The 1958 musical Me Andew staed Dann Kae as Andew Laabee, a teache with a flai fo using unconventional methods in his classes. He uses a musical numbe to teach the Pthagoean theoem, singing and dancing to The Squae of the Hpotenuse :... Paallel lines don t connect, which is just about what ou might epect.though scientific laws ma change and decimals can be moved, the following is constant, and has et to be dispoved: The squae of the hpotenuse of a ight tiangle is equal to the sum of the squaes of the two adjacent sides. The Pthagoean theoem, intoduced in Chapte 9, is used etensivel in the stud of tigonomet, the foundations of which go back at least 3 eas.the wod tigonomet comes fom the Geek wods fo tiangle (tigon) and measuement (met). Toda tigonomet is used in electonics, suveing, and othe engineeing aeas, and is necessa fo futhe couses in mathematics, such as calculus. 14.1 Angles and Thei Measues 14.2 Tigonometic Functions of Angles 14.3 Tigonometic Identities 14.4 Right Tiangles and Function Values 14.5 Applications of Right Tiangles 14.6 The Laws of Sines and Cosines Etension Aea Fomulas fo Tiangles Collaboative Investigation Making a Point about Tigonometic Function Values Chapte 14 Test 749
75 CHAPTER 14 Tigonomet Line AB A B Segment AB A B Ra AB A B Figue 1 Teminal side Vete A Initial side Figue 2 14.1 ANGLES AND THEIR MEASURES Basic Teminolog Degee Measue Angles in a Coodinate Sstem Basic Teminolog A line ma be dawn though the two distinct points A and B. This line is called line AB. The potion of the line between A and B, including points A and B themselves, is segment AB. The potion of the line AB that stats at A and continues though B, and on past B, is called a AB. Point A is the endpoint of the a. See Figue 1. An angle is fomed b otating a a aound its endpoint. The a in its initial position is called the initial side of the angle, while the a in its location afte the otation is the teminal side of the angle. The endpoint of the a is the vete of the angle. See Figue 2. If the otation of the teminal side is counteclockwise, the angle measue is positive. If the otation is clockwise, the angle measue is negative. See Figue 3. C A B Positive angle Figue 3 Negative angle An angle can be named b using the name of its vete. Fo eample, the angle on the ight in Figue 3 can be called angle C. Altenativel, an angle can be named using thee lettes, with the vete lette in the middle. Thus, the angle on the ight also could be named angle ACB o angle BCA. A complete otation of a a gives an angle whose measue is 36. Figue 4 Degee Measue Thee ae two sstems in common use fo measuing the sizes of angles. The most common unit of measue is the degee. (The othe common unit of measue is called the adian.) Degee measue was developed b the Bablonians 4 eas ago. To use degee measue we assign 36 degees to a complete otation of a a. In Figue 4, notice that the teminal side of the angle coesponds to its initial side when it makes a complete otation. 1 One degee, witten 1, epesents of a otation. Theefoe, epesents 9 18 of a complete otation, and epesents 36 = 1 36 = 1 36 9 4 18 2 of a complete otation. Angles of measue 5, 9, and 18 ae shown in Figue 5. 5 angle 9 18 Figue 5
14.1 Angles and Thei Measues 751 Tigonomet, pehaps moe than an othe banch of mathematics, developed as the esult of a continual and fetile intepla of suppl and demand: the suppl of applicable mathematical theoies and techniques available at an given time and the demands of a single applied science, astonom. So intimate was the elation that not until the thiteenth centu was it useful to egad the two subjects as sepaate entities. (Fom The Histo of Tigonomet b Edwad S. Kenned, in Histoical Topics fo the Mathematics Classoom, the Thit-fist Yeabook of N.C.T.M., 1969.) Special angles ae named as shown in the following chat. Name Angle Measue Eample(s) Acute angle Between and 9 6 82 Right angle Eactl 9 9 Obtuse angle Between 9 and 18 97 138 Staight angle Eactl 18 18 If the sum of the measues of two angles is 9, the angles ae called complementa. Two angles with measues whose sum is 18 ae supplementa. EXAMPLE 1 Finding Complement and Supplement Give the complement and the supplement of 5. The complement of 5 is 9-5 = 4. The supplement of 5 is 18-5 = 13. Do not confuse an angle with its measue. The angle itself consists of the vete togethe with the initial and teminal sides.the measue of the angle is the size of the otation angle fom the initial to the teminal side (commonl epessed in degees). Fo eample, if angle A has a 35 otation angle, we sa that m1angle A2 is 35, whee m1angle A2 is ead the measue of angle A. We abbeviate m1angle A2 = 35 as simpl angle A = 35, o just A = 35. Taditionall, potions of a degee have been measued with minutes and seconds. One minute, witten 1 œ, is of a 1 degee. 1 6 One second, 1 fl, is of a minute. 6 1 1 o 6 1 6 1 1 1 o 6 1 6 36 The measue 12 42 38 epesents 12 degees, 42 minutes, 38 seconds.
