Trigonometric Identities And Equations - WebAssign
41088 11 p 795-83610/11/012:06 PMPage 795Trigonometric Identitiesand Equationsy1IC 6cxi-1Although it doesn’t look like it, Figure 1 above shows the graphs of two functions, namelyy cos2 xand1 sin4 xy 1 sin2 xAlthough these two functions look quite different from one another, they are in factthe same function. This means that, for all values of x,cos2 x 1 sin4 x1 sin2 xCHAPTER OUTLINE11.1 Introduction to Identities11.2 Proving Identities11.3 Sum and DifferenceFormulas11.4 Double-Angle andHalf-Angle Formulas11.5 Solving TrigonometricEquationsThis last expression is an identity, and identities are one of the topics we will studyin this chapter.795
41088 11 p 795-836 10/8/01 8:45 AM Page 79611.1Introduction to IdentitiesIn 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 identities. They are identities because they are true for all replacements of the variable for which they aredefined.The eight basic trigonometric identities are listed in Table 1. As we will see,they are all derived from the definition of the trigonometric functions. Since manyof the trigonometric identities have more than one form, we list the basic identityfirst and then give the most common equivalent forms.TABLE 1ReciprocalRatioPythagoreanBasic IdentitiesCommon Equivalent Forms1sin 1sec cos 1cot tan 1csc 1cos sec 1tan cot csc sin sin cos cos cot sin tan cos2 sin2 11 tan2 sec2 1 cot2 csc2 sin2 sin cos2 cos 1 cos2 1 cos2 1 sin2 1 sin2 Reciprocal IdentitiesNote that, in Table 1, the eight basic identities are grouped in categories. For example, since csc 1 (sin ), cosecant and sine must be reciprocals. It is for thisreason that we call the identities in this categoryyreciprocal identities.As we mentioned above, the eight basic(x, y)identities are all derived from the definition of thesix trigonometric functions. To derive the firstrreciprocal identity, we use the definition of sin yto write1r1 csc sin y/ry796θ0xx
41088 11 p 795-836 10/8/01 8:45 AM Page 797797Section 11.1 Introduction to IdentitiesNote that we can write this same relationship between sin and csc assin 1csc because11y sin csc r/yrThe first identity we wrote, csc 1 (sin ), is the basic identity. 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 manner.Examples 1 – 6 show how we use the reciprocal identities to find the value ofone trigonometric function, given the value of its reciprocal.Examples1. If sin 35, then csc , because53csc 115 3 sin 35 32, then sec .2 3(Remember: Reciprocals always have the same algebraic sign.)1If tan 2, then cot .21If csc a, then sin .aIf sec 1, then cos 1.If cot 1, then tan 1.2. If cos 3.4.5.6.Ratio IdentitiesyUnlike the reciprocal identities, the ratio identities do not have any common equivalent forms.Here is how we derive the ratio identity for tan :(x, y)y rsin y tan cos x rxryθ0xx
41088 11 p 795-836 10/8/01 8:45 AM Page 798798CHAPTER 11Trigonometric Identities and EquationsExample 7If sin 34and cos , find tan and cot .55Solution Using the ratio identities we have3 sin 3tan 45 cos 454cos 4 53 cot sin 3 5Note that, once we found tan , we could have used a reciprocal identity to findcot :cot 114 3tan 3 4Pythagorean IdentitiesThe identity cos2 sin2 1 is called a Pythagorean identity because it is derived 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 r is the distance to (x, y) fromthe origin, the relationship between x, y, and r is x2 y2 r2. This relationshipcomes from the Pythagorean Theorem. Here is how we use it to derive the firstPythagorean identity.x2 y2 r 2x2y2 1r2r2 xr2 yr2 1(cos )2 (sin )2 1cos2 sin2 1Divide each side by r 2.Property of exponents.Definition of sin and cos NotationThere are four very useful equivalent forms of the first Pythagorean identity.Two 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 havecos2 sin2 1cos2 1 sin2 cos 1 sin2 Add sin2 to each side.Take the square root of each side.
