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1. CERTIFICATION When you have completed all your work and provided you have maintained an average of at least 60 you will be eligible to write the examination for this course 0 30 SOFAD Answer Key MTH 5108 2 Trigonometric Functions and Equations UNIT 1 CONVERTING FROM DEGREES TO RADIANS AND FROM RADIANS TO DEGREES 11 SETTING THE CONTEXT Having Fun with Your Scientific Calculator Have you ever wondered what the key on your scientific calculator is used for Press this key and you will see the DEG RAD or GRAD modes displayed successively on your calculator even though no value appears on the screen These abbreviations stand for three different units of angular measure namely degres radians and grads Some calculators have a second function 2ndF DRG gt Put you calculator in DEG mode enter 90 and then press 2ndF DRG gt You will see 1 570796327 in RAD mode displayed Press the same keys again and 100 in GRAD mode will be displayed SOFAD 1 1 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations To achieve the objective of this unit you should be able to convert angular measures from degrees to radians and vice versa There are three different units of angular measures degrees radians and grads You are already familiar with degrees For a number of years now you have been using a protractor which allows yo
2. A d MTH 5108 2 s AN L Ld 1 Pi y 9 M n Wr A Md RIGONOM ETRIC y FUNCTIONS SAN AND EQUATIONS ri m Sin A B tan A B cos A B y asin b x A FR sin2x cosx 1 Pid f a x E w gt MTH 5108 2 TRIGONOMETRIC FUNCTIONS AND EQUATIONS Mathematics Project Coordinator Jean Paul Groleau Author Alain Malouin Content Revision Jean Paul Groleau Pedagogical Revision Jean Paul Groleau Translation Claudia de Fulviis Linguistic Revision Johanne St Martin Electronic Publishing P P I inc Cover Page Daniel R my First Edition 2008 O Soci t de formation distance des commissions scolaires du Qu bec All rights for translation and adaptation in whole or in part reserved for all countries ny reproduction by mechanical or electronic means including microre production is forbidden without the written permission of a duly authorized representative of the Soci t de formation distance des commissions scolaires du Qu bec SOFAD Legal Deposit 2008 Biblioth que et Archives nationales du Qu bec Biblioth que et Archives Canada ISBN 978 2 89493 307 7 Answer Key MTH 5108 2 Trigonometric Functions and Equations TABLE OF CONTENTS Introduction to the Program Flowchart 0 4 Program FIowchart nadie tmt vi pata Iq 0 5 How to Use this Guid en 0
3. MTH 5109 1 MTH 5108 2 MTH 5107 2 MTH 5106 1 MTH 5105 1 Geometry IV Trigonometric Functions and Equations You are here Exponential and Logarithmic Functions and Equations Real Functions and Equations Conics Statistics III Optimization Q MTH 4111 2 MTH 4110 1 Complement and Synthesis I The Four Operations on Algebraic Fractions MTH 4109 1 MTH 4108 1 MTH 4107 1 MTH 4106 1 MTH 4105 1 Sets Relations and Functions Quadratic Functions Straight Lines II Factoring and Algebraic Functions Exponents and Radicals Statistics II Trigonometry Geometry III Equations and Inequalities Il Straight Lines Geometry II The Four Operations on Polynomials Statistics and Probabilities Geometry Equations and Inequalities Decimals and Percent The Four Operations on Fractions The Four Operations on Integers e 1 credit 50 hours 2 credits 0 5 1 2 3 Answer Key MTH 5108 2 Trigonometric Functions and Equations HOW TO USE THIS GUIDE Hi My name is Monica and have been 2A E You ll see that with this method math is asked to tell you about this math module E a real breeze What s your name Whether you are you have probably taken a My results on the test registered at an placement test which tells you indicate that should begin adult education exactly which module you with this module center or pur should start with suing distanc
4. SOFAD 1 15 Answer Key
5. Simplify a trigonometric expression using trigonometric identities related to a sum a difference or twice the value of a real number The expression should not consist of more than two terms on each side of the equality and the expression should consist of no more than four trigonometric functions 25 Solving problems that require applying concepts related to sinusoi dal functions Solve problems that require applying concepts related to sinusoi dal functions The solution may involve finding the rule of a sinusoi dal function describing certain characteristics of a sinusoidal function determining the connections between the change in the parameters ofthe rule and the transformation ofthe corresponding graph or comparing certain characteristics of different sinusoidal functions over a given interval SOFAD 0 17 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations This module comprises 25 objectives grouped as follows Unit Objective s 1 Determining the measure of an angle 1 Converting angular measures from degrees to radians and vice versa 2 2 Determining the coordinates of trigonometric points 3 Finding the image of a trigonometric angle under the wrapping function 4 Determining the measure in radians of a trigonometric angle 5 3 Defining trigonometric functions 6 Evaluating the image of a trigonometric function 7 4 Graphing the sine cosine and tan