752 CHAPTER 14 Tigonomet EXAMPLE 2 Calculating with Degee Measue Pefom each calculation. (a) 51 29 +32 46 (b) 9-73 12 51 29' 32 46' DMS 84 15''' 9 ' 73 12' DMS 16 48''' The calculations eplained in Eample 2 can be done with a gaphing calculato capable of woking with degees, minutes, and seconds. (a) Add the degees and the minutes sepaatel. (b) 51 29 + 32 46 83 75 9-73 12 Simplif. Wite 9 as 89 6. 83 + 1 15 84 15 89 6-73 12 16 48 75 = 6 + 15 = 1 15 Angles can be measued in decimal degees. Fo eample, 12.4238 epesents 4238 12.4238 = 12. 1, EXAMPLE 3 Conveting between Decimal Degees and Degees, Minutes, Seconds (a) Convet 74 8 14 to decimal degees. Round to the neaest thousandth of a degee. (b) Convet 34.817 to degees, minutes and seconds. Round to the neaest second. 74 8'14'' 74 8'14'' 34.817 DMS 74.13722222 74.137 34 49'1.2'' (a) 74 8 14 = 74 + 8 14 + 6 36 L 74 +.1333 +.39 = 74.137 1 = 1 6 and 1 = 1 36 Add. Round to thee decimal places. The convesions in Eample 3 can be done on some gaphing calculatos. The second displaed esult was obtained b setting the calculato to show onl thee places afte the decimal point. (b) 34.817 = 34 +.817 = 34 + 1.8172(6 2 = 34 + 49.2 = 34 + 49 +.2 = 34 + 49 + 1.2216 2 = 34 + 49 +1.2 L 34 49 1 1 = 6 1 = 6 Angles in a Coodinate Sstem An angle u (the Geek lette theta) * is in standad position if its vete is at the oigin of a ectangula coodinate sstem and its initial side lies along the positive -ais. The two angles shown in Figues 6(a) and (b) on the net page ae in standad position. An angle in standad position is said to lie in the quadant in which its teminal side lies. Fo eample, an acute angle is in quadant I and an obtuse angle is in quadant II. *The lettes of the Geek alphabet ae identified in a magin note on page 758.
