Pre calculus for grade 11 Quarter 2 Week6 PDF

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11Pre-CalculusQuarter 2 – Module 6:Trigonometric IdentitiesPre-Calculus – Grade 11 Self-Learning Module (SLM) Quarter 2 – Module 6: Trigonometric Identities First Edition, 2020Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. How...

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11 Pre-Calculus Quarter 2 – Module 6: Trigonometric Identities

Pre-Calculus – Grade 11 Self-Learning Module (SLM) Quarter 2 – Module 6: Trigonometric Identities First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.

Development Team of the Module Writers: Marilou S. Pedregosa, Ana Luz Arwena L. Delizo Editors: Nathaniel A. Galopo, Live C. Angga

Reviewers: Reynaldo C. Tagala Illustrator: Layout Artist: Maylene F. Grigana Cover Art Designer: Ian Caesar E. Frondoza Management Team: Allan G. Farnazo, CESO IV – Regional Director Fiel Y. Almendra, CESO V – Assistant Regional Director

Gildo G. Mosqueda, CEO VI - Schools Division Superintendent Diosdado F. Ablanido, CPA – Asst. Schools Division Superintendent Gilbert B. Barrera – Chief, CLMD Arturo D. Tingson Jr. – REPS, LRMS Peter Van C. Ang-ug – REPS, ADM Jade T. Palomar – REPS, Mathematics

Donna S. Panes – Chief, CID Elizabeth G. Torres – EPS, LRMS Judith B. Alba – EPS, ADM Reynaldo C. Tagala – EPS, Mathematics

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11 Pre-Calculus Quarter 2 – Module 6: Trigonometric Identities

Introductory Message For the facilitator: Welcome to the Grade Trigonometric Identities.

11

Pre-Calculus

Self-Learning

Module

(SLM)

on

This module was collaboratively designed, developed and reviewed by educators both from public and private institutions to assist you, the teacher or facilitator in helping the learners meet the standards set by the K to 12 Curriculum while overcoming their personal, social, and economic constraints in schooling. This learning resource hopes to engage the learners into guided and independent learning activities at their own pace and time. Furthermore, this also aims to help learners acquire the needed 21st century skills while taking into consideration their needs and circumstances. In addition to the material in the main text, you will also see this box in the body of the module:

Not es t o t he T eacher This contains helpful tips or strategies that will help you in guiding the learners.

As a facilitator you are expected to orient the learners on how to use this module. You also need to keep track of the learners' progress while allowing them to manage their own learning. Furthermore, you are expected to encourage and assist the learners as they do the tasks included in the module.

i

For the learner: Welcome to the Grade Trigonometric Identities.

11

Pre-Calculus

Self-Learning

Module

(SLM)

on

The hand is one of the most symbolized part of the human body. It is often used to depict skill, action and purpose. Through our hands we may learn, create and accomplish. Hence, the hand in this learning resource signifies that you as a learner is capable and empowered to successfully achieve the relevant competencies and skills at your own pace and time. Your academic success lies in your own hands! This module was designed to provide you with fun and meaningful 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: What I Need to Know

This will give you an idea of the skills or competencies you are expected to learn in the module.

What I Know

This part includes an activity that aims to check what you already know about the lesson to take. If you get all the answers correct (100%), you may decide to skip this module.

What’s In

This is a brief drill or review to help you link the current lesson with the previous one.

What’s New

In this portion, the new lesson will be introduced to you in various ways such as a story, a song, a poem, a problem opener, an activity or a situation.

What is It

This section provides a brief discussion of the lesson. This aims to help you discover and understand new concepts and skills.

What’s More

This comprises activities for independent practice to solidify your understanding and skills of the topic. You may check the answers to the exercises using the Answer Key at the end of the module.

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What I Have Learned

This includes questions or blank sentence/paragraph to be filled in to process what you learned from the lesson.

What I Can Do

This section provides an activity which will help you transfer your new knowledge or skill into real life situations or concerns.

Assessment

This is a task which aims to evaluate your level of mastery in achieving the learning competency.

Additional Activities

In this portion, another activity will be given to you to enrich your knowledge or skill of the lesson learned. This also tends retention of learned concepts.

Answer Key

This contains answers to all activities in the module.

