Generalization) to remember.

This comprises activities for

What I Can Do independent practice to solidify your
(Enrichment Activities)
understanding and skills of the topic.

This is a task which aims to evaluate

Assessment your level of mastery in achieving the
(Post Test)
learning competency.

This contains answers to the following:

• What I Know
Answer Key • What’s In
• What’s More

ii
At the end of this module, you will also find:

References This is a list of all resources used in developing this module.

The following are some reminders in using this module:

1. Use the module with care. Do not put unnecessary mark/s on any part of the
module. Use a separate sheet of paper in answering the exercises.
2. Do not forget to answer What I Know before moving on to the other activities
included in this module.
3. Read the instructions carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.

If you encounter any difficulty in answering the tasks in this module, do not hesitate to
consult your teacher or facilitator. Always bear in mind that you are not alone.

We hope that through this material, you will experience meaningful learning and gain deep
understanding of the relevant competencies. You can do it!

About the Module

This module was designed and written with you in mind. It is here to help you master
about Angle Measures in a Unit Circle, Standard Position of Angles and Its coterminal
Angles. 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 is divided into three lessons, namely:

Lesson 1 – Angle Measures (Linear and Angular) of Arcs in a Unit Circle
Lesson 2 – Conversion of Degree to Radian, and Vice Versa
Lesson 3 – Angle in Standard Position and Its Coterminal Angles
After going through this module, you are expected to:

• illustrate the unit circle and the relationship between linear and angular
measures of arcs in a unit circle;
• convert degree measure to radian measure and vice versa;
• illustrate standard position of angle measures (degrees and radian
forms); and
• illustrate coterminal angles and give one positive and one negative
coterminal angles of the given angle measure.

iii
 What I Know (Pretest)

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

1. What do you call to a ray in an angle that can either rotates in a clockwise or
counterclockwise direction about its endpoint?
A. Initial side B. Positive side C. Stationary side D. Terminal side

2. Standard position is when the …

A. initial side of an angle is on the negative x - axis
B. initial side of an angle is on the negative y - axis
C. initial side of an angle is on the positive x - axis
D. initial side of an angle is on the positive y - axis

3. Which of the following is a pair of equal measures?

𝜋 𝜋
A. ; 45° C. 80°;
6 2
3𝜋 2𝜋
B. ; 250° D. 120°;
2 3

4. What is the equivalent degree measure (in decimal form) for 27° 13´ 30´´?
A. 27.225° B. 27.288° C. 27.325° D. 27.388°

5𝜋
5. Which of the following is the equivalent degree measure for radians?
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A. 90° B. 100° C. 110° D. 120°

6. Which of the following radian measures is in Q3 or the third quadrant?

5𝜋 7𝜋 9𝜋 12𝜋
A. B. C. D.
6 6 6 6

7. Which pair of measurements represents the same angle measure?

3 4𝜋
A. 1200, C. 200°,
4 3
5 7
B. 1500, D. 2400,
6 6

8. If the unit circle is divided into 8 congruent arcs, how long is the arc subtended by
the initial and terminal sides made by the 7 consecutive arcs?
𝜋 𝜋 7𝜋 7𝜋
A. B. C. D.
8 4 4 8

1
9. The illustration/sketch for the standard position of − 100° ?
C.
A. y B. C. D D.

10. Which of the following DOES NOT belong to the group?

3𝜋 3𝜋
A. B. C. 2π D. 3π
4 2

11. The number of coterminal angles of any given angle measure is ______________.
A. exactly 2 B. between 2 & 10 C. between 11& 100 D. infinite

12. What are the negative and positive coterminal angles of − 225°?
A. 600, − 120° B. 125°, − 356° C. 135°, − 585° D. 375°, − 356°

13. Which of the following is NOT coterminal with − 60°?

A. − 420 ° B. − 320° C. 300° D. 660°

14. Which of the following is closest to the measure of the angle shown
at your right?

7𝜋 7𝜋 3𝜋 3𝜋
A. B. C. D.
20 10 8 2

15. Find the coordinate of terminal point on the Unit Circle associated with the rotation
9𝜋
− 2 ?
A. ( −1 , 0 ) B. ( 0 , −1 ) C. ( 0 , 1 ) D. ( 1 , 0 )

Lesson Angles in a Unit Circle

1

What I Need to Know

At the end of this lesson, you are expected to:

o illustrate a unit circle; and

o relate the measure of linear and angular measurement of an arc in a unit circle.

