Analytic Trigonometry

Transcription

1 CHAPTER 5 Analtic Trigonometr 5. Fundamental Identities 5. Proving Trigonometric Identities 5.3 Sum and Difference Identities 5.4 Multiple-Angle Identities 5.5 The Law of Sines 5.6 The Law of Cosines It is no surprise that naturalists seeking to estimate wildlife populations must have an understanding of geometr (a word that literall means earth measurement ). You will learn in this chapter that trigonometr, with its man connections to triangles and circles, enables us to etend the problem-solving tools of geometr significantl. On page 447 we will appl a result called Heron s Formula (which we prove trigonometricall) to estimate the local densit of a deer population. 403

2 404 CHAPTER 5 Analtic Trigonometr Chapter 5 Overview Although the title of this chapter suggests that we are now moving into the analtic phase of our stud of trigonometric functions, the truth is that we have been in that phase for several sections alread. Once the transition is made from triangle ratios to functions and their graphs, one is on analtic soil. But our primar applications of trigonometr so far have been computational; we have not made full use of the properties of the functions to stud the connections among the trigonometric functions themselves. In this chapter we will shift our emphasis more toward theor and proof, eploring where the properties of these special functions lead us, often with no immediate concern for real-world relevance at all. We hope in the process to give ou an appreciation for the rich and intricate tapestr of interlocking patterns that can be woven from the si basic trigonometric functions patterns that will take on even greater beaut later on when ou can view them through the lens of calculus. 5. Fundamental Identities What ou ll learn about Identities Basic Trigonometric Identities Pthagorean Identities Cofunction Identities Odd-Even Identities Simplifing Trigonometric Epressions Solving Trigonometric Equations... and wh Identities are important when working with trigonometric functions in calculus. Identities As ou probabl realize b now, the smbol = means several different things in mathematics.. + = means equalit of real numbers. It is a true sentence = - 6 signifies equivalent epressions. It is a true sentence = 7 is an open sentence, because it can be true or false, depending on whether is a solution to the equation / + = - is an identit. It is a true sentence (ver much like () above), but with the important qualification that must be in the domain of both epressions. If either side of the equalit is undefined, the sentence is meaningless. Substituting - into both sides of the equation in (3) gives a sentence that is mathematicall false i.e., 4 = 7, whereas substituting - into both sides of the identit in (4) gives a sentence that is meaningless. Statements like tan u = sin u/cos u and csc u = /sin u are trigonometric identities because the are true for all values of the variable for which both sides of the equation are defined. The set of all such values is called the domain of validit of the identit. We will spend much of this chapter eploring trigonometric identities, their proofs, their implications, and their applications. Basic Trigonometric Identities Some trigonometric identities follow directl from the definitions of the si basic trigonometric functions. These basic identities consist of the reciprocal identities and the quotient identities. Basic Trigonometric Identities Reciprocal Identities csc u = sec u = sin u sin u = cos u = csc u Quotient Identities tan u = sin u cos u cos u sec u cot u = tan u = cot u = cos u sin u tan u cot u

3 SECTION 5. Fundamental Identities 405 EXPLORATION Making a Point About Domain of Validit. u = 0 is in the domain of validit of eactl three of the basic identities. Which three?. For eactl two of the basic identities, one side of the equation is defined at u = 0 and the other side is not. Which two? 3. For eactl three of the basic identities, both sides of the equation are undefined at u = 0. Which three? Pthagorean Identities Eploration in Section 4.3 introduced ou to the fact that, for an real number t, the numbers cos t and sin t alwas sum to. This is clearl true for the quadrantal angles that wrap to the points, 0 and 0,, and it is true for an other t because cos t and sin t are the (signed) lengths of legs of a reference triangle with hpotenuse (Figure 5.). No matter what quadrant the triangle lies in, the Pthagorean Theorem guarantees the following identit: cos t + sin t =. If we divide each term of the identit b cos t, we get an identit that involves tangent and secant: cos t sin t + cos t cos t = cos t + tan t = sec t If we divide each term of the identit b sin t, we get an identit that involves cotangent and cosecant: cos t sin t sin t + sin t = sin t sin t (cos t, sin t) cos t (, 0) cot t + = csc t These three identities are called the Pthagorean identities, which we restate using the shorthand notation for powers of trigonometric functions. FIGURE 5. B the Pthagorean Theorem, cos t + sin t =. Pthagorean Identities cos u + sin u = + tan u = sec u cot u + = csc u EXAMPLE Using Identities Find sin u and cos u if tan u = 5 and cos u 7 0. SOLUTION We could solve this problem b the reference triangle techniques of Section 4.3 (see Eample 7 in that section), but we will show an alternate solution here using onl identities. First, we note that sec u = + tan u = + 5 = 6, so sec u = 6. Since sec u = 6, we have cos u = /sec u = / 6. But cos u 7 0, so cos u = /6.

