44 CHPTER In Exercises 9 to 96, find the re, to the nerest squre unit, of the sector of circle with the given rdius nd centrl ngle. 9. r = inches, u = p rdins nerest 0 miles, the given city is from the equtor. Use 960 miles s the rdius of Erth. ERTH rdius 960 mi N 94. r =.8 feet,u = p rdins New York City 40 4 N Greenwich Meridin 9. r = 0 centimeters, u = 0.6 rdin Mimi 47 N r Equtor 96. r = 0 feet, u = 6 Ltitude describes the position of point on Erth s surfce in reltion to the equtor. point on the equtor hs ltitude of 0. The north pole hs ltitude of 90. In Exercises 97 nd 98, determine how fr north, to the Longitude est Ltitude north 97.The city of Mimi hs ltitude of 47 N. 98.New York City hs ltitude of 40 4 N. SECTION. The Six Trigonometric Functions Trigonometric Functions of Specil ngles pplictions Involving Right Tringles Right Tringle Trigonometry PREPRE FOR THIS SECTION Prepre for this section by completing the following exercises. The nswers cn be found on pge. PS. Rtionlize the denomintor of PS. Rtionlize the denomintor of.. [P.] [P.] PS. Simplify:, [P.] b PS4. Simplify: b, b [P.] PS. Solve for x. Round your nswer to the nerest hundredth. [.] = x PS6. Solve for x. Round your nswer to the nerest hundredth. [.] = x 8 The Six Trigonometric Functions The study of trigonometry, which mens tringle mesurement, begn more thn 000 yers go, prtilly s mens of solving surveying problems. Erly trigonometry used the length of line segment between two points of circle s the vlue of trigonometric function. In the sixteenth century, right tringles were used to define trigonometric function. We will use modifiction of this pproch.
. RIGHT TRINGLE TRIGONOMETRY 44 When working with right tringles, it is convenient to refer to the side opposite n ngle or the side djcent to (next to) n ngle. Figure.9 shows the sides opposite nd djcent to the ngle u. Figure.0 shows the sides opposite nd djcent to the ngle b. In both cses, the hypotenuse remins the sme. Hypotenuse Opposite side Hypotenuse β djcent side djcent side Opposite side Figure.9 Figure.0 djcent nd opposite sides of u djcent nd opposite sides of b Six rtios cn be formed by using two lengths of the three sides of right tringle. Ech rtio defines vlue of trigonometric function of given cute ngle u. The functions re sine (sin), cosine (cos), tngent (tn), cotngent (cot), secnt (sec), nd cosecnt (csc). Definitions of Trigonometric Functions of n cute ngle Let u be n cute ngle of right tringle. See Figure.9. The vlues of the six trigonometric functions of u re length of opposite side length of djcent side sinu = cosu = length of hypotenuse length of hypotenuse tnu = secu = length of opposite side length of djcent side length of hypotenuse length of djcent side cotu = cscu = length of djcent side length of opposite side length of hypotenuse length of opposite side We will write opp, dj, nd hyp s bbrevitions for the length of the opposite side, djcent side, nd hypotenuse, respectively. EXMPLE Evlute Trigonometric Functions Find the vlues of the six trigonometric functions of u for the tringle given in Figure.. Solution Use the Pythgoren Theorem to find the length of the hypotenuse. hyp hyp = + 4 = = From the definitions of the trigonometric functions, 4 Figure. sinu = opp hyp = cotu = dj opp = 4 cosu = dj hyp = 4 secu = hyp dj = 4 tnu = opp dj = 4 cscu = hyp opp = Try Exercise 6, pge 449