6. domain and range sign of the function y intercept e zeros e intervals of increase or decrease image of an element of the domain element s of the domain associated with a given image phase shift vertical translation O SOFAD 0 13 13 14 15 16 17 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations Determining the rule of a sinusoidal function Determine the rule of a sinusoidal function given relevant information or the graph of the function Comparing the characteristics of two sinusoidal functions Compare the characteristics of two sinusoidal functions given their graph Proving the fundamental identities Prove the fundamental trigonometric identities sin cos 1 e tan t 1 sec t 1 cot t csc f Applying the fundamental identities and trigonometric ratios Apply the fundamental identities and the definitions of the trigonometric ratios to the transformation of simple trigonometric expressions Determining the value of the other trigonometric ratios using a known trigonometric ratio Given the value of a trigonometric ratio at one point within a designated interval determine the value of the other trigonometric ratios at this point using the fundamental identities The interval corresponds to an arc of no more than rn radians and its limits are multiples of 5 A 0 14 SOFAD rj Answer Key MTH 51
7. of two real numbers We will provide the basic formulas that you may use as needed Naturally you will find some ofthe concepts and definitions covered in previous modules very helpful These include definitions of trigonometric ratios and the characteristics of functions In order to be able to prove identities more easily you may need to perform operations on algebraic fractions and polynomials Finally you will be required to solve problems that involve applying concepts related to sinusoidal functions And this in a nutshell is what you will be learning in this second module on trigonometry SOFAD 0 9 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations INTERMEDIATE AND TERMINAL OBJECTIVES OF THE MODULE Module MTH 5108 2 consists of 25 objectives and requires 50 hours of study distributed as follows The terminal objectives are shown in boldface Objectives Number of hours Evaluation 1to3 4 10 4 and 5 4 1096 6 and 7 5 1096 8 9 and 10 5 1096 11 12 13 and 14 5 10 15 to 17 5 1096 18 and 19 5 1096 20 and 21 5 10 22 to 24 5 10 25 5 10 Two hours are allotted for the final evaluation 0 10 SOFAD rj Answer Key MTH 5108 2 Trigonometric Functions and Equations 1 Determining the measure of an angle Determine the measure of an angle in degrees or radians Converting angular measures from degrees
8. you need only apply the method described previously and substitute 3 1416 for n _ 13x180 _ 13x180 _ 74 480 a oot era Now it s your turn 1 10 SOFAD Answer Key MTH 5108 2 Trigonometric Functions and Equations Exercise 1 3 Convert the following radian measures to degrees 1 SE rad 2 SE rad M de 3 E rad adeb od a Sawa MD a GEN 4 32 rad n 5 ag rad VERREM SS S EMEN laesst Did you know that theoretical studies in trigonometry were first carried out by the Babylonians and Greeks Hipparchus and Ptolemy and then pursued by the Arabs and Europeans Regiomontanus Copernicus and Vi te The introduction of logarithms helped to advance trigonometry in the 17th and 18th centuries It was Euler however who developed the definitive theory of this science SOFAD 1 11 Answer Key MTH 5108 2 Trigonometric Functions and Equations e 1 2 PRACTICE EXERCISES 1 Convert the following degree measures to radians a DES ai BE 72 qe e WI ae aD nee B UU acea td een aaa D 924 T E 2 base h VID ans 10 rekin tes in miens DES sense sonne If ui ecc T D DEO 5 P Convert the following radian measures to degrees a pa uu ERN 137 c Im d TT rad S H S
9. 12 1 12 4 Unit 7 b 12 1 12 4 Unit 7 c 12 1 12 4 Unit 7 d 12 1 12 4 Unit 7 e 12 1 12 4 Unit 7 f 12 1 12 4 Unit 7 2 a 12 2 12 17 Unit 7 b 12 2 12 17 Unit 7 c 12 2 12 17 Unit 7 d 12 2 12 17 Unit 7 e 12 2 12 17 Unit 7 f 12 2 12 17 Unit 7 g 12 2 12 17 Unit 7 h 12 2 12 17 Unit 7 3 a 12 3 12 21 Unit 7 b 12 3 12 21 Unit 7 c 12 3 12 21 Unit 7 d 12 3 12 21 Unit 7 e 12 3 12 21 Unit 7 f 12 3 12 21 Unit 7 g 12 3 12 21 Unit 7 4 a 12 4 12 24 Unit 7 b 12 4 12 24 Unit 7 c 12 4 12 24 Unit 7 5 a 12 4 12 24 Unit 7 b 12 4 12 24 Unit 7 c 12 4 12 24 Unit 7 d 12 4 12 24 Unit 7 6 a 12 5 12 30 Unit 7 b 12 5 12 30 Unit 7 c 12 5 12 30 Unit 7 d 12 5 12 30 Unit 7 L e 12 5 12 30 Unit 7 SOFAD 0 25 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations fall your answers are correct you may begin working on this module For each incorrect answer find the related section listed in the Review column Do the review activities for that section before beginning the units listed in the right hand column under the heading Before going on to 0 26 SOFAD rj Answer Key MTH 5108 2 Trigonometric Functions and Equations INFORMATION FOR DISTANCE EDUCATION STUDENTS You now have the learning material for MTH 5108 2 and the relevant homework assignments Enclosed with this package is a letter of introduction from your tutor indicating the various