At the end of this module you will also find:

References

This is a list of all sources 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. Don’t forget to answer What I Know before moving on to the other activities included in the module. 3. Read the instruction 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

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What I Need to Know This module was designed and written with you in mind. It is here to help you master the Conic Sections and System of Nonlinear Equations. 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 to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using. The module is composed of one lesson. Lesson 1 - Trigonometric Identities

After going through this module, you are expected to: 1. identify whether the given equation is an identity or a conditional equation; 2. apply trigonometric identities to solve trigonometric values; and 3. solve situational problems involving trigonometric identities

1

What I Know Find how much you already know about the topic in this module. Take note of the items that you were not able to answer correctly and find the right answer as you go through this module. In all types of test, choose the letter of the best answer. Write the chosen letter on a separate sheet of paper. 1. It is an equation which is satisfied by some values of unknown. A. Identity C. Trigonometric Identity B. Conditional Equation D. Trigonometric Function 2. It is an equation involving trigonometric functions that is true for all values of the independent variable for which the functions are defined. A. Identity C. Trigonometric Identity B. Conditional Equation D. Trigonometric Function 3. The _____________ of two angles is the sum of the product of their cosines and the product of their sines. A. Sine Sum Identity C. Cosine Sum Identity B. Sine Difference Identity D. Cosine Difference Identity 4. The _____________ of two angles is equal to the product of sine of the first and the cosine of the second plus the product of the cosine of the first and the sine of the second. A. Sine Sum Identity C. Cosine Sum Identity B. Sine Difference Identity D. Cosine Difference Identity 5. Which is/are example/s of identity equation? I. II. III. A. I only B. II only C. III only 6. Which is a reciprocal identity? A.

D. I and II

C.

B.

D.

Consider the derivation below of cot (A+B) in answering 7 to 9. Step 1: cot (A+B) =

Step 2: cot (A+B) =

Step 3: cot (A+B)=

Step 4: cot (A+B) =

7. What identity used in the first step? A. Reciprocal Identity B.

Ratio Identity

C. D.

2

Cosine Sum and Sine Sum Identities Cofunction Identity

8. What identity used in the second step? A. Reciprocal Identity B.

Ratio Identity

D.

9. What identity used in the fourth step? A. Reciprocal Identity B.

C.

C.

Ratio Identity

D.

Cosine Sum and Sine Sum Identities Cofunction Identity

Cosine Sum and Sine Sum Identities Cofunction Identity

10. Which of the following is NOT equivalent to sin750. A. sin300cos450 + sin450cos300 C. √ B. D. 2sin750cos750

√

11. If sinx = , where x is in QII,Which of the following is the value of cosx? A. 4 B. C. √ D. √ √ 12. Which of the following is the exact value of sin15 0? √ √

A.

√ √

B.

C.

For items 13-14, Given sinx = , and cos y =

√ √

√ √

D.

.

13. Which of the following are the value of siny and cosx respectively? A.

√

;

√

B.

√

;

√

C.

√

;

√

D.

√

;

√

14. Which of the following is the value of sin(x + y)? A.

(

B.

( )

)

√ √

√

C.

√

( )

D.

(

√

√ )

√

√

15. The range R of a projectile fired at an acute angle θ with the horizontal and with an initial velocity of v meters per second is given by R =

sin(2θ),

m/sec 2

where g is the acceleration due to gravity, which is 9.81 near the Earth’s surface. An archer targets an object 100 meters away from her position. If she positions her arrow at an angle of . How much speed she needed to release the arrow to hit her target? A.

30.01 m/s

B.

33.04 m/s

3

C.

40.03 m/s

D. 50.23 m/s

Lesson

1

Trigonometric Identities

In the previous lessons, we have defined circular functions using the unit circle and also investigated their graphs. This lesson builds on the understanding of the different trigonometric functions by discovery, deriving, and working with trigonometric identities.

What’s In

ACTIVITY 1 Complete the table below and answer the following questions: x (in degrees)

 60

 45

 30

0

sin x cos x tan x csc x sec x cot x

1. How are sin x and csc x related to each other? 2. How are cos x and sec x related to each other? 3. How are tan x and cot x related to each other? 4. How is tan x related to sin x and cos x? 5. How is cot x related to sin x and cosx? 6. What is the value of: a)

sin 2 x  cos2 x ?

b)

csc2 x  cot2 x

c)

sec2 x  tan 2 x

7. Verify the following statements as true or false. a)

sin( 30)   sin 30

b)

cos(45)   cos 45

c)

cot(60)   cot 60 4

30

45

60

What’s New

ACTIVITY 2 Consider the following two groups of equations:

Solve the equations in both group and answer the following questions: 1. What are the solutions (or roots) of the equations in Group A? 2. What are the solutions (or roots) of the equations in Group B? 3. What type of equations are in Group A? What about in B?