444 CHPTER Given the vlue of one trigonometric function of the cute ngle u, it is possible to find the vlue of ny of the remining trigonometric functions of u. EXMPLE Find the Vlue of Trigonometric Function Given tht u is n cute ngle nd cosu = find tnu. 8, Solution cosu = 8 = dj hyp Sketch right tringle with one leg of length units nd hypotenuse of length 8 units. Lbel s u the cute ngle tht hs the leg of length units s its djcent side (see Figure.). Use the Pythgoren Theorem to find the length of the opposite side. 8 Figure. opp Therefore, tnu = opp dj = 9. opp + = 8 opp + = 64 opp = 9 opp = 9 Try Exercise 8, pge 40 Trigonometric Functions of Specil ngles 4 r = 4 C Figure. In Exmple, the lengths of the legs of the tringle were given, nd you were sked to find the vlues of the six trigonometric functions of the ngle u. Often we will wnt to find the vlue of trigonometric function when we re given the mesure of n ngle rther thn the mesure of the sides of tringle. For most ngles, dvnced mthemticl methods re required to evlute trigonometric function. For some specil ngles, however, the vlue of trigonometric function cn be found by geometric methods. These specil cute ngles re 0, 4, nd 60. First, we will find the vlues of the six trigonometric functions of 4. (This discussion is bsed on ngles mesured in degrees. Rdin mesure could hve been used without chnging the results.) Figure. shows right tringle with ngles 4, 4, nd 90. ecuse =, the lengths of the sides opposite these ngles re equl. Let the length of ech equl side be denoted by. From the Pythgoren Theorem, The vlues of the six trigonometric functions of 4 re sin 4 = tn 4 = = = = r = + = r = = cos 4 = cot 4 = = = = sec 4 = = csc 4 = =
. RIGHT TRINGLE TRIGONOMETRY 44 C h = O 0 Figure.4 60 The vlues of the trigonometric functions of the specil ngles 0 nd 60 cn be found by drwing n equilterl tringle nd bisecting one of the ngles, s Figure.4 shows. The ngle bisector lso bisects one of the sides. Thus the length of the side opposite the 0 ngle is one-hlf the length of the hypotenuse of tringle O. Let denote the length of the hypotenuse. Then the length of the side opposite the 0 ngle is The length of the side djcent to the 0 ngle, h, is found by using the. Pythgoren Theorem. = b + h = 4 + h 4 = h h = Subtrct 4 Solve for h. from ech side. The vlues of the six trigonometric functions of 0 re sin 0 = > = tn 0 = sec 0 = > > = = > = = cos 0 = > cot 0 = > > csc 0 = > = = = The vlues of the trigonometric functions of 60 cn be found by gin using Figure.4. The length of the side opposite the 60 ngle is nd the length of the side djcent, to the 60 ngle is The vlues of the trigonometric functions of 60 re. sin 60 = > tn 60 = > > = = cos 60 = > = cot 60 = > > = = sec 60 = > = csc 60 = > = = Tble. on pge 446 summrizes the vlues of the trigonometric functions of the specil ngles 0 p>6,4 p>4, nd 60 p>.
446 CHPTER Study tip Memorizing the vlues given in Tble. will prove to be extremely useful in the remining trigonometry sections. Tble. U 0 ; p 6 4 ; p 4 60 ; p Trigonometric Functions of Specil ngles sin U cos U tn U csc U sec U cot U Question Wht is the mesure, in degrees, of the cute ngle u for which sinu = cosu, tnu = cotu, nd secu = cscu? EXMPLE Evlute Trigonometric Expression Study tip The ptterns in the following chrt cn be used to memorize the sine nd cosine of 0, 4, nd 60. sin 0 = sin 4 = sin 60 = cos 0 = cos 4 = cos 60 = Find the exct vlue of sin 4 + cos 60. Note: sin u =sinusinu =sinu nd cos u =cosucosu =cosu. Solution Substitute the vlues of sin 4 nd cos 60 nd simplify. sin 4 + cos 60 = b + b = 4 + 4 = 4 Try Exercise 4, pge 40 From the definition of the sine nd cosecnt functions, sinucscu = = opp hyp # hyp opp = y rewriting the lst eqution, we find sinu = cscu nd The sine nd cosecnt functions re clled reciprocl functions. The cosine nd secnt re lso reciprocl functions, s re the tngent nd cotngent functions. Tble. shows ech trigonometric function nd its reciprocl. These reltionships hold for ll vlues of u for which both of the functions re defined. or sinucscu = cscu =, provided sinu Z 0 sinu Tble. Trigonometric Functions nd Their Reciprocls sinu = cscu = cscu sinu cosu = secu = secu cosu tnu = cotu = cotu tnu nswer 4.