10. 6 General Introduction kr dd pde BE a UP eua 0 9 Intermediate and Terminal Objectives of the Module 0 10 Diagnostic Test on the Prerequisites 0 19 Answer Key for the Diagnostic Test on the Prerequisites 0 23 Analysis of the Diagnostic Test Results 0 25 Information for Distance Education Students 0 27 UNITS 1 Converting from Degrees to Radians and from Radians to Degrees 1 1 2 The Wrapping Function asus darin 2 1 3 Evaluating a Trigonometric Function for a Number Expressed in Radians manon 3 1 4 Graphing a Trigonometric Function 4 1 5 Graphing a Sinusoidal Function 5 1 6 Fundamental Trigonometric Identities 6 1 7 Proving Simple Trigonometric Identities 7 1 8 Solving Simple Trigonometric Equations of the First or Second Degree 8 1 9 Trigonometric Functions Involving a Sum or a Difference of Two Real Numbers 9 1 10 Solving Problems Using Sinusoidal Functions 10 1 Final Review akan h nawa S ab ela eens al us
11. Graphing the sine cosine and tangent functions Graph the sine cosine and tangent functions in a given interval 9 Determining the characteristics of the sine cosine and tangent functions Using the rule or the graph of the sine cosine and tangent functions determine the characteristics of the functions The following characteristics are studied domain and range image of an element of the domain elements s of the domain associated with a given image maximum and minimum e zeros e period y intercept e intervals of increase or decrease sign of the function equations of the asymptotes 0 12 SOFAD rj Answer Key MTH 5108 2 Trigonometric Functions and Equations 10 Comparing the characteristics of the sine cosine and tangent functions Compare the characteristics of the sine cosine and tangent functions over a given interval 11 Graphing a sinusoidal function Graph a sinusoidal function of the form f x asin b x h k or f x acos b x h k and determine the connections between the change in the parameters of the rule and the transformation of the corresponding graph 12 Determining the characteristics of a sinusoidal function Using the rule or the graph of a sinusoidal function determine the characteristics of the function The following characteristics are studied maximum and minimum amplitude period frequency
12. first path which consists of Modules MTH 3003 2 MTH 314 and MTH 4104 2 MTH 416 leads to a Secondary School Vocational Diploma SSVD and certain college level programs for students who take MTH 4104 2 The second path consisting of Modules MTH 4109 1 MTH 426 MTH 4111 2 MTH 436 and MTH 5104 1 MTH 514 leads to a Secondary School Diploma SSD which gives you access to certain CEGEP programs that do not call for a knowledge of advanced mathematics Lastly the path consisting of Modules MTH 5109 1 M TH 526 and MTH 5111 2 MTH 536 will lead to CEGEP programs that require a thorough knowledge of mathematics in addition to other abilities Good luck Ifthis is your first contact with the mathematics program consult the flowchart on the next page and then read the section How to Use this Guide Otherwise go directly to the section entitled General Introduction Enjoy your work 0 4 SOFAD MTH 5108 2 Trigonometric Functions and Equations PROGRAM FLOWCHART MTH 5112 1 Logic Q MTH 5111 2 Complement and Synthesis Il MTH 5110 1 Introduction to Vectors MTH 526 manb MTH 5104 1 MTH 5103 1 Optimization Il Probability II MTH 5102 1 MTH 5101 1 O MTH 4104 2 MTH 4103 1 MTH 4102 1 C MTH 4101 2 MTH 3003 2 MTH 3002 2 MTH 3001 2 MTH 314 MTH 2008 2 MTH 2007 2 MTH 2006 2 MTH 216 MTH 1007 2 MTH 1006 2 MTH 1005 2 MTH 116 SOFAD
13. in a unit before moving on to the next one Homework Assignments Module MTH 5108 2 comprises three homework assignments The first page of each assignment indicates the units to which the questions refer The assign ments are designed to evaluate how well you have understood the material studied They also provide a means of communicating with your tutor When you have understood the material and have successfully completed the pertinent exercises do the corresponding assignment right away Here are afew suggestions 1 Do a rough draft first and then if necessary revise your solutions before writing out a clean copy of your answer O SOFAD 0 29 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations 2 Copy out your final answers or solutions in the blank spaces of the document to be sent to your tutor It is best to use a pencil 3 Includea clear and detailed solution with the answer if the problem involves several steps 4 Mail only one homework assignment at a time After correcting the assign ment your tutor will return it to you In the section Student s Questions write any questions which you wish to have answered by your tutor He or she will give you advice and guide you in your studies if necessary In this course Homework Assignment 1 is based on units 1 to 5 Homework Assignment 2 is based on units 6 to 10 Homework Assignment 3 is based on units 1 to 10