4 406 CHAPTER 5 Analtic Trigonometr Finall, tan u = 5 sin u cos u = 5 sin u = 5 cos u = 5a 6 b = 5 6 Therefore, sin u = 5 and cos u = Now tr Eercise A r FIGURE 5. Angles A and B are complements in right ABC. B C If ou find ourself preferring the reference triangle method, that s fine. Remember that combining the power of geometr and algebra to solve problems is one of the themes of this book, and the instinct to do so will serve ou well in calculus. Cofunction Identities If C is the right angle in right ABC, then angles A and B are complements. Notice what happens if we use the usual triangle ratios to define the si trigonometric functions of angles A and B (Figure 5.). Angle A: sin A = r tan A = sec A = r cos A = r cot A = csc A = r Angle B: sin B = r tan B = sec B = r cos B = r cot B = csc B = r Do ou see what happens? In ever case, the value of a function at A is the same as the value of its cofunction at B. This alwas happens with complementar angles; in fact, it is this phenomenon that gives a co function its name. The co stands for complement. Cofunction Identities sin a p - ub = cos u cos a p - ub = sin u tan a p - ub = cot u cot a p - ub = tan u sec a p - ub = csc u csc a p - ub = sec u Although our argument on behalf of these equations was based on acute angles in a triangle, these equations are genuine identities, valid for all real numbers for which both sides of the equation are defined. We could etend our acute-angle argument to produce a general proof, but it will be easier to wait and use the identities of Section 5.3. We will revisit this particular set of fundamental identities in that section. Odd-Even Identities We have seen that ever basic trigonometric function is either odd or even. Either wa, the usual function relationship leads to another fundamental identit.

5 SECTION 5. Fundamental Identities 407 Odd-Even Identities sin - = -sin cos - = cos tan - = -tan csc - = -csc sec - = sec cot - = -cot EXAMPLE Using More Identities If cos u = 0.34, find sin u - p/. SOLUTION This problem can best be solved using identities. sin au - p b = -sin a p - ub = -cos u = Sine is odd. Cofunction identit Now tr Eercise 7. Simplifing Trigonometric Epressions In calculus it is often necessar to deal with epressions that involve trigonometric functions. Some of those epressions start out looking fairl complicated, but it is often possible to use identities along with algebraic techniques (e.g., factoring or combining fractions over a common denominator) to simplif the epressions before dealing with them. In some cases the simplifications can be dramatic. EXAMPLE 3 Simplifing b Factoring and Using Identities Simplif the epression sin 3 + sin cos. SOLUTION Solve Algebraicall sin 3 + sin cos = sin sin + cos = sin Pthagorean identit = sin Support Graphicall We recognize the graph of = sin 3 + sin cos (Figure 5.3a) as the same as the graph of = sin (Figure 5.3b). Now tr Eercise 3. [ π, π ] b [ 4, 4] (a) [ π, π ] b [ 4, 4] (b) FIGURE 5.3 Graphical support of the identit sin 3 + sin cos = sin. (Eample 3) EXAMPLE 4 Simplifing b Epanding and Using Identities Simplif the epression 3sec + sec - 4/sin. SOLUTION Solve Algebraicall sec + sec - sin = sec - sin = tan sin = sin cos = cos = sec # sin a + ba - b = a - b Pthagorean identit tan = sin cos (continued)

6 408 CHAPTER 5 Analtic Trigonometr [ π, π ] b [, 4] (a) [ π, π ] b [, 4] (b) FIGURE 5.4 Graphical support of the identit sec + sec - /sin = sec. (Eample 4) Support Graphicall sec + sec - The graphs of = and = sec appear to be identical, as sin epected (Figure 5.4). Now tr Eercise 5. EXAMPLE 5 Simplifing b Combining Fractions and Using Identities cos Simplif the epression - sin - sin cos. SOLUTION cos - sin - sin cos = = cos # cos - sin cos - sin # - sin cos - sin cos cos - sin - sin - sin cos Rewrite using common denominator. = cos - sin + sin - sin cos - sin = Pthagorean identit - sin cos = cos = sec (We leave it to ou to provide the graphical support.) Now tr Eercise 37. We will use these same simplifing techniques to prove trigonometric identities in Section 5.. Solving Trigonometric Equations The equation-solving capabilities of calculators have made it possible to solve trigonometric equations without understanding much trigonometr. This is fine, to the etent that solving equations is our goal. However, since understanding trigonometr is also a goal, we will occasionall pause in our development of identities to solve some trigonometric equations with paper and pencil, just to get some practice in using the identities. EXAMPLE 6 Solving a Trigonometric Equation Find all values of in the interval 30, p that solve cos 3 /sin = cot. SOLUTION cos 3 sin = cot cos 3 sin = cos sin cos 3 = cos Multipl both sides b sin. cos 3 - cos = 0 cos cos - = 0 cos -sin = 0 Pthagorean identit cos = 0 or sin = 0