What is It

Identity and Conditional Equation Some equations are always true. These equations are called identities. An identity is an equation that is true for all acceptable values of the variable, that is, for all values in the domain of the equation. In What’s New activity, the equations in Group B are all identities. When they are solved, a true statement such as 0 = 0 results, which means that the solution set is the all real numbers, an infinitely many solutions. For example, the equation x 2  1   x  1 x  1 leads to a true statement 0 = 0 when solved:

x 2 1   x 1 x 1

Given equation.

x 2  1  x 2 1

Simplify the left side expression using Product of Two Conjugate Binomials

00

Simplify by Addition Property of Equality 5

The truth of some equations is conditional upon the value chosen for the variable. Such equations are called conditional equations. Conditional equations are equations that are true for at least one replacement of the variable and false for at least one replacement of the variable.In other words, if some values of the variable in the domain of the equation do not satisfy the equation (that is, do not make the equation true), then the equation is a conditional equation. Equations in Group A are all conditional equations because not all real values of the variable can satisfy the equation. Consider the equation x 2 1  0 . This has two solutions (or roots) x  1 or x  1 when solved.

Identify whether the given equation is an identity or a conditional equation. For each conditional equation, provide a value of the variable in the domain that does not satisfy the equation. Example.





3 2 (1) x  2  x  3 2 x  3 2x  3 4



(2) sin   cos   1 (3) sin   cos  1 2

(4)

2

1  x 1 2 x  x  1 x 1 x Solution (1) (2) (3)

(4)

This is an identity because this is simply factoring of di ↵ erence of two cubes.

This is a conditional equation. If   0, then the left-hand side of the equation is 0, while the right-hand side is 2.

This is also a conditional equation. If   0, then both sides of the equation are equal to 0. But if    , then the left-hand side of the equation is 0, while the right-hand side is  2 . This is an identity because the right-hand side of the equation is obtained by rationalizing the denominator of the left-hand side.

The Fundamental Trigonometric Identities Recall that if P(x; y) is the terminal point on the unit circle corresponding to , then we have

𝑠𝑖𝑛𝜃 𝑦 𝑐𝑜𝑠 𝜃 𝑥

𝑐𝑠𝑐 𝜃 𝑠𝑒𝑐 𝜃

𝑦

𝑥

a𝜃 𝜃

𝑦 𝑥 𝑥 𝑦

From the definitions, the following reciprocal and quotient identities immediately follow. Note that these identities hold if is taken either as a real number or as an angle.

6

If P(x,y) is the terminal point on the unit circle corresponding to , then . Since sin = y and cos = x, we get Dividing both sides of

by

,

Dividing both sides of

by

,

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

The first two of the negative identities can be obtained from the graphs of the sine and cosine functions, respectively. The third identity can be derived as follows: a

7

The reciprocal, quotient, Pythagorean, and even-odd identities constitute what we call the fundamental trigonometric identities . We can use these identities to simplify trigonometric expressions. Example 1. Simplify:

Solution:

tan  cos sin 

We know that tan  

sin  , so we subtitute this in the cos

expression. Then we have

sin  cos tan  cos cos  sin   1.  sin  sin  sin 

Example 2. Simplify:

Solution:

cos cot

Since cot  

cos , then we have sin 

cos cos   sin  . cot cos sin  Example 3. Simplify: cos2   cos2  tan 2  Solution:





cos2   cos2  tan 2   cos2  1 tan 2   cos2  sec2   cos2   Example 4. If sin    Solution:

1  1. cos2 

3 and cos  0, find cos  . 4

Using the identity sin 2   cos2   1 with cos  0 , we have

sin 2   cos2   1  cos2   1 sin 2   cos  1  sin 2  . So, 2

9 7 7  3 .  cos  1     1   16 16 4  4 We can also use the fundamental trigonometric identities to establish other identities. The proposed trigonometric identities can be verified by either of the following: 1. Work on the expression on both sides of the equality sign until the members of the equation become identical; or better 8

2. Start on the expression on one side of the proposed identity (preferably the complicated side), use and apply some of the already established fundamental trigonometric identities, perform some algebraic manipulations like performing the fundamental operations, factoring, canceling, and multiplying the numerator and denominator by the same quantity, to finally arrive at the expression on the other side of the proposed identity. Example 5. Establish the identity: If sin   sec   cot   1 . Solution:

 1  sin  sec   cot   sin    cot   cos  sin  cot   cos   tan cot 1

Reciprocal Identity Simplify. Quotient Identity Reciprocal Identity

Example 6. Prove: sec x  cos x  sin x tan x Solution:

sec x  cos x 

1  cos x cosx



1  cos 2 x cos x



sin2 x cosx

 sin x

sin x cos x

 sin x tan x There is no unique technique to prove all identities, but familiarity with the di↵ erent techniques may help

More Trigonometric Identities A. Cosine Sum and Difference Identities The cosine of the sum of two angles is equal to the product of their cosines minus the product of their sines.

9

The cosine of the difference of two angles is equal to the product of their cosines plus the product of their sines.

Example 7. Find the exact value of cos105 in terms of radicals. Solution:

cos105  cos60  45  cos60 cos45  sin 60 ...