Learning Area Pre-Calculus Grade Level 11
W2 Quarter 2/4 Date May 14, 2021
I. LESSON TITLE Angles in standard position and coterminal angles
II. MOST ESSENTIAL LEARNING
Illustrate angles in standard position and coterminal angles
COMPETENCIES (MELCs)
III. CONTENT/CORE CONTENT Key concepts of circular functions
IV. LEARNING PHASES AND LEARNING ACTIVITIES
I. Introduction (Time Frame: 10 minutes)
We can see angles in our daily lives. Angles are being used in various fields like engineering, medical imaging,
electronics, astronomy, geography, and a lot more. Added to that, surveyors, pilots, landscapers, designers,
soldiers, and many other professions heavily use angles in a variety of practical tasks. In this lesson, we will be
studying angles in standard position and coterminal angles.
At the end of the session, students are expected to:
1. Illustrate angles in standard position.
2. Describe and illustrate coterminal angles.
3. Determine angle in degrees and radian coterminal to the given angle.
A more general idea of the angle together with the concept of rotation is necessary to
understand functions related to the circle.
An angle is formed by two rays with a common endpoint. One side of the angle rotates
about the common endpoint and the other side remains stationary. The stationary ray is the initial side of the angle, and the
rotating ray is the terminal side.
Since rotation may either be clockwise or counterclockwise,
an arrow is used to indicate its direction. Angles are
measured to determine the amount of rotation the initial side
makes to coincide with the terminal side. This measure is
expressed either in degrees or in radians.
An angle is said to be positive if the ray rotates in a
counterclockwise direction, and the angle is said to
be negative if it rotates in a clockwise direction.
Counterclockwise rotation Clockwise rotation
Positive angle Negative angle
Learning Task #1
Determine which quadrant or axis the following angles lie.
1. 387°
2. 1 250°
3. 2 000°
19𝜋
4. 4 𝑟𝑎𝑑
5. 7𝜋 𝑟𝑎𝑑
6. −450°
7𝜋
7. − 3 𝑟𝑎𝑑
5𝜋
8. 6 𝑟𝑎𝑑
9. −12𝜋 𝑟𝑎𝑑
10. −900°
D. Development (Time Frame: 30 minutes)
Learning Task #2
Consider the following figures,
C D
A B
Guide questions:
1. Where does the initial side of each angle lie? Can you identify these angles?
2. Which of these angles have the same terminal side? Can you name these angles?
3. What do we call these angles whose initial sides lie on the positive x-axis?
4. What do we call these angles having the same terminal sides?
Angles in Standard position
An angle in standard position if it is constructed in a rectangular
coordinate system, such that its vertex falls on the origin and one of
its sides coincides with the positive side of the x-axis. The side which
coincides with the positive side of the x-axis is the initial side and other
side is the terminal side.
Learning Task # 3:
In a rectangular coordinate plane, illustrate the following angles in standard position showing the initial and terminal sides.
7𝜋
1. 450° 6. 6
𝑟𝑎𝑑
11𝜋
2. −225° 7. 𝑟𝑎𝑑
2
13𝜋
3. −380° 8. − 6
𝑟𝑎𝑑
9𝜋
4. 750° 9. − 𝑟𝑎𝑑
4
5. 1550° 10. −4𝜋 𝑟𝑎𝑑
Coterminal Angles
Two angles in standard position that have a common terminal side are called . Observe that the degree measures
of coterminal angles differ by multiples of . The next examples are coterminal angles because both angles have their
initial sides at the positive x-axis and terminate at the same terminal side.
Two angles are coterminal if and only if their degree measures differ by
Similarly, two angles are coterminal if and only if their radian measures differ by
There are infinitely many coterminal angles. One way to find
the measure of an angle that is coterminal with angle is to
add or subtract integer multiples of . For example, angles
measuring and are coterminal.
Illustrative Examples
Give four other examples coterminal with the given angles:
1. 𝜃 = 55°
Any angle coterminal with 55° can be written in the form 55° + 360°𝑘, 𝑤ℎ𝑒𝑟𝑒 𝑘 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟.
𝐼𝑓 𝑘 = 1, 𝑡ℎ𝑒𝑛 55° + 360°(1) = 415°
𝐼𝑓 𝑘 = 2, 𝑡ℎ𝑒𝑛 55° + 360°(2) = 775° Therefore are angles coterminal with .
𝐼𝑓 𝑘 = −1, 𝑡ℎ𝑒𝑛 55° + 360°(−1) = −305°
𝐼𝑓 𝑘 = −2, 𝑡ℎ𝑒𝑛 55° + 360°(−2) = −665°
2. 𝜃 = −310°
Any angle coterminal with −310° can be written in the form −310° + 360°𝑘, 𝑤ℎ𝑒𝑟𝑒 𝑘 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟.
𝐼𝑓 𝑘 = 1, 𝑡ℎ𝑒𝑛 − 310° + 360°(1) = 50°
𝐼𝑓 𝑘 = 2, 𝑡ℎ𝑒𝑛 − 310° + 360°(2) = 410° Therefore are angles coterminal with .
