Trigonometric Identities and Equations

1 Although it doesn t look like it, Figure 1 above shows the graphs of two func-tions, namelyAlthough these two functions look quite different from one another, they are in factthe same function. This means that, for all values of x,This last expression is an identity,and Identities are one of the topics we will studyin this x 1 sin4 x1 sin2 xy cos2xandy 1 sin4x1 sin2 x795 Trigonometric Identities and EquationsIC^6ci-11xyCHAPTER to and and Half-Angle TrigonometricEquations41088_11_p_795-836 10/11/01 2:06 PM Page 795In this section, we will turn our attention to Identities . In algebra, statements suchas 2x x x,x3 x x x, and x (4x) 1 4 are called are iden-tities because they are true for all replacements of the variable for which they eight basic Trigonometric identitiesare listed in Table 1. As we will see,they are all derived from the definition of the Trigonometric functions.

2 Since manyof the Trigonometric Identities have more than one form, we list the basic identityfirst and then give the most common equivalent to IdentitiesTABLE 1 Basic IdentitiesCommon Equivalent FormsReciprocalRatioPythagorean cos 1 sin2 cos2 1 sin2 1 cot2 csc2 sin 1 cos2 1 tan2 sec2 sin2 1 cos2 cos2 sin2 1cot cos sin tan sin cos tan 1cot cot 1tan cos 1sec sec 1cos sin 1csc csc 1sin Reciprocal IdentitiesNote that, in Table 1, the eight basic Identities are grouped in categories. For exam-ple, since csc 1 (sin ), cosecant and sine must be reciprocals. It is for thisreason that we call the Identities in this categoryreciprocal we mentioned above, the eight basicidentities are all derived from the definition of thesix Trigonometric functions. To derive the firstreciprocal identity, we use the definition of sin to write1sin 1y/r ry csc xy00xry(x, y) 41088_11_p_795-836 10/8/01 8:45 AM Page 796 Note that we can write this same relationship between sin and csc as becauseThe first identity we wrote, csc 1 (sin ), is the basic identity.

3 The second one,sin 1 (csc ), is an equivalent form of the first one. The other reciprocal Identities and their common equivalent forms are derivedin a similar 1 6 show how we use the reciprocal Identities to find the value ofone Trigonometric function, given the value of its , then , , then .(Remember:Reciprocals always have the same algebraic sign.) tan 2, then . csc a, then . sec 1, then cos cot 1, then tan IdentitiesUnlike the reciprocal Identities , the ratio identi-tiesdo not have any common equivalent is how we derive the ratio identity for tan :sin cos y rx r yx tan sin 1acot 12sec 2 3cos 32csc 1sin 135 53csc 53sin 35 Examples1csc 1r/y yr sin sin 1csc Section Introduction to Identities797xy0xry(x, y) 41088_11_p_795-836 10/8/01 8:45 AM Page 797If and ,find tan and cot.

4 SolutionUsing the ratio Identities we haveNote that, once we found tan , we could have used a reciprocal identity to find cot :Pythagorean IdentitiesThe identity cos2 sin2 1 is called a Pythagorean identitybecause it is de-rived from the Pythagorean Theorem. Recall from the definition of sin and cos that if (x, y) is a point on the terminal side of and ris the distance to (x, y) fromthe origin, the relationship between x, y,and ris x2 y2 r2. This relationshipcomes from the Pythagorean Theorem. Here is how we use it to derive the firstPythagorean each side by of of sin and cos NotationThere are four very useful equivalent forms of the first Pythagorean of the forms occur when we solve cos2 sin2 1 for cos , while theother two forms are the result of solving for sin .Solving cos2 sin2 1 for cos , we haveAdd sin2 to each the square root of each 1 sin2 cos2 1 sin2 cos2 sin2 1 cos2 sin2 1 (cos )2 (sin )2 1 xr 2 yr 2 1 x2r2 y2r2 1 x2 y2 r2cot 1tan 1 34 43cot cos sin 45 35 43tan sin cos 3545 34cos 45sin 35 Example 7798 CHAPTER1 1 Trigonometric Identities and Equations41088_11_p_795-836 10/8/01 8:45 AM Page 798 Section Introduction to Identities799 Similarly, solving for sin gives ussin2 1 cos2 andIf and terminates in quadrant II,find cos.

5 SolutionWe can obtain cos from sin by using the identityIf , the identity becomesSubstitute for sin .Square to get we know that cos is either or . Looking back to the originalstatement of the problem, however, we see that terminates in quadrant II; there-fore, cos must be and terminates in quadrant IV,find theremaining Trigonometric ratios for .SolutionThe first, and easiest, ratio to find is sec , because it is the reciprocalof cos .Next, we find sin . Since terminates in QIV, sin will be negative. Usingone of the equivalent forms of the Pythagorean identity, we havesec 1cos 112 2cos 12 Example 9cos 45 45 45 Take the square root of the numerator and denominatorseparately. 45 162592535 1 92535cos 1 35 2sin 35cos 1 sin2 sin 35 Example 8sin 1 cos2 41088_11_p_795-836 10/8/01 8:45 AM Page 799 Negative sign because is in for cos.