41088 11 p 795-836 10/8/01 8:45 AM Page 799799Section 11.1 Introduction to IdentitiesSimilarly, solving for sin gives ussin2 1 cos2 andsin 1 cos2 Example 8If sin 3and terminates in quadrant II, find cos .5Solution We can obtain cos from sin by using the identitycos 1 sin2 If sin 3, the identity becomes53cos 1 5 1 1625 Substitute 35 for sin .925Square 3 to get25925Subtract.Take the square root of thenumerator and denominatorseparately.4 5Now we know that cos is either 4 or 4. Looking back to the original55statement of the problem, however, we see that terminates in quadrant II; therefore, cos must be negative.cos 45Example 9If cos 12 and terminates in quadrant IV, find theremaining trigonometric ratios for .Solution The first, and easiest, ratio to find is sec , because it is the reciprocalof cos .sec 11 1 2cos 2Next, we find sin . Since terminates in QIV, sin will be negative. Usingone of the equivalent forms of the Pythagorean identity, we have
41088 11 p 795-836 10/8/01 8:45 AM Page 800800CHAPTER 11Trigonometric Identities and Equationssin 1 cos2 1 1 34Negative sign because is in QIV. 12 2Substitute 12 for cos .14Square 21 to get 14Subtract.Take the square root of thenumerator and denominatorseparately. 3 2Now that we have sin and cos , we can find tan by using a ratio identity.tan sin 3/2 3 cos 1/2Cot and csc are the reciprocals of tan and sin , respectively. Therefore,cot 11 tan 3csc 12 sin 3Here are all six ratios together:sin cos 3212tan 3csc 2 3sec 2cot 1 3The basic identities allow us to write any of the trigonometric functions interms of sine and cosine. The next examples illustrate this.Example 10Write tan in terms of sin .Solution When 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 trigonometric function other than sin . Let’s begin by using a ratio identity to write tan in terms of sin and cos :tan sin cos
41088 11 p 795-836 10/8/01 8:45 AM Page 801801Section 11.1 Introduction to IdentitiesNow we need to replace cos with an expression involving only sin . Sincecos 1 sin2 , we havesin tan cos sin 1 sin2 sin 1 sin2 This 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 functions.Example 11Write sec tan in terms of sin and cos , and thensimplify.The notation sec tan means sec tan .NoteSolution Since sec 1 (cos ) and tan (sin ) (cos ), we havesec tan 1sin cos cos sin cos2 The next examples show how we manipulate trigonometric expressions usingalgebraic techniques.Example 12Add11 .sin cos 131and 4, by first finding a least common denominator, and then writing each expres-Solution We can add these two expressions in the same way we would addsion again with the LCD for its denominator.
41088 11 p 795-836 10/8/01 8:45 AM Page 802802CHAPTER 11Trigonometric Identities and Equations111cos 1sin sin cos sin cos cos sin Example 13 cos sin sin cos cos sin cos sin sin cos The LCD issin cos .Multiply (sin 2)(sin 5).Solution We multiply these two expressions in the same way we would multiply (x 2)(x 5).FOIL(sin 2)(sin 5) sin sin 5 sin 2 sin 10 sin2 3 sin 10Getting Ready for ClassAfter reading through the preceding section, respond in your own words and incomplete sentences.A.B.C.D.State the reciprocal identities for csc , sec , and cot .State the ratio identities for tan and cot .State the three Pythagorean identities.Write tan in terms of sin .PROBLEM SET 11.1Use the reciprocal identities in the following problems.1. If sin 4, find csc .52. If cos 3 2, find sec .3. If sec 2, find cos .4. If csc 13, find sin .125. If tan a (a 0), find cot .6. If cot b (b 0), find tan .Use a ratio identity to find tan if:347. sin and cos 558. sin 2 5 and cos 1 5Use a ratio identity to find cot if:5129. sin and cos 131310. sin 2 13 and cos 3 13
41088 11 p 795-836 10/8/01 8:45 AM Page 803Section 11.1 Problem SetUse the equivalent forms of the Pythagorean identity onProblems 11 – 20.3and terminates in QI.11. Find sin if cos 55and terminates in QI.12. Find sin if cos 131and terminates in QII.13. Find cos if sin 314. Find cos if sin 3 2 and terminates in QII.415. If sin and terminates in QIII, find cos .5416. If sin and terminates in QIV, find cos .517. If cos 3 2 and terminates in QI, find sin .118. If cos and terminates in QII, find sin .219. If sin 1 5 and QII, find cos .20. If cos 1 10 and QIII, find sin .Find the remaining trigonometric ratios of if:1221. cos and terminates in QI131222. sin and terminates in QI13123. sin and terminates in QIV2124. cos and terminates in QIII325. cos 2 13 and QIV26. sin 3 10 and QII27. sec 3 and QIII28. sec 4 and QIIWrite each of the following in terms of sin and cos ,and then simplify if possible:30. sec cot 29. csc cot 31. csc tan 32. sec tan csc sec csc sin 35.csc 37. tan sec 39. sin cot cos 33.csc sec cos 36.sec 38. cot csc 40. cos tan sin 803Add and subtract as indicated. Then simplify youranswers if possible. Leave all answers in terms of sin and or cos .41.1sin cos sin 42.sin cos sin cos 43.11 sin cos 44.11 cos sin 45. sin 1cos 46. cos 1 sin sin Multiply.47.48.1sin 1 cos cos 49. (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)2Review ProblemsThe problems that follow review material we covered inSection 10.1.Convert to radian measure.61. 