14.1 Angles and Thei Measues 753 Figue 6(c) shows anges of angle measues fo each quadant when 6 u 6 36. Angles in standad position having thei teminal sides along the -ais o -ais, such as angles with measues 9, 18, 27, and so on, ae called quadantal angles. Vete Q I Teminal side Initial side Q II 18 Q II 9 < < 18 Q III 18 < < 27 9 27 (a) (b) (c) Figue 6 Q I < < 9 Q IV 27 < < 36 36 A complete otation of a a esults in an angle of measue 36. If the otation is continued, angles of measue lage than 36 can be poduced. The angles in Figue 7(a) have measues 6 and 42. These two angles have the same initial side and the same teminal side, but diffeent amounts of otation. Angles that have the same initial side and the same teminal side ae called coteminal angles. As shown in Figue 7(b), angles with measues 11 and 83 ae coteminal. (a) 6 42 Coteminal angles Figue 7 11 (b) 83 Coteminal angles 188 98 EXAMPLE 4 Finding Measues of Coteminal Angles Figue 8 Find the angle of least possible positive measue coteminal with each angle. (a) 98 (b) -75 (a) Add o subtact 36 fom 98 as man times as needed to get an angle with measue geate than but less than 36. Because 285 75 98 2 # 36 = 98 72 = 188, an angle of 188 is coteminal with an angle of 98. See Figue 8. (b) See Figue 9. Use a otation of Figue 9 36 + 1-75 2 = 285.
754 CHAPTER 14 Tigonomet Sometimes it is necessa to find an epession that will geneate all angles coteminal with a given angle. Fo eample, because an angle coteminal with 6 can be obtained b adding an appopiate intege multiple of 36 to 6, we can let n epesent an intege, and the epession 6 n # 36 will epesent all such coteminal angles. Table 1 shows a few possibilities. Table 1 Value of n Angle Coteminal with 6 2 1 6 + 2 # 36 = 78 6 + 1 # 36 = 42 6 + # 36 = 6 (the angle itself) -1 6 + 1-12 # 36 = 3 14.1 EXERCISES Give (a) the complement and (b) the supplement of each angle. 1. 3 2. 6 3. 45 4. 55 5. 89 6. 2 7. If an angle measues degees, how can we epesent its complement? 8. If an angle measues degees, how can we epesent its supplement? Pefom each calculation. 9. 62 18 +21 41 1. 75 15 +83 32 11. 71 58 +47 29 12. 9-73 48 13. 9-51 28 14. 18-124 51 15. 9-72 58 11 16. 9-36 18 47 Convet each angle measue to decimal degees. Use a calculato, and ound to the neaest thousandth of a degee. 17. 2 54 18. 38 42 19. 91 35 54 2. 34 51 35 21. 274 18 59 22. 165 51 9 Convet each angle measue to degees, minutes, and seconds. Use a calculato, and ound to the neaest second. 23. 31.4296 24. 59.854 25. 89.94 26. 12.3771 27. 178.5994 28. 122.6853 Find the angle of least positive measue coteminal with each angle. 29. -4 3. -98 31. -125 32. -23 33. 539 34. 699 35. 85 36. 1 Give an epession that geneates all angles coteminal with the given angle. Let n epesent an intege. 37. 3 38. 45 39. 6 4. 9 Sketch each angle in standad position. Daw an aow epesenting the coect amount of otation. Find the measue of two othe angles, one positive and one negative, that ae coteminal with the given angle. Give the quadant of each angle. 41. 75 42. 89 43. 174 44. 234 45. 3 46. 512 47. -61 48. - 159