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 What’s In
• Angles in trigonometry differ from
angles in Euclidean geometry in the
sense of motion. An Angle in geometry is
defined as a union of rays (that is, static)
and has a measure between 0° and
180°. An angle in trigonometry is a
rotation of a ray, and, therefore, has no
limit. It has positive and negative
Figure 1: Angle form between two rays directions and measures (Garces, I.
J.,2016).

• The measure of an angle is a number which indicates the amount of

rotation that separates the rays of the angle.
• Angle measure is positive if the rotation is counterclockwise, while negative
for clockwise.
• Generally, we use lower Greek letters such as α (alpha), β (beta), γ (gamma),
and ϴ (theta) to label angles.

(Cited from the book, Precalculus Version [ π ]= 3 Corrected Edition by Stitz, C, 2013)

What’s New (Presentation of the Lesson)

Activity 1.1 The Relation of Circumference to the Length of the Radius

Instruction: You will have a Hands-On activity to explore the relationship between
circumference and length of the radius. You will be given 20 points if you can perform
the activity properly and arrive at the expected result.

Materials needed: 1 whole sheet of bond paper (or any paper), compass, string,
protractor and a ruler.

STEP 1: Draw a circle with a radius of at least 2 inches. (use a compass)

STEP 2: Cut a piece of string with a length equal to the radius of the circle. Place
the string along the circle.

STEP 3: Mark points at each end to locate the endpoints of an arc equal in length to
the radius of a circle.

STEP 4: Draw rays from the center of the circle through the endpoints of the arc,
forming a central angle. Using a protractor, find the degree measure of
this central angle.

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 STEP 5: Repeat steps 1 – 4 two or three times, using a circle with a different
radius each time.
(Adapted from Amsco’s Mathematics B, Teacher’s Manual by Keenan, Edward, 2002)

After following the 5 steps above, answer the following questions.

1. Compare the result of trial 1, trial 2 and trial 3 by observing on the measure of the
central angle with an intercepted arc equal to the length of its radius.

2. As you change the size of the circle, will the measure of the central angle remain
the same? Why or Why not?

Rubrics for scoring:

Points Description
16 -20 Followed the 5 steps and answered 2 questions
11 -15 Followed the 5 steps and answered 1 question
6 - 10 Followed the 5 steps but did not answer any of the questions
1-5 Followed 3 steps only
0 Followed NONE of the steps

NOTE: For modular class: Insert your output in your portfolio and submit on the
scheduled date of collection.
For online class: Take a picture of your output and send it to our Google Classroom
or messenger for checking.

What Is It (Discussion)

Angular Measure
(Cited from the book of Stitz, C. et al, 2013)

There are types of angular measures: degrees (deg), radian (rad), and gradian
. (grad). We had the degrees which are very familiar to you during your Junior High
Mathematics and now we will have radians.

The circumference of a circle compared to the radius brings us to our new angle
measure, called radian. Radian is the standard unit used to measure angles in
Mathematics. One radian is the measure of a central angle of a circle that
intercepts an arc equal in length to the radius of that circle. It allows us to treat
the trigonometric functions as functions with the set of real numbers as domain,
rather than angles (Stitz, et. al 2013).

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 Figure 2: α has a radian measure 1, 2013 Figure 3: Precalculus Version [ π ]= 3 Corrected Edition by Stitz, 2013)

, year :

According to Stitz (2013) that getting a better feel for radian measure, we note that an angle
with radian measure means the responding arc length s equals the radius (r ) of the circle,
hence s = r . When the radian measure is 2, we have s = 2r; when the measure is 3, s = 3r,
and so forth. Thus, the radian measure of an angle θ tells us how many “radius
lengths” we need to sweep out along the circle to subtend the angle θ.

Since one revolution sweeps out the entire circumference 2π r, one revolution has radian
2π𝑟
measure = 2π, we divide the circumference by the length of the radius. From this, we
𝑟
can find the radian measure of other central angles. This is further related to the concept
of a unit circle.