120 62. 330 63. 135 64. 270 Convert to degree measure.65.67.65466.5668.4334.Extending the ConceptsRecall from algebra that the slope of the line through(x1, y1) and (x2, y2) ism y2 y1x2 x1
41088 11 p 795-836 10/8/01 8:45 AM Page 804804CHAPTER 11Trigonometric Identities and Equations71. Find the slope of the line y mx.72. Find tan if is the angle formed by the line y mxand the positive x-axis. (See Figure 2.)It is the change in the y-coordinates divided by the changein the x-coordinates.69. The line y 3x passes through the points (0, 0) and(1, 3). Find its slope.70. Suppose the angle formed by the line y 3x and thepositive x-axis is . Find the tangent of . (SeeFigure 1.)yy ymxy 3x(1, m)θ(1, 3)0xFIGURE 2θx0FIGURE 111.2Proving IdentitiesNext we want to use the eight basic identities and their equivalent forms to verifyother trigonometric identities. To prove (or verify) that a trigonometric identity istrue, we use trigonometric substitutions and algebraic manipulations to either:1. Transform the right side into the left side.Or:2. Transform the left side into the right side.The main thing to remember in proving identities is to work on each side of theidentity separately. We do not want to use properties from algebra that involve bothsides of the identity — such as the addition property of equality. We prove identitiesin order to develop the ability to transform one trigonometric expression into another. When we encounter problems in other courses that require the use of thetechniques used to verify identities, we usually find that the solution to these problems hinges upon transforming an expression containing trigonometric functionsinto a less complicated expression. In these cases, we do not usually have an equalsign to work with.Example 1Verify the identity: sin cot cos .Proof To prove this identity we transform the left side into the right side:
41088 11 p 795-836 10/8/01 8:45 AM Page 805805Section 11.2 Proving Identitiessin cot sin cos sin sin cos sin cos Example 2Ratio identityMultiply.Divide out common factor sin .Prove: tan x cos x sin x(sec x cot x).Proof We begin by applying the distributive property to the right side of theidentity. Then we change each expression on the right side to an equivalent expression involving only sin x and cos x.sin x(sec x cot x) sin x sec x sin x cot x sin x 1cos x sin x cos xsin xsin x cos xcos x tan x cos xMultiply.Reciprocal and ratioidentitiesMultiply and divide outcommon factor sin x.Ratio identityIn this case, we transformed the right side into the left side.Before we go on to the next example, let’s list some guidelines that may beuseful in learning how to prove identities.Probably the best advice is to remember that these are simply guidelines. Thebest way to become proficient at proving trigonometric identities is to practice. Themore identities you prove, the more you will be able to prove and the more confident you will become. Don’t be afraid to stop and start over if you don’t seem to begetting anywhere. With most identities, there are a number of different proofs thatwill lead to the same result. Some of the proofs will be longer than others.Guidelines for Proving Identities1. It is usually best to work on the more complicated side first.2. Look for trigonometric substitutions involving the basic identities that mayhelp simplify things.3. Look for algebraic operations, such as adding fractions, the distributiveproperty, or factoring, that may simplify the side you are working with orthat will at least lead to an expression that will be easier to simplify.4. If you cannot think of anything else to do, change everything to sines andcosines and see if that helps.5. Always keep and eye on the side you are not working with to be sure youare working toward it. There is a certain sense of direction that accompanies a successful proof.