. RIGHT TRINGLE TRIGONOMETRY 447 Integrting Technology The vlues of the trigonometric functions of the specil ngles 0, 4, nd 60 shown in Tble. re exct vlues. If n ngle is not one of these specil ngles, then grphing clcultor often is used to pproximte the vlue of trigonometric function. For instnce, to find sin.4 on TI-8/TI-8 Plus/TI-84 Plus clcultor, first check tht the clcultor is in degree mode. Then use the sine function key SIN to key in sin(.4) nd press ENTER. See Figure.. To find sec., first check tht the clcultor is in rdin mode. TI-8/TI-8 Plus/TI-84 Plus clcultor does not hve secnt function key, but becuse the secnt function is the reciprocl of the cosine function, we cn evlute sec. by evluting /(cos.). See Figure.6. Line of sight Select Degree in the Mode menu. Norml Sci Eng Flot 046789 Rdin Degree Funcsin(.4) Cong Sequ.7989644 Rel Full Select Rdin in the Mode menu. Norml Sci Eng Flot 046789 Rdin Degree Func/(cos(.)) Cong Sequ.77694 Rel Full Line of sight ngle of elevtion Horizontl line ngle of depression Figure. Figure.6 When you evlute trigonometric function with clcultor, be sure the clcultor is in the correct mode. Mny errors re mde becuse the correct mode hs not been selected. Figure.7 pplictions Involving Right Tringles Some pplictions concern n observer looking t n object. In these pplictions, ngles of elevtion or ngles of depression re formed by line of sight nd horizontl line. If the object being observed is bove the observer, the cute ngle formed by the line of sight nd the horizontl line is n ngle of elevtion. If the object being observed is below the observer, the cute ngle formed by the line of sight nd the horizontl line is n ngle of depression. See Figure.7. EXMPLE 4 Solve n ngle-of-elevtion ppliction 64. ft Figure.8 h From point feet from the bse of redwood tree, the ngle of elevtion to the top of the tree is 64.. Find the height of the tree to the nerest foot. Solution From Figure.8, the length of the djcent side of the ngle is known ( feet). ecuse we need to determine the height of the tree (length of the opposite side), (continued)
448 CHPTER we use the tngent function. Let h represent the length of the opposite side. tn 64. = opp dj The height of the tree is pproximtely 9 feet. Try Exercise 6, pge 4 = h h = tn 64. L 8.9 Use clcultor to evlute tn 64.. Note The significnt digits of n pproximte number re every nonzero digit the digit 0, provided it is between two nonzero digits or it is to the right of nonzero digit in number tht includes deciml point For exmple, the pproximte number 0 hs significnt digits. 700 hs significnt digits. 47.0 hs significnt digits. 0.00 hs significnt digits. 0.00840 hs significnt digits. ecuse the cotngent function involves the sides djcent to nd opposite n ngle, we could hve solved Exmple 4 by using the cotngent function. The solution would hve been cot 64. = dj opp = h h = cot 64. L 8.9 feet The ccurcy of clcultor is sometimes beyond the limits of mesurement. In Exmple 4 the distnce from the bse of the tree ws given s feet (three significnt digits), wheres the height of the tree ws shown to be 8.9 feet (six significnt digits). When using pproximte numbers, we will use the conventions given below for clculting with trigonometric functions. Rounding Convention: Significnt Digits for Trigonometric Clcultions ngle Mesure to the Nerest Degree Tenth of degree Hundredth of degree Significnt Digits of the Lengths Two Three Four EXMPLE Solve n ngle-of-depression ppliction 7 x 60 mi Distnce mesuring equipment (DME) is stndrd vionic equipment on commercil irplne. This equipment mesures the distnce from plne to rdr sttion. If the distnce from plne to rdr sttion is 60 miles nd the ngle of depression is, find the number of ground miles from point directly below the plne to the rdr sttion. Solution From Figure.9, the length of the hypotenuse is known (60 miles). The length of the side opposite the ngle of 7 is unknown. The sine function involves the hypotenuse nd the opposite side, x, of the 7 ngle. sin 7 = x 60 x = 60 sin 7 L 4.87 Rounded to two significnt digits, the plne is 0 ground miles from the rdr sttion. Figure.9 Try Exercise 8, pge 4