A central angle of the unit circle that intercepts

an arc of the circle with length 1 unit is said to
have a measure of one radian, written 1 rad. See
Figure 4

Figure 4: Representation of 1- radian measure,2016

The unit circle is the circle of radius 1 centered

at the origin in the xy-plane. See figure 5

Figure 5: Unit Circle, 2013

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 What’s More

Activity 1.2: DIG DEEP

A. Instruction. In your Hands-On activity 1.1, found in What’s New, use the same
circles but this time paste along the whole circumference as many as cut string equal
to the length of its radius. As much as possible use two (2) different colors of string
and paste it alternately.
Note: For modular class: Insert your output in your portfolio and submit on the scheduled date
of collection.
For online class: Take a picture of your output and send it to our Classroom or messenger
for checking.

After doing it, briefly answer the following questions:

1. How many strings with length equal to the radius swept the whole circumference
in each circle?
2. Suppose the length of the radius in each circle is 1 unit, what is the total arc
length of the circle? Justify.

Rubrics for scoring:

Points Description
16 - 20 Followed the instructions completely with accuracy and answered
2 questions correctly.

11- 15 Followed the instructions completely with accuracy and answered

1 question correctly.

6 - 10 Followed 50% of the instructions with accuracy but did not answer any
of the questions.

1-5 Followed 25% of the instructions and did not answer any of the
questions.

0 Followed NONE of the steps.

B. NOW IT’S YOUR TURN!

Instruction: Fill-in the blanks to complete the sentences. Write your answers on a
separate sheet of paper.

1. ________ is the measure of a central angle of a circle that intercepts an arc equal
in length to the radius.
2. A unit circle is the circle centered at coordinate ____ with a length radius of
____unit.

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 3. If the total arc length (s) of a unit circle is 2π, then half of this is _____.
3
4. The arc length (s) of a quarter of a unit circle is _____, while 𝜋 is an arc length
2
for _____ of the unit circle.
5. The equation of a unit circle is _______________.
6. The total measurement (in degrees) of the central angle of one revolution of the
unit circle is _____, and it is equal to ____ measurement (in radian)

What I Need to Remember

• The measure of an angle in radians is defined as the length of the arc cut
off by one radius length.
• Radian is the standard unit used to measure central angles in
Mathematics.
• One radian is the linear measure of a central angle of a circle that intercepts
an arc length equal to the radius of that circle. It is approximately equal to
57.3°.
• The unit circle is the circle of radius 1 centered at the origin in the xy-plane.

Lesson Conversion of Degrees to Radian

2
Measure, and Vice Versa

What I Need to Know

At the end of this lesson, you are expected to:

o identify the conversion factor to be used; and

o convert angle measures, degrees to radian and radian to degrees.

What’s In
Ferris wheel is an entertaining ride and will make you
ponder about how such a colossal moving structure
was built. To make 12 equidistant spokes, we needed
a consistent rule to follow. Therefore, we idealized the
unit circle and transformed the intervals in terms
of pi (π) and marked them on your sketch.

Image: https://creazilla.com/nodes/36695-ferris-wheel-clip

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 What’s New

Activity 2.1: The Equivalent Degrees and Radian Measures of an Angle

Instruction: Fill-in the blank in the table and write your answer on a separate sheet of
paper, applying your basic operation in multiplication.

Equivalent
1 revolution/1 complete turn
measures
(360 degrees)/ 2π radian
Degree Form Radian Form (degrees to
(radius = 1 unit)
radian, vice
versa)
1
of 360°= _____ 1
2 of 2π = ______ 180° = ______
2
1 1 𝜋
of 360°= _____ of 2π = ______ _____ =
360 °
2π 4 4 2
1
of 360°= _____ 1
6 of 2π = ______ 60° = ______
6
1 1
of 360°= ____ of 2π = _____ _____ = ______
12 12

After completing the table above, answer the following questions:

1. If 180° is half of a circle, how many radians are there?
2. If 90° is a quarter of a circle, how many radians are there?
3. If 270° is three-quarters of a circle, how many radians are there?
4. If 45° is ________________(hint: part of a circle) of a circle, so there are _______
radians.

What Is It

Conversion of Degree Measurement to Radian Measurement and Vice Versa

(Adapted from the book of Kennan, M. 2017)

Being able to convert degrees to radians is beneficial because radians are commonly used
in trigonometric functions, but most people are more familiar with degrees. It is easy to
convert between degree measurement to radian measurement since one revolution sweeps
2𝜋𝑟
out the entire circumference 2πr, one revolution has radian measure = 2𝜋, from this we
𝑟
can find the radian measure of other central angles (degrees form), so it follows that 360°
equals 2π radian.