41088 11 p 795-836 10/8/01 8:45 AM Page 806806CHAPTER 11Trigonometric Identities and EquationsExample 3Prove:cos4 t sin4 t 1 tan2 t.cos2 tProof In this example, factoring the numerator on the left side will reduce theexponents there from 4 to 2.cos4 t sin4 t(cos2 t sin2 t)(cos2 t sin2 t) cos2 tcos2 t 1 (cos2 t sin2 t)cos2 tPythagoreanidentity sin2 tcos2 t cos2 tcos2 tSeparate intotwo fractions. 1 tan2 tExample 4Factor.Prove: 1 cos Ratio identitysin2 .1 cos Proof We begin by applying an alternative form of the Pythagorean identity tothe right side to write sin2 as 1 cos2 . Then we factor 1 cos2 and reduce tolowest terms.sin2 1 cos2 1 cos 1 cos (1 cos )(1 cos )1 cos 1 cos Example 5Pythagorean identityFactor.Reduce.Prove: tan x cot x sec x csc x.Proof We begin by rewriting the left side in terms of sin x and cos x. Then wesimplify by finding a common denominator, changing to equivalent fractions, andadding, as we did when we combined rational expressions in Chapter 4.sin xcos x cos xsin xsin x sin xcos x cos x cos x sin xsin x cos xsin2 x cos2 x cos x sin x1 cos x sin xtan x cot x 11 cos x sin x sec x csc x Change to expressionsin sin x and cos x.LCDAdd fractions.Pythagorean identityWrite as separatefractions.Reciprocal identities
41088 11 p 795-836 10/8/01 8:45 AM Page 807807Section 11.2 Proving IdentitiesExample 6Prove:1 cos Asin A 2 csc A.1 cos Asin AProof The LCD for the left side is sin A(1 cos A).sin A1 cos Asin Asin A1 cos A 1 cos A 1 cos Asin Asin A 1 cos Asin A1 cos ALCD sin2 A (1 cos A)2sin A(1 cos A)Add fractions. sin2 A 1 2 cos A cos2 Asin A(1 cos A)Expand (1 cos A)2. 2 2 cos Asin A(1 cos A)Pythagorean identity 2(1 cos A)sin A(1 cos A)Factor out 2. 2sin AReduce. 2 csc AReciprocal identityExample 7Prove:cos t1 sin t .cos t1 sin tProof The trick to proving this identity is to multiply the numerator and denominator on the right side by 1 sin t.cos tcos t1 sin t 1 sin t1 sin t 1 sin tMultiply numerator anddenominator by 1 sin t. cos t(1 sin t)1 sin2 tMultiply out thedenominator. cos t(1 sin t)cos2 tPythagorean identity 1 sin tcos tReduce.Note that it would have been just as easy for us to verify this identity by multiplying the numerator and denominator on the left side by 1 sin t.
