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Radian & Degree Measureand The Unit CircleLearner's Module in Pre- CalculusQuarter 2 ● Module 1 (Weeks 1 and 2)GERTRUDES D. BAGANO DeveloperDepartment of Education • Cordillera Administrative RegionNAME:________________________ GRADE AND SECTION ____________TEACHER: ____________________ SCOR...

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Radian & Degree Measure and The Unit Circle Learner's Module in Pre- Calculus Quarter 2 ● Module 1 (Weeks 1 and 2)

GERTRUDES D. BAGANO Developer Department of Education • Cordillera Administrative Region NAME:________________________ GRADE AND SECTION ____________ TEACHER: ____________________ SCORE _________________________

Republic of the Philippines DEPARTMENT OF EDUCATION Cordillera Administrative Region SCHOOLS DIVISION OF BAGUIO CITY No. 82 Military Cut-off, Baguio City

Published by: DepEd Schools Division of Baguio City Curriculum Implementation Division Learning Resource Management and Development System

COPYRIGHT NOTICE 2020

Section 9 of Presidential Decree No. 49 provides: “No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency of office wherein the work is created shall be necessary for exploitation of such work for profit.” This material has been developed for the implementation of K-12 Curriculum through the DepEd Schools Division of Baguio City - Curriculum Implementation Division (CID). It can be reproduced for educational purposes and the source must be acknowledged. Derivatives of the work including creating an edited version, an enhancement or a supplementary work are permitted provided all original work is acknowledged and the copyright is attributed. No work may be derived from this material for commercial purposes and profit.

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PREFACE This module is a project of the DepEd Schools Division of Baguio City through the Curriculum Implementation Division (CID) which is in response to the implementation of the K to 12 Curriculum. This Learning Material is a property of the Department of Education, Schools Division of Baguio City. It aims to improve students’ performance specifically in Mathematics. Date of Development Resource Location Learning Area Grade Level Learning Resource Type Language Quarter/Week

November 2020 DepEd Schools Division of Baguio City Mathematics 11 Module English Q2/W1 & W2

Learning Competency Codes

STEM_PC11AG- IIa -1 STEM_PC11AG- IIa -2 STEM_PC11AG- IIa -3 1. Illustrate the unit circle and the relationship between the linear and angular measures of a central angle in a unit circle. 2. Convert degree measure to radian measure and vice versa. 3. Illustrate angles in standard position and coterminal angles.

Learning Competencies

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ACKNOWEDGEMENT The developer wishes to express her gratitude to those who helped in the development of this learning material. The fulfillment of this learning material would not be possible without these people who gave their support, helping hand, understanding, and wisdom. Development Team Developer: Gertrudes D. Bagano Layout Artists: Jopie B. Ferrer School Learning Resources Management Committee Ma. Joan D. Andayan School Principal Pia P. Duligas Assistant School Head Genevieve C. Tudlong School LR Coordinator Aurea D. Daweng Master Teacher Specialist Kathy M. Papcio Learning Area Coordinator Quality Assurance Team Francisco C. Copsiyan Leticia A. Hidalgo Niño E. Martinez

EPS – Mathematics PSDS – District 7 Head Teacher VI (Mathematics)

Learning Resource Management Section Staff Loida C. Mangangey EPS – LRMDS Christopher David G. Oliva Project Development Officer II – LRMDS Priscilla A. Dis-iw Librarian II Lily B. Mabalot Librarian I Ariel Botacion Admin. Assistant CONSULTANTS JULIET C. SANNAD, EdD Chief Education Supervisor – CID CHRISTOPHER C. BENIGNO, PhD OIC – Asst. Schools Division Superintendent MARIE CAROLYN B. VERANO, CESO V Schools Division Superintendent

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TABLE OF CONTENTS COPYRIGHT NOTICE ................................................................................................ii PREFACE .................................................................................................................. iii ACKNOWEDGEMENT ...............................................................................................iv TABLE OF CONTENTS ……………………………………………………………………v TITLE PAGE ………………………………………………………………………………...1 What I Need to Know ................................................................................................. 2 What I Know ............................................................................................................... 3 What’s In .................................................................................................................... 4 What’s New ................................................................................................................ 5 What Is It …………………………...………………………………………………………..5 Lesson 1.1. Trigonometric Definition of an Angle ............................................. 5 Lesson 1.2. The Degree Measure .................................................................... 7 Lesson 1.3. Angles in Standard Position .......................................................... 9 Lesson 1.4. Radian Measure and Arc Length ................................................ 13 Lesson 1.5. Converting Degree Measure to Radian Measure ........................ 17 Lesson 1.6: The Unit Circle ............................................................................. 18 What’s More ............................................................................................................. 29 What I Have Learned ............................................................................................... 31 What I Can Do .......................................................................................................... 32 Post-Assessment ..................................................................................................... 33 Additional Activities .................................................................................................. 34 Answer Key .............................................................................................................. 35 References ............................................................................................................... 36

