Trigonometric Identities,Inverses, And Equations

cob19537 ch07 653-661.qxd1/26/118:56 AMPage 653Precalculus—CHAPTER CONNECTIONSTrigonometricIdentities, Inverses,and EquationsCHAPTER OUTLINE7.1 Fundamental Identities and Families of Identities 6547.2 More on Verifying Identities 6617.3 The Sum and Difference Identities 6697.4 The Double-Angle, Half-Angle, andProduct-to-Sum Identities 6807.5 The Inverse Trig Functions and Their Applications 6957.6 Solving Basic Trig Equations 711This chapter will unify much of what we’velearned so far , and lead us to some intriguing,sophisticated, and surprising applications oftrigonometr y. Defining the trig functions helpeus study a number of new r elationships notpossible using algebra alone. Their graphs gaveus insights into how the functions wer e relatedto each other , and enabled a study of periodicphenomena. W e will now use identities tosimplify complex expr essions and show howtrig functions often work together to modelnatural events. One such “event” is a river’sseasonal dischar ge rate, which tends to begreater during the annual snow melt. In thischapter, we’ll lear n how to pr edict thedischar ge rate during specific months of thyear, infor mation of gr eat value to fisheriesoceanographers, and other scientists.䊳This application appears as Exer cises 53and 54 in Section 7.77.7 General Trig Equations and Applications 721Trigonometric equations, identities, and substitutions also play a vital role in a study of calculus, helping tosimplify complex expressions, or rewrite an expression in a form more suitable for the tools of calculus.These connections are explored in the Connections to Calculus feature following Chapter 7.Connectionsto Calculus653

cob19537 ch07 653-661.qxd1/26/114:50 PMPage 654Precalculus—6547–000CHAPTER 7 Trigonometric Identities, Inverses, and Equations7.1Fundamental Identities and Families of IdentitiesLEARNING OBJECTIVESIn this section, we begin laying the foundation necessary to work with identitiessuccessfully. The cornerstone of this effort is a healthy respect for the fundamentalidentities and vital role they play. Students are strongly encouraged to do more thanmemorize them — they should be internalized, meaning they must become a naturaland instinctive part of your core mathematical knowledge.In Section 7.1 you will seehow we can:A. Use fundamentalidentities to helpunderstand and recognizeidentity “families”B. Verify other identitiesusing the fundamentalidentities and basicalgebra skillsC. Use fundamentalidentities to express agiven trig function interms of the other fiveA. Fundamental Identities and Identity FamiliesAn identity is an equation that is true for all elements in the domain. In trigonometry,some identities result directly from the way the functions are defined. For instance,the reciprocal relationships we first saw in Section 6.2 followed directly from their definitions. We call identities of this type fundamental identities. Successfully workingwith other identities will depend a great deal on your mastery of these fundamentaltypes. For convenience, the definitions of the trig functions are reviewed here, followedby the fundamental identities that result.Given point P(x, y) is on the terminal side of angle in standard position, withr 2x2 y2 the distance from the origin to (x, y), we haveyrrcsc ; y 0yxrrsec ; x 0xcos sin ytan ; x 0xxcot ; y 0yFundamental Trigonometric IdentitiesReciprocalidentitiesRatioidentitiessin cos sec tan csc cos cot sin 1csc 1cos sec 1tan cot sin EXAMPLE 1䊳Pythagoreanidentitiestan cos2 sin2 11 tan2 sec2 cot2 1 csc2 Proving a Fundamental IdentityUse the coordinate definitions of the trigonometric functions to prove the identitycos2 sin2 1.SolutionWORTHY OF NOTEThe Pythagorean identities areused extensively in future courses.See Example 1 of the Connectionsto Calculus feature at the end ofthis chapter.䊳We begin with the left-hand side.y 2x 2cos2 sin2 a b a brr22yx 2 2rr2x y2 r2substituteyxfor cos , for sin rrsquare termsadd termsNoting that r 2x2 y2 implies r2 x2 y2, we have r2 1r2substitute x 2 y 2 for r 2, simplfyNow try Exercises 7 through 10654䊳7–2

cob19537 ch07 653-661.qxd1/26/114:50 PMPage 655Precalculus—7–3Section 7.1 Fundamental Identities and Families of Identities655The fundamental identities seem to naturally separate themselves into the threegroups or families listed, with each group having additional relationships that canbe inferred from the definitions. For instance, since sin is the reciprocalof csc , csc must be the reciprocal of sin . Similar statements can be maderegarding cos and sec as well as tan and cot . Recognizing these additional“family members” enlarges the number of identities you can work with, and willhelp you use them more effectively. In particular, since they are reciprocals:sin csc 1, cos sec 1, and tan cot 1. See Exercises 11 and 12.EXAMPLE 2䊳Identifying Families of IdentitiesStarting with cos2 sin2 1, use algebra to write four additional identities thatbelong to the Pythagorean family.Solution䊳cos2 sin2 11) sin2 1 cos2 2) sin 21 cos2 cos2 sin2 13) cos2 1 sin2 4) cos 21 sin2 original identitysubtract cos2 take square rootoriginal identitysubtract sin2 take square rootFor the identities involving a radical, the choice of sign will depend on thequadrant of the terminal side.Now try Exercises 13 and 14A. You’ve just seen howwe can use fundamentalidentities to help understandand recognize identity“families”䊳The fact that each new equation in Example 2 represents an identity gives us moreoptions when attempting to verify or prove more complex identities. For instance,since cos2 1 sin2 , we can replace cos2 with 1 sin2 , or replace 1 sin2 with cos2 , any time they occur in an expression. Note there are many other membersof this family, since similar steps can be performed on the other Pythagorean identities.In fact, each of the fundamental identities can be similarly rewritten and there are a variety of exercises at the end of this section for practice.B. Verifying an Identity Using AlgebraNote that we cannot prove an equation is an identity by repeatedly substituting inputvalues and obtaining a true equation. This would be an infinite exercise and we mighteasily miss a value or even a range of values for which the equation is false. Instead weattempt to rewrite one side of the equation until we obtain a match with the other side,so there can be no doubt. As hinted at earlier, this is done using basic algebra skillscombined with the fundamental identities and the substitution principle. For now we’llfocus on verifying identities by using algebra. In Section 7.2 we’ll introduce someguidelines and ideas that will help you verify a wider range of identities.