14. test and answer key They tell big hep s i i i l m glad Now db i 2 This is great never thouqht that would ve got to youl like mathematics as much as this gt 0 8 SOFAD 1 2 3 Answer Key MTH 5108 2 Trigonometric Functions and Equations GENERAL INTRODUCTION TAKING TRIGONOMETRY A LITTLE FURTHER You have now reached the second to last module of the mathematics program In this module you will learn the concepts you will need if you wish to continue studying mathematics at the college level This module will allow you to expand your knowledge of trigonometry The first few units deal with various angular measures the trigonometric circle and a new function called the wrapping function We will be returning to familiar territory since we will be calculating sines cosines tangents and so on Note however that we will not be calculating these ratios in terms of a right triangle Rather we will be calculating them for real numbers on the trigonometric circle A scientific calculator will definitely come in handy In subsequent units you will learn how to graph the six trigonometric functions and sinusoidal functions You will also be required to state certain characteristics of the resulting curves The last few units will give you the opportunity to prove both simple and complex trigonometric identities and to calculate the image ofreal numbers that can be expressed as a sum or a difference
15. ways in which you can communicate with him or her e g by letter or telephone as well as the times when he or she is available Your tutor will correct your work and help you with your studies Do not hesitate to make use of his or her services if you have any questions DEVELOPING EFFECTIVE STUDY HABITS Learning by correspondence is a process which offers considerable flexibility but which also requires active involvement on your part It demands regular study and sustained effort Efficient study habits will simplify your task To ensure effective and continuous progress in your studies it is strongly recommended that you draw up a study timetable that takes your work habits into account and is compatible with your leisure and other activities develop a habit of regular and concentrated study O SOFAD 0 27 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations The following guidelines concerning theory examples exercises and assign ments are designed to help you succeed in this mathematics course Theory To make sure you grasp the theoretical concepts thoroughly 1 Read the lesson carefully and underline the important points 2 Memorize the definitions formulas and procedures used to solve a given problem this will make the lesson much easier to understand 3 At the end of the assignment make a note of any points that you do not understand using the sheets provided for th
16. 08 2 Trigonometric Functions and Equations 18 19 20 21 Simplifying and factoring trigonometric expressions Perform the four operations on trigonometric expressions simplify trigonometric expressions and factor trigonometric expressions Proving a simple trigonometric identity Prove a simple trigonometric identity The expression should consist of no more than two terms on each side of the equality Each term should contain no more than two trigonometric ratios The definitions of trigonometric ratios and the fundamental identities will not be provided during examinations Finding the image of an angle given in radian measure for a given trigonometric function Using a calculator find the image of an angle given in radian measure for a given trigonometric function In addition given the value of a trigonometric function expressed as a real number determine the corresponding trigonometric angle over a given interval or in R The limits of the interval are multiples of n Solving a simple trigonometric equation of the first or second degree Solve a simple trigonometric equation of the first or second degree over a given interval or in R using the trigonometric circle or a calculator The solution may require simple factorization The limits of the interval must be multiples of x O SOFAD 0 15 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations 22 Verifying trigonometric i
17. YO s ti ti ia DER Mc Ep Rn TERRIER 1 8 SOFAD ro Answer Key MTH 5108 2 Trigonometric Functions and Equations 3 20 T0 T5 pec 4 190 45 35 Did you know that trigonometry acquired its name in the early 17th century when the German astronomer Pitiscus entitled one of his works Trigonometria libri quinque However this branch of mathematics had been known since the 3rd century BCE Now let us perform the inverse operation that is convert radian angular measures to degrees Where the measure is given in terms of zx use proportions mrad 180 n rad The resulting proportion is mrad _ 180 n rad x mrad x x n rad x 180 x n rad x 180 mrad _ nx 180 T Example 4 What is the value in degrees of an angle of T rad IT x 180 x 27 x 180 108 SOFAD 1 9 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations What happens ifthe angular measure is not expressed in terms of x You guessed it The calculator will do all the work for us Example 5 Given an angle of 1 3 rad express this value in degrees First make sure that the calculator is in RAD mode Then do the following 1 3 2nd F DRG p gt 82 7606 2nd F DRG b 74 4845 Therefore 1 3 rad 74 48 In this example you can convert 1 3 rad to degrees without using the DRG key D 4 Do you know how to calculate this In fact