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To derive the conversion factor for degrees to radian:

360° = 2π radian (divide both sides of the equation by 360)

2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛
1° = (simplify the right side of the equation)
360

𝜋
1° = (conversion factor for degrees to radian)
180

To derive the conversion factor for radian to degrees:

2π radian = 360° (divide both sides of the equation by 2π)

360°
1 radian = (simplify the right side of the equation)
2𝜋

180
1 radian = (conversion factor for radian to degrees)
𝜋

𝜋
Multiply by Converting degrees to radian, and vice
180
versa
1.To convert a degree measure to radian,
𝜋
degrees radians multiply it by .
180
2.To convert a radian measure to degree,
180
multiply it by
180 𝜋
Multiply by
𝜋
Figure 6: Diagram in converting degrees to radian
and vice versa
(https://images.app.goo.gl/MPE7mjcMmghLyWUn8)

Example 1:

Express 75° and 240° in radians.

5
𝜋 5𝜋 5𝜋
Solution: 75 ( 180 ) = 75° = radians
12 12
12

𝜋 4𝜋 4𝜋
240 ( )= 240° = radians
180 3 3
3

9
But there are times that the given degree measure is in the degree/minute/second form (or
DMS mode). However, instead of using minutes and seconds, we shall usually convert them
to fractions of 1 degree. For instance, if the measure of an angle is 27° 13´ 30”. We define
the minutes and seconds as follows:

1
1 minute, written 1´, = of 1° or 1° = 60´
60
1
1 second, written 1´, = of 1´ or 1´ = 60" or 1° = 3600"
60

Example 2:
To convert 27° 13´ 30” to radian measure, first we must express it in degree
measure containing a portion of a decimal, we have:

Solution:
1° 1°
Step 1: 27° + 13´ ( ) + 30” ( ) (hint: all similar units are cancelled out)
60´ 3600”

27°+ 0.216° + 0.008 (express the portion of decimal to nearest thousandths)

27.224° (the degree measure containing a portion of the decimal)

𝜋 3403 Note: Use the fraction key in dividing 27.224

Step 2: 27.224 (180 ) = 𝜋
22500 and 180, then press the “equal” key to get
3403
.
22500

Example 3:
5
What is the degree measure of an angle that has a measure 𝜋 radians?
9

Solution: 20
5
9
𝜋 ( 180
𝜋
) = 100° 5
9
𝜋 = 100°

10
 What’s More

Activity 2.2: NOW IT’S YOUR TURN!

Instruction: (3 points for each item). Convert the following angle measures from degrees
to radians or vice versa. Write your answer on a separate sheet of paper (show your
solution)

4𝜋
1. 150° = ________ radians 5. = ________ °
3
7𝜋
2. 225° = ________ radians 6. = ________ °
4
7𝜋
3. − 60° = ________ radians 7. − = ________ °
6
3. 55.55° = _______ radians 8. 110° 50´ 30" = ________ radians

What I Need to Remember

• Angle measurement in degrees must be converted to radian measure for

it is necessary in dealing with trigonometric functions.

𝜋
• To convert a degree measure to radian, multiply it by .
180

180
• To convert a radian measure to a degree, multiply it by .
𝜋

• To convert degree measures with DMS mode, express it first in a portion

of degree with decimal.

Lesson Standard Position of Angles and

3 Coterminal Angles

What I Need to Know

At the end of this lesson, you are expected to:

o illustrate standard position of angle measures (degrees and radian form);

o identify the quadrant which angles terminate and;
o illustrate coterminal angles and give one positive and one negative coterminal
angle of the given angle measures.

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 What’s In

Instruction: Find and encircle 8 words which are related to our lesson in this module.
Write the words in a separate sheet of paper. (Hints: words begin with
letters P, Q, T, A, I, M and C)

LOOK FOR ME!

What’s New (Presentation of the Lesson)

Activity 3.1 ILLUSTRATE ME!
Instruction: Do and follow the steps given below on a sheet of bond paper.