cob19537 ch07 653-661.qxd1/26/118:56 AMPage 656Precalculus—6567–4CHAPTER 7 Trigonometric Identities, Inverses, and EquationsEXAMPLE 3䊳Using Algebra to Help Verify an IdentityUse the distributive property to verify that sin 1csc sin 2 cos2 is anidentity.Solution䊳Use the distributive property to simplify the left-hand side.sin 1csc sin 2 sin csc sin2 1 sin2 cos2 distributesubstitute 1 for sin csc 1 sin2 cos2 Since we were able to transform the left-hand side into a duplicate of the right,there can be no doubt the original equation is an identity.Now try Exercises 15 through 24䊳Often we must factor an expression, rather than multiply, to begin the verificationprocess.EXAMPLE 4䊳Using Algebra to Help Verify an IdentityVerify that 1 cot2 sec2 cot2 is an identity.Solution䊳The left side is as simple as it gets. The terms on the right side have a commonfactor and we begin there.cot2 sec2 cot2 cot2 1sec2 12 cot2 tan2 1cot tan 2 2 12 1factor out cot2 substitute tan2 for sec2 1power property of exponentscot tan 1Now try Exercises 25 through 32䊳Examples 3 and 4 show you can begin the verification process on either the left orright side of the equation, whichever seems more convenient. Example 5 shows howthe special products 1A B2 1A B2 A2 B2 and/or 1A B2 2 A2 2AB B2can be used in the verification process.EXAMPLE 5Solution䊳Using a Special Product to Help V erify an IdentityUse a special product and fundamental identities to verify that1sin cos 2 2 1 ⴚ 2 sin cos is an identity.䊳Begin by squaring the left-hand side, in hopes of using a Pythagorean identity.1sin cos 2 2 sin2 2 sin cos cos2 cos2 sin2 2 sin cos 1 ⴚ 2 sin cos binomial squarerewrite termssubstitute 1 for cos2 sin2 Now try Exercises 33 through 38B. You’ve just seen how wecan verify other identities usingthe fundamental identities andbasic algebra skills䊳Another common method used to verify identities is simplification by combiningsin2 ACAD BC. For sec cos , the right-handterms, using the model BDBDcos sin2 cos2 1 sec . See Exercises 39side immediately becomes, which givescos cos through 44.

cob19537 ch07 653-661.qxd1/26/114:51 PMPage 657Precalculus—7–5Section 7.1 Fundamental Identities and Families of Identities657C. Writing One Function in Terms of AnotherAny one of the six trigonometric functions can be written in terms of any of the otherfunctions using fundamental identities. The process involved offers practice in working with identities, highlights how each function is related to the other, and has practical applications in verifying more complex identities.EXAMPLE 6䊳Writing One Trig Function in Terms of AnotherWrite the function cos in terms of the tangent function.Solution䊳Begin by noting these functions share “common ground” via sec , since1. Starting with sec2 ,sec2 1 tan2 and cos sec sec2 1 tan2 sec 21 tan2 Pythagorean identitysquare rootsWe can now substitute 21 tan2 for sec in cos cos 1 21 tan 21.sec substitute 21 tan2 for sec Note we have written cos in terms of the tangent function.Now try Exercises 45 through 50WORTHY OF NOTEAlthough identities are valid whereboth expressions are defined, thisdoes not preclude a difference inthe domains of each function. Forexample, the result of Example 6 isindeed an identity, even though the left side is defined at while the2right side is not.EXAMPLE 7䊳Example 6 also reminds us of a very important point — the sign we choose for thefinal answer is dependent on the terminal side of the angle. If the terminal side is in QIor QIV we chose the positive sign since cos 7 0 in those quadrants. If the angle terminates in QII or QIII, the final answer is negative since cos 6 0 in those quadrants.Similar to our work in Section 6.7, given the value of cot and the quadrant of ,the fundamental identities enable us to find the value of the other five functions at . Infact, this is generally true for any given trig function and angle .䊳Using a Known Value and Quadrant Analysis to F ind Other Function Values 9with the terminal side of in QIV, find the value of the other40five functions of . Use a calculator to check your answer.Given cot Solution䊳40follows immediately, since cotangent and tangent9are reciprocals. The value of sec can be found using sec2 1 tan2 .The function value tan sec2 1 tan2 40 2 1 a b9811600 81811681sec2 8141sec 9Pythagorean identitysubstitute square 40for tan 94081, substitutefor 1981combine termstake square roots