18. ae Ma b TES EE ernennen nenne 2 c yy IT d rn E EE xy2 x e P kk ik ra oo a a 2ab b f EST EEE due suisse l l RE ed 063 809 99 0 9 9 000 0 0 0 8 084199 2 03 9 0 sise ie na diese 49 9 80 8 NU 0905 VE 2x y y x2 y g Eee 4 Calculate the following products a xx L TD b xx bn die a b be _ x TRES 5 Calculate the following quotients 1 y a _ E a L Domen b D eee b x ax L b a _ c D kk ANANA Aa p 1 d din ide PERDER 14 2 SOFAD 0 21 Answer Key MTH 5108 2 Trigonometric Functions and Equations 6 Calculate the following sums and differences The result must be reduced to its simplest form a y ls rv s S s aa as E c TE 1 er E d en e NEED 0 22 SOFAD Answer Key MTH 5108 2 Trigonometric Functions and Equations l a b c d e c e g h c e ANSWER KEY FOR THE DIAGNOSTIC TEST ON THE PREREQUISITES a x ay alax y xy y yy 1 x y x y x y xy y y x2 1 y x 1 x 1 a b a b a b a b a b a b ax x xa 1 m m n m mn b ab a b ab ab x 1 x x x2 d 2 3y xy 2xy 3xy m n m n m2 n 2r s 2r s 4
19. ase a central angle of 1 radian intercepts an arc of a circle 1 unit in length D 4 Ina unit circle an angle of intercepts an arc two units in length If your answer is 2 rad then you have understood the explanation given above Let us continue Since the circumference of a unit circle is 21 C 2nr 2r x l the central angle which intercepts the entire circumference measures 2n rad T Complete the following statement The central angle which intercepts half of a circle s circumference measures CUM rad a central angle which intercepts one quarter of a circle s circumference measures rad Solution The central angle which intercepts half of a circle s circumference measures T rad since i the circumference i x 2n m and a central angle which I 2 MP n I 1 intercepts one quarter of a circle s circumference measures rad since 1 the circumference i x 2T D SOFAD 1 3 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations If we compare the two units of angular measure we obtain the following 360 2n rad 180 z rad o _ L 90 5 rad We will use the second equality or n rad 180 to do conversions When used in a proprotion this equality allows us to convert from degrees to radians and vice versa Example 1 An angle measures 210 What is the measure of this angle in radians Solution mrad 180 210 The proporti
20. d sa vu DE an saf EE 11 1 Answer Key for the Final Review 2222222222sssseeeeeessnennnnnneeennnnn 11 10 Terminal Objectives cuoc bv yode aa ot bn TIR ap MU Rd Ip NIIS 11 17 Self Evaluation Test aa ea 11 21 Answer Key for the Self Evaluation Test 11 33 Analysis of the Self Evaluation Test Results 11 41 Final Evaluation eisen 11 43 Answer Key for the Exercises ccccccceccceecceceeeeessessseseeeeceeeeeeeeess 11 45 KE AY a mund nnd n MINE kl ya ki MM MM 11 151 List of Symbols een 11 155 Bibliography ENDE 11 156 Review Activities za panqa b aaah unas ERES 12 1 SOFAD 0 3 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations INTRODUCTION TO THE PROGRAM FLOWCHART WELCOME TO THE WORLD OF MATHEMATICS This mathematics program has been developed for adult students enrolled either with Adult Education Services of school boards or in distance education The learning activities have been designed for individualized learning If you encounter difficulties do not hesitate to consult your teacher or to telephone the resource person assigned to you The following flowchart shows where this module fits into the overall program It allows you to see how far you have come and how much further you still have to go to achieve your vocational objective There are three possible paths you can take depending on your goal The
21. dentities related to a sum a difference or twice the 23 value of a real number Using simple examples verify the trigonometric identities related to a sum a difference or twice the value of a real number sin A B sin A cos B cos A sin B sin A B sin A cos B cos A sin B cos A B cos A cos B sin A sin B cos A B cos A cos B sin A sin B _ tan A tan B 2 A tan A B T tan A tan p Where 1 tan A tan B 0 _ tanA tanB tan A B 1 tan tan B sin 2A 2sin A cos A cos 2A cos A sin A where 1 tan A tan B z 0 tan 2A n where 1 tan A z 0 tan Proving the cofunction identities the odd even identities the double angle identities or a reduction formula Using the trigonometric identies related to a sum or a difference of real numbers prove the cofunction identities the odd even identities the double angle identities or a reduction formula when the proof involves identities related to the sine or cosine A or B is a multiple of 5 or a variable when the proof involves identities related to the tangent function A or Bisa multiple of or a variable N B The formulas will be provided during examinations 0 16 SOFAD rj Answer Key MTH 5108 2 Trigonometric Functions and Equations 24 Simplifying a trigonometric expression using trigonometric identi ties related to a sum a difference or twice the value of a realnumber