41088 11 p 795-836 10/8/01 8:45 AM Page 808808CHAPTER 11Trigonometric Identities and EquationsGetting Ready for ClassAfter reading through the preceding section, respond in your own words and incomplete sentences.A. What is an identity?B. In trigonometry, how do we prove an identity?cos4t sin4t?cos2t1 cos Asin A D. What is a first step in simplifying the expression,?1 cos Asin AC. What is a first step in simplifying the expression,PROBLEM SET 11.2Prove that each of the following identities is true:1. cos tan sin 2. sec cot csc 3. csc tan sec 4. tan cot 1cot Atan A sin A cos A5.6.sec Acsc A7. sec cot sin 18. tan csc cos 19. cos x(csc x tan x) cot x sin x10. sin x(sec x csc x) tan x 111. cot x 1 cos x(csc x sec x)12. tan x(cos x cot x) sin x 113. cos2 x(1 tan2 x) 114. sin2 x(cot2 x 1) 115. (1 sin x)(1 sin x) cos2 x16. (1 cos x)(1 cos x) sin2 xcos4 t sin4 t cot2 t 117.sin2 tsin4 t cos4 t sec2 t csc2 t18.sin2 t cos2 tcos2 19. 1 sin 1 sin cos2 20. 1 sin 1 sin 21.1 sin4 cos2 1 sin2 22.1 cos4 sin2 1 cos2 23. sec2 tan2 124. csc2 cot2 125. sec4 tan4 1 sin2 cos2 26. csc4 cot4 1 cos2 sin2 27. tan cot sin2 cos2 sin cos 28. sec csc sin cos sin cos 29. csc B sin B cot B cos B30. sec B cos B tan B sin B31. cot cos sin csc 32. tan sin cos sec 33.cos x1 sin x 2 sec x1 sin xcos x34.cos x1 sin x 01 sin xcos x35.11 2 csc2 x1 cos x1 cos x
41088 11 p 795-836 10/8/01 8:45 AM Page 809809Section 11.3 Sum and Difference Formulas36.37.38.39.40.41.42.43.44.45.46.11 2 sec2 x1 sin x1 sin x1 sec xcos x 1 1 sec xcos x 1csc x 11 sin x csc x 11 sin xcos t1 sin t 1 sin tcos tsin t1 cos t 1 cos tsin t2(1 sin t)1 sin t cos2 t1 sin tsin2 t1 cos t(1 cos t)2 1 cos tsec 1tan tan sec 1cot csc 1 cot csc 1Show that sin(A B) is, in general, not equal tosin A sin B by substituting 30 for A and 60 for Bin both expressions and simplifying.Show that sin 2x 2 sin x by substituting 30 for xand then simplifying both sides.Review ProblemsThe problems that follow review material we covered inSection 10.2. Reviewing these problems will help youwith some of the material in the next section.11.3Give the exact value of each of the following:47. sin349. cos651. tan 4553. sin 9048. cos350. sin652. cot 4554. cos 90Extending the ConceptsProve each identity.sec4 y tan4 y 155.sec2 y tan2 ycsc2 y cot2 y 156.csc4 y cot4 ysin3 A 8 sin2 A 2 sin A 457.sin A 21 cos3 A cos2 A cos A 158.1 cos A1 tan3 t sec2 t tan t59.1 tan t1 cot3 t csc2 t cot t60.1 cot tsec B1 sin B 61.sin B 1cos3 Bsin3 B1 cos B 62.csc B1 cos BSum and Difference FormulasThe expressions sin(A B) and cos(A B) occur frequently enough in mathematics that it is necessary to find expressions equivalent to them that involve sines andcosines of single angles. The most obvious question to begin with isNoteA counterexample is anexample that shows that a statement is not, in general, true.sin(A B) sin A sin B?The answer is no. Substituting almost any pair of numbers for A and B in the formula will yield a false statement. As a counterexample, we can let A 30 and
41088 11 p 795-836 10/8/01 8:45 AM Page 810810CHAPTER 11Trigonometric Identities and EquationsB 60 in the formula above and then simplify each side.sin(30 60 ) sin 30 sin 60sin 90 1 1 3 221 32The formula just doesn’t work. The next question is, what are the formulas forsin(A B) and cos(A B)? The answer to that question is what this section is allabout. Let’s start by deriving the formula for cos(A B).We begin by drawing A in standard position and then adding B and B to it.These angles are shown in Figure 1 in relation to the unit circle. The unit circle isthe circle with its center at the origin and with a radius of 1. Since the radius of theunit circle is 1, the point through which the terminal side of A passes will have coordinates (cos A, sin A). [If P2 in Figure 1 has coordinates (x, y), then by the definition of sin A, cos A, and the unit circle, cos A x r x 1 x and sin A y r y 1 y. Therefore, (x, y) (cos A, sin A).] The points on the unit circle throughwhich the terminal sides of the other angles in Figure 1 pass are found in the samemanner.(cos (A B), sin (A B))P1P2(cos A, sin A)A BBAP3(1, 0)–BP4(cos ( B), sin ( B)) (cos B, sin B)FIGURE 1To derive the formula for cos(A B), we simply have to see that line segmentP1P3 is equal to line segment P2P4. (From geometry, they are chords cut off byequal central angles.)P1 P3 P2 P4