7 SECTION 5. Fundamental Identities 409 We reject the possibilit that sin = 0 because it would make both sides of the original equation undefined. The values in the interval 30, p that solve cos = 0 (and therefore cos 3 /sin = cot are p/ and 3p/. Now tr Eercise 5. EXAMPLE 7 Solving a Trigonometric Equation b Factoring Find all solutions to the trigonometric equation sin + sin =. SOLUTION Let = sin. The equation + = can be solved b factoring: + = + - = = 0 - = 0 or + = 0 = or = - So, in the original equation, sin = / or sin = -. Figure 5.5 shows that the solutions in the interval 30, p are p/6, 5p/6, and 3p/. (a) (b) FIGURE 5.5 (a) sin = / has two solutions in 30, p: p/6 and 5p/6. (b) sin = - has one solution in 30, p: 3p/. (Eample 7) To get all real solutions, we simpl add integer multiples of the period, p, of the periodic function sin : 0.7 FIGURE 5.6 There are two points on the unit circle with -coordinate 0.7. (Eample 8) = p 6 + np or = 5p 6 + np or = 3p + np n = 0,,, Á Now tr Eercise 57. You might tr solving the equation in Eample 7 on our grapher for the sake of comparison. Finding all real solutions still requires an understanding of periodicit, and finding eact solutions requires the savv to divide our calculator answers b p. It is likel that anone who knows that much trigonometr will actuall find the algebraic solution to be easier!

8 40 CHAPTER 5 Analtic Trigonometr 4 3 = = cos 3 4 FIGURE 5.7 Intersecting the graphs of = cos and = 0.7 gives two solutions to the equation cos t = 0.7. (Eample 8) EXAMPLE 8 Solving a Trig Equation with a Calculator Find all solutions to the equation cos t = 0.7, using a calculator where needed. SOLUTION Figure 5.6 shows that there are two points on the unit circle with an -coordinate of 0.7. We do not recognize this value as one of our special triangle ratios, but we can use a graphing calculator to find the smallest positive and negative values for which cos = 0.7 b intersecting the graphs of = cos and = 0.7 (Figure 5.7). The two values are predictabl opposites of each other: t L Using the period of cosine (which is p), we get the complete solution set: np n = 0,,, 3, Á 6. Now tr Eercise 63. QUICK REVIEW 5. (For help, go to Sections A., A.3, and 4.7.) Eercise numbers with a gra background indicate problems that the authors have designed to be solved without a calculator. In Eercises 4, evaluate the epression... cos - a 3 5 b 3. sin - a 3 b cos - 4 a - 5 b 4. sin - 5 a - 3 b In Eercises 5 8, factor the epression into a product of linear factors. 5. a - ab + b 6. 4u + 4u v - 5v - 3 In Eercises 9, simplif the epression. 9. a 0. + b / + / + SECTION 5. EXERCISES In Eercises 4, evaluate without using a calculator. Use the Pthagorean identities rather than reference triangles. (See Eample.). Find sin u and cos u if tan u = 3/4 and sin u Find sec u and csc u if tan u = 3 and cos u Find tan u and cot u if sec u = 4 and sin u Find sin u and tan u if cos u = 0.8 and tan u 6 0. In Eercises 5 8, use identities to find the value of the epression. 5. If sin u = 0.45, find cos p/ - u. 6. If tan p/ - u = -5.3, find cot u. 7. If sin u - p/ = 0.73, find cos -u. 8. If cot -u = 7.89, find tan (u - p/). In Eercises 9 6, use basic identities to simplif the epression. 9. tan cos 0. cot tan. sec sin p/ -. cot u sin u + tan - cos u csc sin u sin u + tan u + cos u cos - cos 3 sec u In Eercises 7, simplif the epression to either or sin csc - 8. sec - cos - 9. cot - cot p/ - 0. cot - tan -. sin - + cos -. sec - - tan In Eercises 3 6, simplif the epression to either a constant or a basic trigonometric function. Support our result graphicall. tan p/ - csc 3. csc + tan 4. + cot 5. sec + csc - tan + cot 6. sec u - tan u cos v + sin v