22. e education Now the module you have in your de se eid wrieh By carefully correcting this test using the F hands is divided into three CORTES Me test ONIE corresponding answer key and record sections The first section i prerequisites ing your results on the analysis sheet TUN J Lh J 0 6 SOFAD rj Answer Key MTH 5108 2 Trigonometric Functions and Equations E In that case before you start the review Boas UR eea a lie activities in the module the results analysis chart refers you to a review happens then activity near the end of the module you can tell if you re well enough prepared to do all the activities in the module The starting line shows where the learning activities begin Exactly The second section contains the learning activities It s the main part of the module D The little white question mark indicates the questions for which answers are given in the text The target precedes the objective to be met The memo pad signals a brief reminder of concepts which you have already studied The boldface question mark gt indicates practice exercises which allow you to try out what you have Just learned Look pose at the box to the right t explains the symbols used to Identity the various activities The calculator symbol reminds you that you will need to use your calculator The sheaf of wheat indicates a
23. gent functions 8 Determining the characteristics of the sine cosine and tangent functions 9 Comparing the characteristics of the sine cosine and tangent functions 10 5 Graphing a sinusoidal function 11 Determining the characteristics of a sinusoidal function 12 Determining the rule of a sinusoidal function 13 Comparing the characteristics of two sinusoidal functions 14 6 Proving fundamental identities 15 Applying fundamental identities and trigonometric ratios 16 Determining the value of the other trigonometric ratios using a known trigonometric ratio 17 7 Simplifying and factoring trigonometric expressions 18 Proving a simple trigonometric identity 19 8 Finding the image of an angle given in radian measure for a given trigonometric function 20 Solving a simple trigonometric equation of the first or second degree 21 9 Verifying trigonometric identities related to a sum a difference or twice the value of a real number 22 Proving the cofunction identities the odd even identities the double angle identities or a reduction formula 23 Simplifying a trigonometric expression using trigonometric identities related to a sum a difference or twice the value of a real number 24 10 Solving problems that require applying concepts related to sinusoidal functions 25 0 18 SOFAD rj Answer Key MTH 5108 2 Trigonometric Functions and Equations DIAGNOSTIC TEST ON THE PREREQUISITES Instructions 1 Answer as ma
24. is purpose Your tutor will then be able to give you pertinent explanations 4 Try to continue studying even if you run into a problem However if a major difficulty hinders your progress contact your tutor before handing in your assignment using the procedures outlined in the letter of introduction Examples The examples given throughout the course are applications of the theory you are studying They illustrate the steps involved in doing the exercises Carefully study the solutions given in the examples and redo the examples yourself before starting the exercises 0 28 SOFAD rj Answer Key MTH 5108 2 Trigonometric Functions and Equations Exercises The exercises in each unit are generally modeled on the examples provided Here are a few suggestions to help you complete these exercises 1 Write up your solutions using the examples in the unit as models It is important not to refer to the answer key found on the coloured pages at the back of the module until you have completed the exercises 2 Compare your solutions with those in the answer key only after having done all the exercises Careful Examine the steps in your solutions carefully even if your answers are correct 3 Ifyou find a mistake in your answer or solution review the concepts that you did not understand as well as the pertinent examples Then redo the exercise 4 Make sure you have successfully completed all the exercises
25. lso use a graphing calculator for this type of conversion To convert 90 to radians proceed as follows MODE Select Radian ENTER and CLEAR to get back to the screen 9 0 2na AN GLE 1 ENTER 1 570796327 should appear on the screen L gt to degrees proceed as follows Conversely to convert Select degree and CLEAR Q 2nd x 2 JO ena AN GLE 3 ENTER 90 should appear on the screen As you may have noticed up to now all radian measures have been given in terms of n But is this always the case Let us look at the following example 1 6 SOFAD ro Answer Key MTH 5108 2 Trigonometric Functions and Equations Example 3 Given an angle of 18 15 30 express this value in radians e First convert 18 15 30 to decimal form Convert the seconds to minutes 30 Son 0 5 Add 0 5 to 15 15 0 5 15 5 Convert the minutes to degrees 15 5 0 258 Add 0 258 to 18 18 0 258 18 258 Therefore 18 15 30 18 258 e Convert 18 258 to radians _ 18 258 xmrad _ 18258mrad _ x A805 180000 0 3187 rad N B n was replaced by its value 3 14159 You may be wondering why r was replaced by its numerical value in the example above whereas it was not in the others There is in fact a rule of thumb that allows you to determine whether the result will be expressed in terms of n or otherwise IQ Ingeneral radian