Materials needed: 1 whole sheet of bond paper, protractor, ruler, colored pens

STEP 1: Make an xy-plane and draw all the given angle measures below. Make
sure that one (initial side) of the rays of an angle always coincide with
the positive x-axis. Use different color of pens in each angle.

5𝜋 3𝜋
45°, π , − 315° , − 𝜋 , 405°, , 1050°, − , − 30°
4 4

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 Note: 1) Positive angle measure rotates counterclockwise while negative rotates
clockwise. (Refer to previous module 1).
2) Initial side is a stationary ray of an angle and always coincide with the positive
x-axis while terminal side is the rotating ray of any angle.

Rubrics for scoring the illustrations.

Points Description
16 -20 Illustrated all the given angle measures properly
11 -15 Illustrated 7 out of 9 items of the given angle measures properly
6 -10 Illustrated 5 out of 9 items of the given angle measures properly
1-5 Illustrated 2 out of 9 items of the given angle measures properly
0 Illustrated NONE of the given angle measures.

What Is It (Discussion)

When is an angle said to be drawn in standard position?

Standard Position of Angles

(Cited from the book of Cabral, E. et al, 2010)

At this point, we also extend our allowable rotation to include angles which encompasses
.more than one revolution. For example, 450°, we start with an initial side, rotate
counterclockwise one complete revolution (to take care of the “first” 360°) then continue
with an additional 90°, as shown below:

Figure 1: Standard position of 450°

13
According to Cabral, E (2010), an angle is said to be in standard position if its vertex is
the origin and its initial side coincides with the positive x-axis. And if the terminal side
of an angle lies on one of the coordinate axes, it is called a quadrantal angle. Two angles
in standard position are called coterminal if they share the same terminal side.

Another illustration of angles in their Standard Position, as shown in the figure 2 below:

Figure 2: Angles in Standard Position

Two angles in standard position are COTERMINAL if their sides coincide. In figure 2, the
angles in ( a ) and ( c ) are coterminal.

Some more illustrations of coterminal angles drawn in one xy-plane, as in figure 3 below:

α = 120°

β = − 240°

Figure 3: Two different angles in standard position that are coterminal.

Examples of finding coterminal angles (positive and negative) both in degrees and radian.

a) Find angles that are coterminal with the angle ϴ = 30° in standard position.
𝜋
b) Find angles that are coterminal with the angle ϴ= in standard position.
3

14
Solutions:

a) To find positive angles that are coterminal with ϴ, we add any multiple of 360°or
ϴ ± 𝟑𝟔𝟎𝒌, where k stands for any integer except zero.

Thus, 30° + (360°)1 = 390°, k = 1

30° + (360°)2 = 750°, k = 2
30° + (360°)3 = 1110°, k = 3

So, 390°, 750°, 1110° are one of the positive coterminal angles with ϴ = 30°
and there are infinite number of positive coterminal angles since k is any
integer except zero.

To find negative angles that are coterminal with ϴ, we subtract any multiple of
360°.

Thus, 𝟑𝟎° − 𝟏(360°) = − 330°, k = − 1

𝟑𝟎° − 𝟐(360°) = − 690°, k = − 2
𝟑𝟎° − 𝟑(360°) = − 1050°,k = − 3
So, − 330°, − 690°, − 1050° are one of the negative coterminal angles with
ϴ = 30° and there are also infinite number of negative coterminal angles
since k is any integer except zero.

Therefore 30°, 390°, 750°, 1110°, − 330°, − 690°, − 1050° are coterminal angles.
There are more angle measures which are still coterminal with the six highlighted
angle measures above. Observe that the degree measure of coterminal angles
differ by multiples of 360°.

b) To find positive angles that are coterminal with ϴ, we add any multiple of 2π or
ϴ ± 𝟐𝛑𝒌, where k stands for any integer except zero.

𝜋 7𝜋
Thus, + 2π(1) = ,k=1
3 3

𝜋 13𝜋
+ 2π(2) = ,k=2
3 3

𝜋 19𝜋
+ 2π(3) = ,k=3
3 3

7𝜋 13𝜋 19𝜋
So, , , are one of the positive coterminal angles (radian form)
3 3 3
𝜋
with ϴ = and there are also infinite number of negative coterminal angles
3
since k is any integer except zero.