. RIGHT TRINGLE TRIGONOMETRY 449 EXMPLE 6 Solve n ngle-of-elevtion ppliction n observer notes tht the ngle of elevtion from point to the top of spce shuttle is 7.. From point 7. meters further from the spce shuttle, the ngle of elevtion is.9. Find the height of the spce shuttle. y x 7. Figure.40.9 7. m Solution From Figure.40, let x denote the distnce from point to the bse of the spce shuttle, nd let y denote the height of the spce shuttle. Then () tn 7. = y nd () x Solving Eqution () for x, x = Eqution (), we hve y tn 7. = y cot 7., tn.9 = y x + 7. nd substituting into Note The intermedite clcultions in Exmple 6 were not rounded off. This ensures better ccurcy for the finl result. Using the rounding convention stted on pge 448, we round off only the lst result. tn.9 = y - y tn.9 cot 7. =tn.9 7. y - tn.9 cot 7. =tn.9 7. tn.9 7. y = - tn.9 cot 7. L 6.99 To three significnt digits, the height of the spce shuttle is 6. meters. Try Exercise 68, pge 4 y y cot 7. + 7. y = tn.9 y cot 7. + 7. Solve for y. EXERCISE SET. In Exercises to, find the vlues of the six trigonometric functions of U for the right tringle with the given sides.... 6. 8 7. 4. 7 4 9 7. 8. 9. 0. 6 0 0.8
40 CHPTER.. 6. cos p 4 tn p 6 + tn p 7. csc p 4 - secp cos p 6 6 In Exercises to, let U be n cute ngle of right tringle for which sinu Find.. tnu 4. secu. cosu In Exercises 6 to 8, let U be n cute ngle of right tringle for which tnu 4 Find. 8. tn p 4 + secp 6 sin p In Exercises 9 to 0, use clcultor to find the vlue of the trigonometric function to four deciml plces. 9. tn 40. sec 88 4. cos 6 0 4. cot 0 4. cos 4.7 44. tn 8. 6. sinu 7. cotu 8. secu 4. sec.9 46. sin p 47. tn p 7 In Exercises 9 to, let be n cute ngle of right tringle for which sec Find. 9. cosb 0. cotb. cscb In Exercises to 4, let U be n cute ngle of right tringle for which cosu Find.. sinu. secu 4. tnu In Exercises to 8, find the exct vlue of ech expression.. sin 4 + cos 4 6. csc 4 - sec 4 48. sec p 49. csc. 0. sin 0.4 8. Verticl Height from Slnt Height -foot ldder is resting ginst wll nd mkes n ngle of with the ground. Find the height to which the ldder will rech on the wll.. Distnce cross Mrsh Find the distnce cross the mrsh shown in the ccompnying figure. 7. sin 0 cos 60 - tn 4 8. 9. sin 0 cos 60 + tn 4 csc 60 sec 0 + cot 4 m C 0. sec 0 cos 0 - tn 60 cot 60. sin p. csc p + cosp 6 6 - secp. Width of Rmp skteborder wishes to build jump rmp tht is inclined t 9.0 ngle nd tht hs mximum height of.0 inches. Find the horizontl width x of the rmp.. sin p 4 + tnp 6 4. sin p cos p 4 - tnp 4.0 in.. sec p cos p - tnp 6 9.0 x
. RIGHT TRINGLE TRIGONOMETRY 4 4. Time of Closest pproch t :00 P.M., bot is. miles due west of rdr sttion nd trveling t mph in direction tht is 7. south of n est west line. t wht time will the bot be closest to the rdr sttion? from the building. Find the height of the building to the nerest tenth of foot. Trnsit 7.8 ft 7.. mi. ft. Plcement of Light For best illumintion of piece of rt, lighting specilist for n rt gllery recommends tht ceiling-mounted light be 6 feet from the piece of rt nd tht the ngle of depression of the light be 8. How fr from wll should the light be plced so tht the recommendtions of the specilist re met? Notice tht the rt extends outwrd 4 inches from the wll. 60. Width of Lke The ngle of depression to one side of lke, mesured from blloon 00 feet bove the lke s shown in the ccompnying figure, is 4. The ngle of depression to the opposite side of the lke is 7. Find the width of the lke. 4 in. 8 4 7 6 ft 00 ft 6. Height of the Eiffel Tower The ngle of elevtion from point 6 meters from the bse of the Eiffel Tower to the top of the tower is 68.9. Find the pproximte height of the tower. 7. Distnce of Descent n irplne trveling t 40 mph is descending t n ngle of depression of 6. How mny miles will the plne descend in 4 minutes? 6. stronomy The moon Europ rottes in nerly circulr orbit round Jupiter. The orbitl rdius of Europ is pproximtely 670,900 kilometers. During revolution of Europ round Jupiter, n stronomer found tht the mximum vlue of the ngle u formed by Europ, Erth, nd Jupiter ws 0.06. Find the distnce d between Erth nd Jupiter t the time the stronomer found the mximum vlue of u. Round to the nerest million kilometers. 8. Time of Descent submrine trveling t 9.0 mph is descending t n ngle of depression of. How mny minutes, to the nerest tenth, does it tke the submrine to rech depth of 80 feet? 9. Height of uilding surveyor determines tht the ngle of elevtion from trnsit to the top of building is 7.8. The trnsit is positioned. feet bove ground level nd feet Erth = 0.06 d Jupiter Not drwn to scle. Europ r = 670,900 km