26. measures are given in terms of t ifthe simplified fraction by which x is multiplied is a fraction whose denominator is less than 100 Otherwise use a calculator to perform the operations SOFAD 1 7 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations Given that in Example 3 18 15 30 18 258 when rounded off to the nearest thousandth this value obviously cannot be converted to a simplified fraction with a denominator of less than 100 The operations are therefore performed using a calculator 18 00 18 15 00 2nd F DEG 18 258 2nd F DRG p gt 0 319 The result is 18 15 30 0 319 rad Observations 1 We converted 18 15 30 to decimal form to make sure we could not express this result using a simplified fraction with a denominator of less than 100 2 Remember to put your calculator in DEG mode when you enter the data to be converted to radians otherwise the result will be incorrect Let us look at one last detail before you go on to the exercises If you are wondering why the result is expressed in terms of x when it would be so easy to express it in decimal form the answer is simple As you will see later on it is because the main points on the trigonometric circle are generally expressed in terms of x Exercise 1 2 Convert the following angular measures expressed in degrees minutes and seconds to radians MEL 1 E 22 MAT
27. ny questions as you can 2 Do not use a calculator 3 Write your answers on the test paper 4 Do not waste any time If you cannot answer a question go on to the next one immediately 5 When you have answered as many questions as you can correct your answers using the answer key which follows the diagnostic test 6 To be considered correct your answers must be identical to those in the answer key In addition the various steps in your solution should be equivalent to those shown in the answer key 7 Transcribe your results on the chart which follows the answer key This chart will help you analyze your diagnostic text results 8 Doonly the review activities that apply to each of your incorrect answers 9 If all your answers are correct you may begin working on this module O SOFAD 0 19 Answer Key MTH 5108 2 Trigonometric Functions and Equations 1 Find the prime factors of the following polynomials 2 Calculate the following products a MMN m babe 2 dE CC E d 2 yl y eme e m num n D p s rt s e asina uii URBC RADI a nn era der craie dd O h ZE ANS so baka bk kk kk kk ok ti ee en ke Do ue 0 20 SOFAD Answer Key MTH 5108 2 Trigonometric Functions and Equations 3 Reduce each of the following algebraic fractions to their simplest form 6a _ a b _ a ha ES idiots voee dua Gas e
28. on can be written as follows mrad _ 180 x 210 180 x x 210 x n rad _ 210 x trad x 180 _ Zm rad x x The answer is m rad In future we will write out only the step marked with an asterisk however ifyou have any doubts feel free to use the original proportion 1 4 SOFAD ro Answer Key MTH 5108 2 Trigonometric Functions and Equations Exercise 1 1 Convert the following degree measures to radians 1 Bo A DO Gita ea Dic JO 4 sl ansa H De OU Posse pena ain desig PA E Had 0 225 mea qa Gusan tn Example 2 Given an angle of 37 30 express this value in radians First convert 30 to a decimal fraction expressed in degrees If 1 60 then 30 30 0 5 An angle of 37 30 is therefore an angle of 37 5 in decimal notation Hence 37 5 Xnrad 5r 1809 24 rad X Observations You can use your calculator to convert 37 30 to 37 5 as follows 37 00 30 2nd F GDEG 37 5 N B Some calculators use the symbol rather than DEG but the result is the same To go back a step simply press the key on certain models or 2nd F eDEG on others SOFAD 1 5 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations If your calculator does not function in this way refer to the user manual or consult a resource person if possible You can a
29. r s m n m 2 m n n m 2mn n 2x y 2x 2 2x y y 4x 4xy y 2 2 2 6a _9u b b cannot be simplified 3a a aa b a x y x y a a b d ue a 2 2 DS _ myt 1 swa SOFAD 0 23 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations f a2 2ab b _ a b a bla b ab a2 b a b a b a b a b a b 2x y y x2 y _ 2x y y3 x y _ Xyy y y x y _ x y g xy Xy gt a 7 Xy 3 x 4 a xix df ay x2 y 1_ Xy _ 1 b yar z xy y c a b _be__ arb ee_b ac a b ac a b 4 i 2 2 a_l b b 5 a LD a e p L 2 W Eid x By pe dy Dou s ab o8 D b 1 1 ab 1 4 b b x yTX d D I I ton x 1 2 n y xty y X x at Y_xty 6 a y FIE yT y y D as olg x y x y X Xy x xy l x _1 x l y_2 x y o tay Fd 1 y l y l y 1 1 a b a b Gb Et aba a bann 2a __ gs a _ a b Ya b q p a 1 x y x y L y x y x y x y Xx X X E yMa y X y X y _ 2y 2y x y x y x y x y x y 0 24 SOFAD rj Answer Key MTH 5108 2 Trigonometric Functions and Equations ANALYSIS OF THE DIAGNOSTIC TEST RESULTS Answers Review Questions Correct Incorrect Section Page Before going on to 1 a