41088 11 p 795-836 10/8/01 8:45 AM Page 811811Section 11.3 Sum and Difference FormulasSquaring both sides gives us(P1 P3)2 (P2 P4 )2Now, applying the distance formula, we have[cos(A B) 1]2 [sin(A B) 0]2 (cos A cos B)2 (sin A sin B)2Let’s call this Equation 1. Taking the left side of Equation 1, expanding it, and thensimplifying by using the Pythagorean identity, gives usLeft side of Equation 1cos2(A B) 2 cos(A B) 1 sin2(A B) 2 cos(A B) 2Expand squares.Pythagorean identityApplying the same two steps to the right side of Equation 1 looks like this:Right side of Equation 1cos2 A 2 cos A cos B cos2 B sin2 A 2 sin A sin B sin2 B 2 cos A cos B 2 sin A sin B 2Equating the simplified versions of the left and right sides of Equation 1, we have 2 cos(A B) 2 2 cos A cos B 2 sin A sin B 2Adding 2 to both sides and then dividing both sides by 2 gives us the formulawe are after.cos(A B) cos A cos B sin A sin BThis is the first formula in a series of formulas for trigonometric functions ofthe sum or difference of two angles. It must be memorized. Before we derive theothers, let’s look at some of the ways we can use our first formula.Example 1Find the exact value for cos 75 .Solution We write 75 as 45 30 and then apply the formula forcos(A B).cos 75 cos (45 30 ) cos 45 cos 30 sin 45 sin 30 2 3 21 2222 6 24
1 cot2 csc2 cos2 1 sin2 1 tan 2 sec sin 1 cos2 cos 2 2 sin 1 sin 2 1 cos2 cot cos sin tan sin cos tan 1 cot cot 1 tan cos 1 sec sec 1 cos sin 1 csc csc 1 sin Reciprocal Identities Note that, in Table 1, the eight basic identities are grouped in categories. For exam-ple,
identities related to odd and . Topic: Verifying trig identities with tables, unit circles, and graphs. 9. verifying trigonometric identities worksheet. verifying trigonometric identities worksheet, verifying trigonometric identities worksheet
25 More Trigonometric Identities Worksheet Concepts: Trigonometric Identities { Addition and Subtraction Identities { Cofunction Identities { Double-Angle Identities { Half-Angle Identities (Sections 7.2 & 7.3) 1. Find the exact values of the following functions using the addition and subtraction formulas (a) sin 9ˇ 12 (b) cos 7ˇ 12 2.
7 Trigonometric Identities and Equations 681 7.1Fundamental Identities 682 Fundamental Identities Uses of the Fundamental Identities 7.2Verifying Trigonometric Identities 688 Strategies Verifying Identities by Working with One Side Verifying Identities by Working with Both Sides 7
654 CHAPTER 7 Trigonometric Identities, Inverses, and Equations 7–000 Precalculus— 7.1 Fundamental Identities and Families of Identities In this section, we begin laying the foundation necessary to work with identities successfully. The cornerstone of this effort is a healthy respect for the fundamental identities and vital role they play.
Use the basic trigonometric identities to verify other identities. Find numerical values of trigonometric functions. 7 ft 5 ft Transform the more complicated side of the equation into the simpler side. Substitute one or more basic trigonometric identities to simplify expressions. Factor or multiply to simplify expressions.
Pre-Calculus Mathematics 12 – 6.1 – Trigonometric Identities and Equations The FUN damental TRIG onometric Identities In trigonometry, there are expressions and equations that are true for any given angle. These are called identities. An infinite number of trigonometric identities exist, and we are going to prove many of these
Section 7.3 Double Angle Identities . 431 Section 7.4 Modeling Changing Amplitude and Midline. 442 Section 7.1 Solving Trigonometric Equations with Identities In the last chapter, we solved basic trigonometric equations.
for trigonometric functions can be substituted to allow scientists to analyse data or solve a problem more efficiently. In this chapter, you will explore equivalent trigonometric expressions. Trigonometric Identities Key Terms trigonometric identity Elizabeth Gleadle, of Vancouver, British Columbia, holds the Canadian women’s