15
 To find negative angles that are coterminal with ϴ, we subtract any multiple of 2π.

𝜋 5𝜋
Thus, − 𝟏(2𝜋) =− , k = −1
3 3

𝜋 11𝜋
− 𝟐(2𝜋) =− , k = −2
3 3

𝜋 17𝜋
− 𝟑(2𝜋) =− , k = −3
3 3

5𝜋 11𝜋 17𝜋
So, − , − , − are one of the negative coterminal angles (radian
3 3 3
𝜋
form with ϴ = and there are also in infinite number of negative
3
coterminal angles since k is any integer except zero.

𝜋 7𝜋 13𝜋 19𝜋 5𝜋 11𝜋 17𝜋

Therefore , , , ,− , − ,− are coterminal angles.
3 3 3 3 3 3 3
There are more angle measures which are still coterminal with the five highlighted
angle measures(radian) in solution (b). Observe that the radian measure of
coterminal angles differ by multiples of 2π.

Two angles are coterminal if and only if their degree measure differ
by 360k , where k ∈ Z and k is any integer except zero.
Similarly, two angles are coterminal if and only if their radian
measures differ by 2πk, k ∈ Z.
Note: Recall your Mathematics 7 for the meaning of the symbols, ∈ and Z (about SET)

Other examples on finding coterminal angles:

c) Find angle measure coterminal with − 380° that has measure:

between 0° and 360° and between − 360° and 0°.

Solutions:

A negative angle moves in a clockwise direction, the angle − 380° lies in QIV.

1) − 380° + 2(360°) = 340°(coterminal with − 380° which is between 0° and 360°)

2) − 380° +1(360°) = −20° (coterminal with − 380° which is between − 360° and 0°)

16
 What’s More
Activity 3.2: NOW IT’S YOUR TURN!

A. Instruction: The measures of two angles in standard position are given. Determine
whether the paired angles are coterminal or not by solving using
multiples of 360° or 2π.

1) 70° , 1140° 3) − 30° , 330°

5𝜋 17𝜋 32𝜋 11𝜋

2) , 4) ,
6 6 2 3

B. Instruction: Find and illustrate the standard position of the described coterminal
angle with the given angle.

1) Biggest negative angle measure coterminal with 375°

8𝜋
2) Smallest positive angle measure coterminal with − ?
3

Rubrics for scoring illustrations in Activity B

Points Description
15 Illustrated all the given items properly and neatly.
12 -14 Illustrated all the given items properly.
9 -11 Illustrated 1 out of 2 given items properly and neatly
6-8 Illustrated 1 out of 2 given items properly.
1-5 Illustrated all the given items neatly only.

What I Need to Remember

• An angle is in Standard Position if it is drawn in the xy-plane with its

vertex at the origin and its initial side on the positive x – axis.
• Quadrantal angle is angle in standard position where its terminal side
coincides with any coordinate axes.
• Coterminal angles are angles in standard position that have a common
terminal side.
• Two angles are coterminal if and only if their degree measures differ
by 360k, where k ∈ Z except zero.
• Two angles are coterminal if and only if their radian measures differ by
2πk, where k ∈ Z except zero.
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 What I Can Do

Activity 3.2: LET’S GET REAL

A. Instruction: Do what is being asked in Set A and Set B. Write your answer on
separate sheet of paper (show your solutions).

Set A: Feel to experience!

(Direction: Starting point is located at bottommost

compartment on the picture on the left and spokes are placed
equally from each other)

1) If you and your friends are riding on a Ferris wheel, and

stops when a group of girls ride on the compartment
directly below your compartment, how far did you and
your friends move from the starting point?

2) If your parents are riding and the rotation of the Ferris

wheel stops at two-thirds of one rotation, how far are your
parents from the starting point? (Express your answer in
radian)

Set B: Solve the problem

1. Suppose you are jogging around the Fuente Osmena rotunda, and stops at
three-fifths of the rotunda to drink water, how far did you jog from your
starting point? (Express your answer in radian)

B. Classify each angle measure according to where its terminal side lies and give one (1)
(positive and negative) coterminal angles. (show your solutions) Then illustrate and
label the corresponding standard position of each angle.

1) 135° 3) − 200°

5𝜋 10𝜋
2) 4) −
4 6

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 Assessment (Posttest)

Instruction: Choose the letter of the correct answer. Write your chosen answer on
a separate sheet of paper.