𝐼𝑓 𝑘 = −1, 𝑡ℎ𝑒𝑛 − 310° + 360°(−1) = −670°
𝐼𝑓 𝑘 = −2, 𝑡ℎ𝑒𝑛 − 310° + 360°(−2) = −1 030°
5𝜋
3. 𝜃= 𝑟𝑎𝑑
3
5𝜋 5𝜋
Any angle coterminal with 𝑟𝑎𝑑 can be written in the form + 2𝜋𝑘, 𝑤ℎ𝑒𝑟𝑒 𝑘 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟.
3 3
5𝜋 11𝜋
𝐼𝑓 𝑘 = 1, 𝑡ℎ𝑒𝑛 3
+ 2𝜋(1) = 3
𝑟𝑎𝑑
Therefore are angles coterminal with .
5𝜋 17𝜋
𝐼𝑓 𝑘 = 2, 𝑡ℎ𝑒𝑛 3
+ 2𝜋(2) = 3
𝑟𝑎𝑑
5𝜋 𝜋
𝐼𝑓 𝑘 = −1, 𝑡ℎ𝑒𝑛 3
+ 2𝜋 (−1) = − 3 𝑟𝑎𝑑
5𝜋 7𝜋
𝐼𝑓 𝑘 = −2, 𝑡ℎ𝑒𝑛 3
+ 2𝜋 (−2) = − 3
𝑟𝑎𝑑
E. Engagement (Time Frame: 15 minutes)
Learning Task #4.
Find the angle between 0° 𝑎𝑛𝑑 360° (if in degrees) or between 0 𝑟𝑎𝑑 𝑎𝑛𝑑 2𝜋 𝑟𝑎𝑑 (if in radians) that is coterminal with the
given angle.
1. 595°
2. 876° 88′ 44"
3. −900° 56′40"
29𝜋
4. 𝑟𝑎𝑑
6
5. −20 𝑟𝑎𝑑
6. 695°
3𝜋
7. − 𝑟𝑎𝑑
2
8. 4000°
9. 10 𝑟𝑎𝑑
55𝜋
10. 𝑟𝑎𝑑
4
A. Assimilation (Time Frame: 15 minutes)
Learning Task #5.
Find the angle between − 360° 𝑎𝑛𝑑 0° (if in degrees) or between −2𝜋 𝑟𝑎𝑑 𝑎𝑛𝑑 0 𝑟𝑎𝑑 (if in radians) that is coterminal with the
given angle.
1. 595°
2. 876° 88′ 44"
3. −900° 56′40"
29𝜋
4. 𝑟𝑎𝑑
6
5. −20 𝑟𝑎𝑑
6. 695°
3𝜋
7. − 𝑟𝑎𝑑
2
8. 4000°
9. 10 𝑟𝑎𝑑
55𝜋
10. 𝑟𝑎𝑑
4
V. ASSESSMENT (Time Frame: 30 minutes)
(Learning Activity Sheets for Enrichment, Remediation, or Assessment to be given on Weeks 3 and 6)
A. Illustrate the following angles measured represented by each rotation in a Cartesian plane. Express your answers both
in degrees and in radian measures.
1. 3.5 rotations clockwise
2. 1.5 rotations counterclockwise
3. 2.25 rotations clockwise
4. 2 rotations counterclockwise
5. 7.75 rotations clockwise
B. If each angle in standard position, determine a coterminal angle that is between 0° 𝑎𝑛𝑑 360° (if in degrees) or
between 0 𝑟𝑎𝑑 𝑎𝑛𝑑 2𝜋 𝑟𝑎𝑑 (if in radians). State the quadrant or axis in which the terminal side lies.
1. 1005°45′45"
2. −269°75′123"
3. −979°
17𝜋
4. 𝑟𝑎𝑑
3
21𝜋
5. − 𝑟𝑎𝑑
4
VI. REFLECTION (Time Frame: 5 minutes)
● Communicate your personal assessment as indicated in the Learner’s Assessment Card.
Personal Assessment on Learner’s Level of Performance
Using the symbols below, choose one which best describes your experience in working on each given task. Draw it in the column
for Level of Performance (LP). Be guided by the descriptions below:
- I was able to do/perform the task without any difficulty. The task helped me in understanding the target content/ lesson.
- I was able to do/perform the task. It was quite challenging, but it still helped me in understanding the target content/lesson.
? – I was not able to do/perform the task. It was extremely difficult. I need additional enrichment activities to be able to do/perform this
task.
Learning Task LP Learning Task LP Learning Task LP Learning Task LP
Number 1 Number 3 Number 5 Number 7
Number 2 Number 4 Number 6 Number 8
VII. REFERENCES Precalculus DepEd Learner’s Material
DepEd Advanced Algebra, Trigonometry and Statistics
Prepared by: Joralie J. Mendoza Checked by: Severa C. Salamat
Iriz D. Pinuela