14.2 Tigonometic Functions of Angles 755 Q P(, ) O Figue 1 θ 14.2 TRIGONOMETRIC FUNCTIONS OF ANGLES Tigonometic Functions Undefined Function Values Tigonometic Functions The stud of tigonomet coves the si tigonometic functions defined in this section. To define these si basic functions, stat with an angle u in standad position. Choose an point P having coodinates 1, 2 on the teminal side of angle u. (The point P must not be the vete of the angle.) See Figue 1. A pependicula fom P to the -ais at point Q detemines a tiangle having vetices at O, P, and Q.The distance fom P1, 2 to the oigin, 1, 2, can be found fom the distance fomula. = 21-2 2 + 1-2 2 Distance fomula = 2 2 + 2 Notice that >, because this distance is neve negative. The si tigonometic functions of angle u ae called sine, cosine, tangent, cotangent, secant, and cosecant. In the following definitions, we use the customa abbeviations fo the names of these functions: sin, cos, tan, cot, sec, csc. Definitions of the Tigonometic Functions Let 1, 2 be a point othe than the oigin on the teminal side of an angle u in standad position. The distance fom the point to the oigin is = 2 2 + 2. The si tigonometic functions of u ae defined as follows. sin U cos U tan U 1 2 csc U 1 2 sec U 1 2 cot U 1 2 17 θ 8 (8, 15) 15 Figue 11 = 8 = 15 = 17 Although Figue 1 shows a second quadant angle, these definitions appl to an angle u. Due to the estictions on the denominatos in the definitions of tangent, cotangent, secant, and cosecant, quadantal angles will have some undefined function values. EXAMPLE 1 Finding Function Values of an Angle The teminal side of an angle u in standad position passes though the point 18, 152. Find the values of the si tigonometic functions of angle u. Figue 11 shows angle u and the tiangle fomed b dopping a pependicula fom the point 18, 152 to the -ais. The point 18, 152 is 8 units to the ight of the -ais and 15 units above the -ais, so that = 8 and = 15. = 2 2 + 2 = 28 2 + 152 = 264 + 225 = 2289 = 17 Let 8 and 15. Squae 8 and 15. Add. Find the squae oot.
756 CHAPTER 14 Tigonomet The values of the si tigonometic functions of angle u can now be found with the definitions given in the bo, whee = 8, = 15, and = 17. sin u = = 15 17 csc u = = 17 15 cos u = = 8 17 sec u = = 17 8 tan u = = 15 8 cot u = = 8 15 EXAMPLE 2 Finding Function Values of an Angle 4 3 5 θ ( 3, 4) Figue 12 = 3 = 4 = 5 The teminal side of an angle u in standad position passes though the point 1-3, -42. Find the values of the si tigonometic functions of u. As shown in Figue 12, = -3 and = -4. Find the value of. = 21-32 2 + 1-42 2 = 225 = 5 Remembe that 7 O. Then use the definitions of the tigonometic functions. sin u = -4 5 =-4 5 csc u = 5-4 =-5 4 cos u = -3 5 =-3 5 sec u = 5-3 =-5 3 tan u = -4-3 = 4 3 cot u = -3-4 = 3 4 OP = OP = (, ) θ P (, ) O Q Q Figue 13 P The si tigonometic functions can be found fom an point on the teminal side of the angle othe than the oigin. To see wh an point ma be used, efe to Figue 13, which shows an angle u and two distinct points on its teminal side. Point P has coodinates 1, 2 and point P (ead P-pime ) has coodinates 1, 2. Let be the length of the hpotenuse of tiangle OPQ, and let be the length of the hpotenuse of tiangle OP Q. Because coesponding sides of simila tiangles ae in popotion, =, and thus sin u = is the same no matte which point is used to find it. Simila esults hold fo the othe five functions. Undefined Function Values If the teminal side of an angle in standad position lies along the -ais, an point on this teminal side has -coodinate. Similal, an angle with teminal side on the -ais has -coodinate fo an point on the teminal side. Because the values of and appea in the denominatos of some of the tigonometic functions, and because a faction is undefined if its denominato is, some of the tigonometic function values of quadantal angles will be undefined. EXAMPLE 3 Finding Function Values and Undefined Function Values Find values of the tigonometic functions fo each angle. Identif an that ae undefined. (a) an angle of 9 (b) an angle in standad position with teminal side though 1-3, 2