1. Which of the following is to be multiplied if we convert radian measure to degree

measure?
𝜋 𝜋 180 360
A. B. C. D.
180 360 𝜋 𝜋

2. Which of the following is TRUE about a unit circle?

A. circumference is 2πr C. radius is 1 unit
B. circumference varies D. radius varies

3. Which of the following is NOT used as a symbol to label an angle?

A. Θ B. μ C. γ D. β

4. Which of the following arc length is the shortest?

5𝜋 5𝜋 5𝜋 5𝜋
A. B. C. D.
6 4 3 2
5. Which of the following radian measures is greater than 180°?
𝜋 𝜋 7𝜋 7𝜋
A. B. C. D.
8 5 8 5

6. Which of the following is the standard position of an angle?

A. B. C. D.

𝜋
7. How many equal arcs is the circle to be divided if each arc measures ?
20
A. 10 B. 20 C. 30 D. 40

8. Which of the following angle measures had its terminal side that coincides in any
of the coordinate axes?
𝜋 𝜋 7𝜋 10𝜋
A. − B. − C. D.
3 4 2 3
9. Which of the following radian measures is three-fourths of a revolution starting
from point ( 1, 0 ), and rotates in a clockwise direction?
𝜋 3𝜋
A. − B. − 𝜋 C. − 2𝜋 D. −
2 2

19
 19𝜋
10. Which of the following is coterminal with ?
6
5𝜋 8𝜋
I. − 150° II. 780° III. − 6 IV.
6
A. I & II B. I & III C. II & III D. III & IV
17𝜋 31𝜋
11. What is the value of ϴ, if two of its coterminal angles are − 6 and 6 ?
A. 210° B. 240° C. 270° D. 300°

12. Which of the following is coterminal angles in its standard position?

A. C.

B. D.

For items 13-14, refer to the color wheel on the

picture found at your right.
13. What is the radian measure from orange to blue?
𝜋
A. C. π
8
5𝜋
B. D. 2π
4

14. What is the radian measure covered by any 3

different colors? commons.wikimedia.org/wiki/File:Intermediate.Colours.j
3𝜋 3𝜋 pg
A. C.
2 6
3𝜋 3𝜋
B D.
4 8
15. Mercadejas family has a total income of ₱ 30,000, 60% of it is for food. If this is
represented through the pie graph, what is the radian measure equivalent to this
portion?
6𝜋 6𝜋 6𝜋 6𝜋
A. B. C. D.
5 4 3 2

20
REFERENCES

Book
Cabral E.A, E.T. (2010). Precalculus. Ateneo de Manila University Press. Manila 2010.
pp 234 – 235
Garces, I. J., Precalculus: Teaching Guide for Senior High School. Sunshine Interlinks
Publishing House, Incorporated.Quezon City. 2016. pp. 122 - 127
Keenan, E., Gantert, A. Amsco’s Mathematics B: Teacher’s Manual).AMSCO School
Publications, Incorporated.New York. 2002. pp 181-186

PDF File
Zeager, J. Precalculus: Version ⌊𝜋⌋ = 3Corrected editon . Lorain County Community
College. 2010. p 701
Activities
Keenan, E., Gantert, A. Amsco’s Mathematics B: Teacher’s Manual ).AMSCO School
Publications, Incorporated.New York. 2002. p 186

Figures
Figure 1: Angles form between rays

Figure 2: Zeager, J. Precalculus: Version ⌊𝜋⌋ = 3Corrected editon. Lorain County

Community College. 2010. p 701

Figure 3: Zeager, J. Precalculus: Version ⌊𝜋⌋ = 3Corrected editon . Lorain County

Community College. 2010. p 701

Figure 4: Garces, I. J. Precalculus:Teaching Guide for Senior High School . Sunshine

Interlinks Publishing House, Incorporated.Quezon City. Quezon City 2016. p 128

Figure 5: Zeager, J. Precalculus: Version ⌊𝜋⌋ = 3Corrected editon . Lorain County

Community College. 2010. p 701

Figure 6: https://images.app.goo.gl/MPE7mjcMmghLyWUn8

Images
Image 1: retrieved from https://creazilla.com/nodes/36695-ferris-wheel-clipart on
October 18, 2020
Image 2: retrieved from commons.wikimedia.org/wiki/File:Intermediate.Colours.jpg
October 20, 2020

23
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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