9 SECTION 5. Fundamental Identities 4 In Eercises 7 3, use the basic identities to change the epression to one involving onl sines and cosines. Then simplif to a basic trigonometric function. 7. sin tan + cot 8. sin u - tan u cos u + cos p/ - u 9. sin cos tan sec csc sec - tan sec + tan 30. sec tan 3. csc + tan sec sec csc 3. sec + csc In Eercises 33 38, combine the fractions and simplif to a multiple of a power of a basic trigonometric function e.g., 3 tan sin + sin + sec tan + sin sin sec - - cot - sin cos sec + sec sin cos + - cos sin - sin cos sin In Eercises 39 46, write each epression in factored form as an algebraic epression of a single trigonometric function e.g., sin + 3sin cos + cos sin + sin 4. - sin + - cos 4. sin - cos cos - sin sin + csc tan sin csc cot 46. sec - sec + tan In Eercises 47 50, write each epression as an algebraic epression of a single trigonometric function e.g., sin sin tan a sin + tan a sin tan cos sec + In Eercises 5 56, find all solutions to the equation in the interval 30, p. You do not need a calculator. 5. cos sin - cos = 0 5. tan cos - tan = tan sin = tan 54. sin tan = sin 55. tan = sin = In Eercises 57 6, find all solutions to the equation. You do not need a calculator cos - 4 cos + = sin + 3 sin + = sin u - sin u = sin t = cos t 6. cos sin = 6. sin + 3 sin = In Eercises 63 68, find all solutions to the trigonometric equation, using a calculator where needed. 63. cos = cos = sin = tan = cos = sin = 0.4 In Eercises 69 74, make the suggested trigonometric substitution, and then use Pthagorean identities to write the resulting function as a multiple of a basic trigonometric function , +, - 9, 36 -, + 8, - 00, = cos u = tan u = 3 sec u = 6 sin u = 9 tan u = 0 sec u Standardized Test Questions 75. True or False If sec - p/ = 34, then csc = 34. Justif our answer. 76. True or False The domain of validit for the identit sin u = tan u cos u is the set of all real numbers. Justif our answer. You should answer these questions without using a calculator. 77. Multiple Choice Which of the following could not be set equal to sin as an identit? (A) cos a p - b (B) cos a - p b (C) - cos (D) tan sec (E) -sin Multiple Choice Eactl four of the si basic trigonometric functions are (A) odd. (B) even. (C) periodic. (D) continuous. (E) bounded. 79. Multiple Choice A simpler epression for sec u + sec u - is (A) sin u. (B) cos u. (C) tan u. (D) cot u. (E) sec u. 80. Multiple Choice How man numbers between 0 and p solve the equation 3 cos + cos =? (A) None (B) One (C) Two (D) Three (E) Four

10 4 CHAPTER 5 Analtic Trigonometr Eplorations 8. Write all si basic trigonometric functions entirel in terms of sin. 8. Write all si basic trigonometric functions entirel in terms of cos. 83. Writing to Learn Graph the functions = sin and = -cos in the standard trigonometric viewing window. Describe the apparent relationship between these two graphs and verif it with a trigonometric identit. 84. Writing to Learn Graph the functions = sec and = tan in the standard trigonometric viewing window. Describe the apparent relationship between these two graphs and verif it with a trigonometric identit. 85. Orbit of the Moon Because its orbit is elliptical, the distance from the Moon to the Earth in miles (measured from the center of the Moon to the center of the Earth) varies periodicall. On Frida, Januar 3, 009, the Moon was at its apogee (farthest from the Earth). The distance of the Moon from the Earth each Frida from Januar 3 to March 7 is recorded in Table 5.. Table 5. Distance from Earth to Moon Date Da Distance Jan 3 0 5,966 Jan ,344 Feb 6 4 5,784 Feb 3 40,385 Feb 0 8 5,807 Feb ,35 Mar 6 4 6,0 Mar ,390 Mar ,333 Mar ,347 (a) Draw a scatter plot of the data, using da as and distance as. (b) Use our calculator to do a sine regression of on. Find the equation of the best-fit sine curve and superimpose its graph on the scatter plot. (c) What is the approimate number of das from apogee to apogee? Interpret this number in terms of the orbit of the moon. (d) Approimatel how far is the Moon from the Earth at perigee (closest distance)? (e) Since the data begin at apogee, perhaps a cosine curve would be a more appropriate model. Use the sine curve in part (b) and a cofunction identit to write a cosine curve that fits the data. 86. Group Activit Divide our class into si groups, each assigned to one of the basic trigonometric functions. With our group, construct a list of five different epressions that can be simplified to our assigned function. When ou have finished, echange lists with our cofunction group to check one another for accurac. Etending the Ideas 87. Prove that sin 4 u - cos 4 u = sin u - cos u. 88. Find all values of k that result in sin + = k sin having an infinite solution set. 89. Use the cofunction identities and odd-even identities to prove that sin p - = sin. 3Hint: sin p - = sin p/ - - p/ Use the cofunction identities and odd-even identities to prove that cos p - = -cos. 3Hint: cos p - = cos p/ - - p/.4 9. Use the identit in Eercise 89 to prove that in an ABC, sin A + B = sin C. 9. Use the identities in Eercises 89 and 90 to find an identit for simplifing tan p -. Source: The World Almanac and Book of Facts 009.