14.2 Tigonometic Functions of Angles 757 (, 1) 9 (a) θ ( 3, ) (b) Figue 14 (a) Fist, select an point on the teminal side of a 9 angle. We select the point 1, 12, as shown in Figue 14(a). Hee = and = 1. Veif that = 1. Then, use the definitions of the tigonometic functions. sin 9 = 1 1 = 1 csc 9 = 1 1 = 1 (b) Figue 14(b) shows the angle. Hee, = -3, =, and = 3, so the tigonometic functions have the following values. sin u = 3 = csc u = 3 1undefined2 Undefined Function Values cos 9 = 1 = sec 9 = 1 1undefined2-3 cos u = 3 = -1 sec u = 3-3 = -1 tan 9 = 1 1undefined2 cot 9 = 1 = tan u = -3 = cot u = -3 1undefined2 If the teminal side of a quadantal angle lies along the -ais, the tangent and secant functions ae undefined. If it lies along the -ais, the cotangent and cosecant functions ae undefined. Because the most commonl used quadantal angles ae, 9, 18, 27 and 36, the values of the functions of these angles ae summaized in Table 2. Table 2 Function Values of Quadantal Angles U sin U cos U tan U cot U sec U csc U 1 Undefined 1 Undefined 9 1 Undefined Undefined 1 18-1 Undefined -1 Undefined 27-1 Undefined Undefined -1 36 1 Undefined 1 Undefined 14.2 EXERCISES In Eecises 1 4, sketch an angle u in standad position such 7. 1, 22 that u has the least possible positive measue, and the given point is on the teminal side of u. 8. 1-4, 2 1. 1-3, 42 2. 1-4, -32 3. 15, -122 4. 1-12, -52 9. 11, 232 Find the values of the tigonometic functions fo the angles in standad position having the following points on thei teminal sides. Identif an that ae undefined. Rationalize denominatos when applicable. 5. 1-3, 42 6. 1-4, -32 1. 1-223, -22 11. 13, 52 12. 1-2, 72 13. 1-8, 2
758 CHAPTER 14 Tigonomet 14. 1, 92 15. Fo an nonquadantal angle u, sin u and csc u will have the same sign. Eplain wh this is so. 16. If cot u is undefined, what is the value of tan u? 17. How is the value of intepeted geometicall in the definitions of the sine, cosine, secant, and cosecant functions? 18. If the teminal side of an angle u is in quadant III, what is the sign of each of the tigonometic function values of u? Suppose that the point 1, 2 is in the indicated quadant. Decide whethe the given atio is positive o negative. (Hint: It ma be helpful to daw a sketch.) 19. II, 2. II, 21. III, 22. III, 23. IV, 24. IV, 25. IV, 26. IV, Use the appopiate definition to detemine each function value. If it is undefined, sa so. 27. cos 9 28. sin 9 29. tan 9 3. cot 9 31. sec 9 32. csc 9 33. sin 18 34. sin 27 35. tan 18 36. cot 27 37. sin1-27 2 38. cos1-27 2 39. tan 4. sec1-18 2 41. cos 18 42. cot The Geek Alphabet a alpha b beta g gamma d delta P epsilon z zeta h eta u theta i iota k kappa l lambda m mu n nu j i o omicon p pi ho s sigma t tau upsilon f phi chi c psi v omega 14.3 TRIGONOMETRIC IDENTITIES Recipocal Identities Signs of Function Values In Quadants Pthagoean Identities Quotient Identities Recipocal Identities The definitions of the tigonometic functions on page 755 wee witten so that functions diectl above and below one anothe ae ecipocals of each othe. Because sin u = and csc u =, 1 sin u = csc u and csc u = 1 sin u. Also, cos u and sec u ae ecipocals, as ae tan u and cot u. The ecipocal identities hold fo an angle u that does not lead to a zeo denominato. Recipocal Identities sin U 1 csc U csc U 1 sin U Identities ae equations that ae tue fo all meaningful values of the vaiable. When studing identities, be awae that vaious foms eist. Fo eample, sin U 1 csc U can also be witten cos U 1 sec U sec U 1 cos U csc U 1 sin U and You should become familia with all foms of these identities. tan U 1 cot U cot U 1 tan U 1sin U21csc U2 1.