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Radian & Degree Measure and The Unit Circle Learner's Module in Pre- Calculus Quarter 2 ● Module 1 (Weeks 1 and 2)

GERTRUDES D. BAGANO Developer

Department of Education • Cordillera Administrative Region 1

What I Need to Know This module is a lesson in Angles in Trigonometry, Chapter 1 of Pre- Calculus Second Quarter and is divided into the following subtopics. Lesson 1.1. Trigonometric Definition of an Angle Lesson 1.2. The Degree Measure Lesson 1.3. Angles in Standard Position Lesson 1.4. Radian Measure and Arc Length Lesson 1.5: Conversion of Degree Measure to Radian Measure, and Vice Versa Lesson 1.6. The Unit Circle HOW TO USE THIS MODULE This module discusses lessons in identifying conic section. To make the most out of them, you need to do the following: 1. Scan the list of Learning Objectives to get an idea of the knowledge and skills you are expected to gain and develop as you study the module. These outcomes are based on the content standards, performance standards, and learning competencies of Pre- Calculus 7. 2. Take the What I Know or the Pre- Assessment. Your score will determine your knowledge of the lessons in the module. 3. Each Lesson aims to develop one of the learning objectives set for the module. It starts with an activity that will help you understand the lesson and meet the required competencies. 4. Write down points for clarification in your lecture notebook. You may discuss these points with your teacher. 5. Perform all activities. 6. At the end of the module, take the Post-Assessment to evaluate your overall understanding about the lessons. LEARNING OBJECTIVES: Through the different activities in the module, the learners will be able to: 1. define and explore angles in relation to trigonometry; 2. define degree measure of an angle; 3. covert decimal degree (DD) measures of angles into decimal, minute, second (DMS) and vice versa; 4. visualize angle measures beyond 3600 or less than – 3600; 5. draw angles in standard form; 6. find the co-terminals of angles; 7. define quadrantal angles; 8. define radian and arc length; 9. convert degree measure to radian measure and vice versa;

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10. define unit circle; 11. locate quadrantal angles and special angles in the unit circle; and 12. illustrate the terminal points of the angle in the unit circle.

What I Know PRE- ASSESSMENT Directions: Choose the letter that corresponds to the correct answer. Write your answers on your lecture notebook. A. Write T if the statement is true, otherwise, write F. 1. If two angles are co-terminal, then their initial and terminal sides are the same. 2. An angle whose terminal side lies in a quadrant is called a quadrantal angle. 3. The degree measures of quadrantal angles are divisible by 900. 𝜋 4. The radian equivalent of 600 is . 6 5. The supplementary angle of 600 is 300.

B. Choose the letter that corresponds to the correct answer. 1. Find the measure of a central angle that intercepts an arc length of 6m in a circle with radius of 2.5 m. A. 2.4 rad B. 2.40 C. 0.42 rad D. 0.420 0 2. Which of the following is a co-terminal angle of 45 ? A. 3150 B. – 3150 C. 1350 D. – 1350 3. Find a positive angle co-terminal with 300. A. 3300 B. 3000 C. 1500 D. 3900 4. What is 900

3𝜋 2

𝑟𝑎𝑑 in degree measure?

A. B. 1800 C. 2700 5. Which of the following is NOT a quadrantal angle? A. 300 B. 1800 C. 2700 6. What is the terminal point of the central angle 300? 1 √3 ) 2 3𝜋

A. (2 ,

7. What is 450

4

√3 1

1 √3 ) 2

B. ( 2 , 2)

C. (− 2 ,

D. 3600 D. 3600 D. (−

√3 1 , ) 2 2

in degree measure?

A. B. 1350 C. 2250 8. In which Quadrant does the angle 4950 lie? A. Quadrant I C. Quadrant II B. Quadrant III D. Quadrant IV

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D. 3150

9. In which Quadrant does the angle A. Quadrant I C. Quadrant III 10. What is the terminal point of 1

A. (− 2 , −

√3 ) 2

B.