6 Square to get that we have sin and cos , we can find tan by using a ratio and csc are the reciprocals of tan and sin , respectively. Therefore,Here are all six ratios together:The basic Identities allow us to write any of the Trigonometric functions interms of sine and cosine. The next examples illustrate tan in terms of sin .SolutionWhen we say we want tan written in terms of sin , we mean thatwe want to write an expression that is equivalent to tan but involves no trigono-metric function other than sin . Let s begin by using a ratio identity to write tan in terms of sin and cos :tan sin cos Example 10tan 3 cot 1 3cos 12sec 2 sin 32csc 2 3cot 1tan 1 3csc 1sin 2 3tan sin cos 3/21/2 3 Take the square root of the numerator and denominator separately. 32 341412 1 1412 1 12 2sin 1 cos2 800 CHAPTER1 1 Trigonometric Identities and Equations41088_11_p_795-836 10/8/01 8:45 AM Page 800 Section Introduction to Identities801 NoteThe notation sec tan means sec tan.

7 Now we need to replace cos with an expression involving only sin . Since, we haveThis last expression is equivalent to tan and is written in terms of sin only. (In aproblem like this it is okay to include numbers and algebraic symbols with sin just no other Trigonometric functions.)Here is another example. This one involves simplification of the product of twotrigonometric sec tan in terms of sin and cos , and thensimplify. SolutionSince sec 1 (cos ) and tan (sin ) (cos ), we haveThe next examples show how we manipulate Trigonometric expressions usingalgebraic SolutionWe can add these two expressions in the same way we would add and , by first finding a least common denominator, and then writing each expres-sion again with the LCD for its 1cos .Example 12 sin cos2 sec tan 1cos sin cos Example 11 sin 1 sin2 sin 1 sin2 tan sin cos cos 1 sin2 41088_11_p_795-836 10/8/01 8:45 AM Page 801 Use the reciprocal Identities in the following ,find csc.

8 ,find sec . sec 2,find cos . ,find sin . ,find cot .tan a (a 0)csc 1312cos 3 2sin ,find tan .Use a ratio identity to find tan Use a ratio identity to find cot cos 3 13sin 2 13cos 1213sin 513cos 1 5sin 2 5cos 45sin 35cot b (b 0)Multiply (sin 2)(sin 5).SolutionWe multiply these two expressions in the same way we would multi-ply (x 2)(x 5).FOIL(sin 2)(sin 5) sin sin 5 sin 2 sin 10 sin2 3 sin 10 Getting Ready for ClassAfter reading through the preceding section, respond in your own words and incomplete the reciprocal Identities for csc , sec , and cot . the ratio Identities for tan and cot . the three Pythagorean tan in terms of sin .Example 13 cos sin sin cos cos sin cos sin cos sin The LCD is sin cos.

9 1sin 1cos 1sin cos cos 1cos sin sin 802 CHAPTER1 1 Trigonometric Identities and EquationsPROBLEM SET 10/8/01 8:45 AM Page 802 Section Problem Set803 Use the equivalent forms of the Pythagorean identity onProblems 11 sin if cos and terminates in sin if cos and terminates in cos if sin and terminates in cos if sin and terminates in sin and terminates in QIII,find cos . sin and terminates in QIV,find cos . and terminates in QI,find sin . and terminates in QII,find sin . and QII,find cos . and QIII,find sin .Find the remaining Trigonometric ratios of terminates in terminates in terminates in terminates in 3 and 4 and QIIW rite each of the following in terms of sin and cos ,and then simplify if cot cot tan tan csc sec csc cot cos tan sin cos sec sin csc csc sec sec csc sin 3 10 cos 2 13cos 13sin 12sin 1213cos 1213 cos 1 10 sin 1 5cos 12cos 3 2 45 45 3 2 13 513 35 Add and subtract as indicated.

10 Then simplify youranswers if possible. Leave all answers in terms of sin and or cos . (sin 4)(sin 3)50.(cos 2)(cos 5)51.(2 cos 3)(4 cos 5)52.(3 sin 2)(5 sin 4)53.(1 sin )(1 sin )54.(1 cos )(1 cos )55.(1 tan )(1 tan )56.(1 cot )(1 cot )57.(sin cos )258.(cos sin )259.(sin 4)260.(cos 2)2 Review ProblemsThe problems that follow review material we covered inSection to radian Convert to degree the ConceptsRecall from algebra that the slope of the line through (x1,y1) and (x2,y2) ism y2 y1x2 x14 35 45 6 61cos cos 1sin sin cos 1sin sin 1cos 1cos 1sin 1sin 1cos cos sin sin cos sin cos 1sin 41088_11_p_795-836 10/8/01 8:45 AM Page 803804 CHAPTER1 1 Trigonometric Identities and EquationsNext we want to use the eight basic Identities and their equivalent forms to verifyother Trigonometric Identities .