30. review designed to reinforce what you have just learned A row of sheaves near the end of the module indicates the final review which helps you to interrelate all the learning activities in the module Lastly the finish line indicates h that it is time to go on to the nee test to verify how well you have understood the learning activities SOFAD 0 7 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations There are aleo many Tun IDE F A Did you know that Must memorize what the sage says in this module For examp when you see the drawing of a sage 1t introduces a Did you know that Yes for example short tidbits No it s not part of the learn ne Te ae he E ing activity It s just there to teresting and relieve tension at giv you a breathier the same time A A i And the whole module has es which are designed espe They are so stimulating tha b di k ially for those who love math even if you don t have to do een arrangeq to make hem you ll still wantto learning easier For example words in bold statements in boxes are important i i i i face italics appear in the _ points to remember like definitions for view wish iterrelatas the diferant glossary at the end of the mulas and rules l m telling you the for parts of the module mat makes everything much easier There is also a self evaluation Thanks Monica you ve been a
31. ss e 1 12 SOFAD ro Answer Key MTH 5108 2 Trigonometric Functions and Equations SOFAD 1 13 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations 1 3 REVIEW ACTIVITY 1 Complete the following statements There are three different units of angular measure One corresponds to 150 of the circumference One A corresponds to 540 of the circumference One corresponds to a central angle which intercepts an arc whose length is the same as the length of the radius 2 Complete the following table Angle Acute Right Flat Reentering Round Degrees 45 180 360 3 Radians gt Se 3 a Ifan angle measures n its measure in can be found by applying the formula Bau b Ifan angle measures n degrees its measure in can be found by applying the formula A 1 14 SOFAD ro Answer Key MTH 5108 2 Trigonometric Functions and Equations 14 THE MATH WHIZ PAGE Meet the Challenge Until now we have worked with the unit circle that is a circle whose radius measures one unit Given a circle that is not a unit circle can you find its radius if its central angle of 2 5 rad intercepts an arc of 25 cm N B Acentral angle intercepts an arc of the same measure
32. to radians and vice versa Given the central angles of a circle convert angular measures from degrees to radians and vice versa convert angula measures fom radians to degrees Determining the coordinates of trigonometric points Using the unit circle and the wrapping function determine the coordinates of the trigonometric points Finding the image of a trigonometric angle under the wrapping function Find the image of a trigonometric angle t under the wrapping function Determine the reference angle t 0 lt t 27 corresponding nr NN nm oy MA or to angle Angle is expressed in radians as nm 2 3 4 6 where n is a whole number Determining the measure in radians of a trigonometric angle Determine the measure in radians ofa trigonometric angleina given interval using the coordinates of a trigonometric point The interval is expressed as nm nn 2r where n is a whole number SOFAD 0 11 rj Answer Key MTH 5108 2 Trigonometric Functions and Equations 6 Defining trigonometric functions Define the trigonometric sine cosine tangent cotangent secant and cosecant functions in the context of a unit circle and the wrapping function 7 Evaluating the image of a trigonometric function Evaluate the image of a trigonometric function associated with a trigonometric angle The angle is expressed in radians as nm NA NA RR oy BZ where n is a whole number 2 3 4 6 8
33. u to measure an angle in degrees You probably know that a degree corresponds to the measure of the angle between two consecutive radii when a circle is divided into 360 equal parts In other words an angle of 1 intercepts an arc equal in length to of the circle s 360 circumference One degree 1 is subdivided into 60 minutes and one minute 1 is subdivided into 60 seconds 60 These angular units should not be confused with the units of time of the same name Divisions of degrees can also be represented using decimal notation for example 12 30 is equivalent to 12 5 The second unit of angular measure is the grad As you may have noted from the statement of the objective we will not be using grads in this module Note 1 however that one gradis equal to 100 ofa circle s circumference and the symbol for grad is gr The third unit of angular measure is the radian One radian is the measure of a central angle which intercepts an arc of a circle equal in length to the radius of the circle In other words a central angle which intercepts an arc equal in length to the radius of the circle measures one radian The symbol for radian is rad 1 2 SOFAD 1 2 3 Answer Key MTH 5108 2 Trigonometric Functions and Equations Fig 1 1 Central angle of 1 radian intercepting an arc of the same length as the radius To make our task easier we will use a circle whose radius is one unit in length In this c

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