31𝜋

6 √3 (− 2

31𝜋 6

lie? B. Quadrant II D. Quadrant IV

? 1

, − 2)

1 √3 ) 2

C. (− 2 ,

D. (−

√3 1 , ) 2 2

Chapter 1. Angles in Degree and Radian Measure

In Euclidean Geometry, an angle is defined as the figure formed by two rays intersecting at a common endpoint called the vertex of the angle. An angle is represented by the symbol ∠. The angle below is ∠ KIM Looking Back The word angle has been derived from the Latin word ‘ Angulus’, which means “ a little bending”. The concept of angle was first used by Eudemos, who defined an angle as a deviation from a straight line.

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What’s New The Minute Hand of a Clock From 12 noon to 12:15 PM, how many degrees does the minute hand of a clock travel? The initial and final positions of the minute hand of the clock from 12:00 to 12:15 determine an angle whose measure is -900. Do you know why it is negative?

What Is It Lesson 1.1. Trigonometric Definition of an Angle This section discusses the trigonometric definition of an angle.

Definition An angle is formed by rotating a ray about its endpoint (called the vertex) from some initial position (called the initial side) to some terminal position (called the terminal side). The measure of an angle is the amount of rotation covered between its initial side and its terminal side. If an angle is formed by a rotation in the counterclockwise direction, then the angle measure is positive; and if the rotation is in the clockwise direction, then the angle measure is negative.

Figure 1 shows us an angle formed by a rotating ray. The initial side is KI, the terminal side is IM, and the vertex is point I. The curve arrow tells us the direction and amount of rotation. 5

Figure 1. A Rotation Angle

Table 1: Commonly Used Greek Letter in Naming Angles Name Symbol Alpha 𝛼 Beta 𝛽 Gamma 𝛾 Delta 𝛿 Theta 𝜃 Mu 𝜇 Phi 𝜙 𝑜𝑟 𝜑 𝜆 Lambda Omega 𝜔

We name angles using the capital letters in the English alphabet. We also use the Greek letter. These also denote the measure of angles. That is, 𝜃 may refer to the angle and its measure. Example 1. Which of the following angles have positive measures? Which have negative measures?

a.

c.

b

d.

Solution: Angles 𝛼, 𝛽 , and 𝐴 have positive measures while angle 𝜆 has a negative measure.

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Practice Exercise 1: Complete the following table. Use your lecture notebook to show your answer.

Initial Side

Given Angle

Terminal Side

Vertex

Direction

1.

2.

3.

Lesson 1.2. The Degree Measure This section discusses and illustrates degree measure of an angle. Explore: Scuba Diving Scuba divers use a compass to navigate a straight line. The ring on the compass, called bezel, is turned so that the index needle is aligned over the compass needle. For as long as these are aligned, the divers are swimming in a straight line to their desired location. To return to their starting point, the divers swim in the direction that is 1800 from the original direction. Marina has been swimming such that her compass points 300. Where should the needle point when she swims back to the dock where she started? Returning to the starting point requires the diver to swim in the direction that is 1800 from the original direction. Thus, Marina’s compass needle should point at 2100. 7

As seen in the scuba diving compass, a circle measures 360°. A complete rotation of XY about point X in a counterclockwise direction is equal to 360° or equivalently, 1° = 1

360

rotation.

Definition 1

One degree is the measure of an angle formed by rotating a ray 360 of a complete rotation. The symbol for degree is denoted by (°). An angle measure may also be written in a decimal degree (DD) form. An alternative way is to write it in the degree- minute –second (DMS). That is, each degree is divided into 60 minutes and each minute is divided into 60 seconds. The notation used for minutes and seconds are (‘) and (“) respectively. In symbols, 1° = 60’ and 1’ = 60” In the degree- minute- second (DMS) form, the notation d° m’ s’’ indicates a measure of d° + m’ + s’’. Example 2. Write 24° 12’ 20’’ in decimal degree (DD). Solution: 24° + 12’ •

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60′

+ 20’’ •

= 24° + 0.20 + 0.0056 = 24. 20560

1′

60′′

•

Looking Back This idea of dividing the whole circle into 360 parts came from the Babylonians who believed that the seasons repeated every 360 days.

1°

60′′

Example 3. Change 61. 315° to degree – minute –second (DMS) form. Solution: 61. 315° = 61° + 0.315° = 61° + 0.3150 • = 61° + 18.9’

60′ 10

= 61° + 18’ + 0.9’ • = 61° + 18’ + 54’’ = 61° 18’ 54’’

60′′ 1′

You may also use your calculator in converting decimal degree (DD) form to degree minute – second (DMS) form and vice versa. How? 1. Set your calculator to the degree mode. 2. Keying in of an angle in the DMS form and displaying the same angle in its DD form and vice versa. 8

Lesson 1.3. Angles in Standard Position This section discusses and illustrates angles in standard position. It also reviews supplementary and complementary angles.

Quick Review on Complementary and Supplementary Angles Two angles are said to be complementary if the sum of their measures is 900. Two angles are said to be supplementary if the sum of their measures is 1800. Example: Find the complement and supplement of 39.30. Solution: a. The complement angle of 39.3° is: 90° – 39.3° = 50.70 b. The supplement of 39.30 is : 180° – 39.3° = 140.70°

An angle is said to be in standard position if the following conditions are satisfied: 1. The initial side of the angle is the positive 𝑥 – 𝑎𝑥𝑖𝑠. 2. The vertex of the angle is the origin (0, 0). 3. The angle rotates counterclockwise.

Figure 2. Angle in Standard Position

Note: An angle can have more than one rotation. There is no reason the rotation stops at 36°0. By continuing the rotation, an angle can have more than 360°. Similarly, more than one complete rotation in clockwise direction results to an angle whose measure is less than – 360°. Hence, many different angles have the same initial sides and terminal sides. Such angles are called co- terminal angles. Given an angle 𝜃, there is an unlimited number of angles coterminal with 𝜃. 9

Definition Coterminal angles are angle that have the same initial and terminal sides. Quadrantal angles are angles in standard position in which their terminal sides lie on the coordinate axes (x or y- axis). Note: If an angle is NOT a quadrantal angle, then it is said to be in a certain quadrant where its terminal side lies.

Figure 3. Co- terminal Angles Angles 𝛼 and 𝛽 have the same initial and terminal sides, thus, the angles are co- terminal.

Figure 4. Quadrantal Angle Angle 𝛼 is an angle in standard position and its terminal side lies in the coordinate axis (y – axis). Such is an example of a quadrantal angle.

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Figure 5. Location of an Angle

Example 4. Refer to the figure at the right. a. Which angles are in standard position? b. Which angles are coterminal? Solutions: a. Angles in standard position are as follows: a. Angle 𝛼 b. Angle 𝜇 c. Angle 𝜆 b. Angle 𝜇 and Angle 𝜆 are coterminal angles. Note: Although angles 𝛼 and 𝛽 have the same terminal side AC, their initial side, AD and AB, respectively, are different rays. Example 5. Sketch each of the following angles in standard position and give its location. a. 00 c. - 1800 0 b. 90 d. - 4200

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Solutions: a.

b.

As you notice, the terminal side of the angle lies in the coordinate axis (x – axis). Thus, the angle is quadrantal. The location of the angle is in the positive x – axis.

c.

The terminal side of the angle lies in the coordinate axis (y – axis), thus the angle is quadrantal. The location of the angle is in the positive y – axis.

d.

The terminal side of the angle lies in the coordinate axis (negative x – axis). The location of the angle is in the negative x – axis.

The angle rotates clockwise, thus, the angle is negative. Also, the angle rotates more than 3600.

Practice Exercise 2: Sketch each angle in standard position and give its co-terminal angle 𝜃 where 00 ≤ 𝜃 ≤ 3600. Show your solutions and final answers in your lecture notebook. Check your answers using the key answer. 1. 4370 2. – 1600 3. 9900

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Lesson 1.4. Radian Measure and Arc Length This section discusses and illustrates the radian measure and arc length of a circle.

Definition A central angle is an angle whose vertex is at the origin and the center of a circle. The intercepted arc of a central angle is the arc of the circle swept out by the angle. The central angle is also said to be subtended by the arc. One radian (denoted as 1 rad) is the measure of the central angle of a circle subtended by an arc whose length is equal to the radius of the circle.

From your geometry class, it is said that the measure of a central angle is equal to the measure of its intercepted arc. In trigonometry, the angles in the unit circle are the rotation angles (specifically the angles in standard position) and these angles have the same measure with that of its intercepted arc but is usually (in trigonometry) expressed in radian measure. How is radian measured? Do the following illustration...