4 CHPTER 6. stronomy Venus rottes in nerly circulr orbit round the sun. The lrgest ngle formed by Venus, Erth, nd the sun is 46.. The distnce from Erth to the sun is pproximtely 49 million kilometers. See the following figure. Wht is the orbitl rdius r of Venus? Round to the nerest million kilometers..9 6.4 0 ft Venus 66. Height of uilding Two buildings re 40 feet prt. The ngle of elevtion from the top of the shorter building to the top of the other building is. If the shorter building is 80 feet high, how high is the tller building? Erth 46. 49,000,000 km r Sun 67. Height of the Wshington Monument From point on line from the bse of the Wshington Monument, the ngle of elevtion to the top of the monument is 4.00. From point 00 feet wy from nd on the sme line, the ngle to the top is 7.77. Find the height of the Wshington Monument. 6. re of n Isosceles Tringle Consider the following isosceles tringle. The length of ech of the two equl sides of the tringle is, nd ech of the bse ngles hs mesure ofu. Verify tht the re of the tringle is = sin u cos u. 7.77 4.00 00.0 ft 64. re of Hexgon Find the re of the hexgon. (Hint: The re consists of six isosceles tringles. Use the formul from Exercise 6 to compute the re of one of the tringles nd multiply by 6.) b 4 in. 4 in. 60 60 4 in. 68. Height of Tower The ngle of elevtion from point to the top of tower is.. From point, which is on the sme line but. feet closer to the tower, the ngle of elevtion is 6.. Find the height of the tower. 6. Height of Pyrmid The ngle of elevtion to the top of the Egyptin pyrmid of Cheops is 6.4, mesured from point 0 feet from the bse of the pyrmid. The ngle of elevtion from the bse of fce of the pyrmid is.9. Find the height of the Cheops pyrmid. 6... ft 69. Length of Golf Drive The helipd of the urj l rb hotel is meters bove the surrounding beches. golfer drives golf bll off the edge of the helipd s shown in the
. RIGHT TRINGLE TRIGONOMETRY 4 following figure. Find the length (horizontl distnce ) of the drive. Helipd.0 Horizontl line C D meters 70. Size of Sign From point, t street level nd 0 feet from the bse of building, the ngle of elevtion to the top of the building is.. lso, from point the ngle of elevtion to the top of neon sign, which is top the building, is.9.. Determine the height of the building. b. How tll re the letters in the sign? C = 4 ft C =.6 is t ground level CD =. 7. n Eiffel Tower Replic Use the informtion in the ccompnying figure to estimte the height of the Eiffel Tower replic tht stnds in front of the Pris Ls Vegs Hotel in Ls Vegs, Nevd. Tony Crddock/Getty Imges..9 0 ft Street level 6. ft 46. 7. Rdius of Circle circle is inscribed in regulr hexgon with ech side 6.0 meters long. Find the rdius of the circle. 7. The Petrons Towers The Petrons Towers in Kul Lumpur, Mlysi, re the world s tllest twin towers. Ech tower is 48 feet in height. The towers re connected by skybridge t the forty-first floor. Note the informtion given in the following figure.. Determine the height of the skybridge. 74. re of Tringle Show tht the re of the tringle given in the figure is = b sin u. c b. Determine the length of the skybridge. b
44 CHPTER 7. Find Mximum Length Find the length of the longest piece of wood tht cn be slid round the corner of the hllwy in the figure following. Round to the nerest tenth of foot. 76. Find Mximum Length In Exercise 7, suppose tht the hll is 8 feet high. Find the length of the longest piece of wood tht cn be tken round the corner. Round to the nerest tenth of foot. ft ft SECTION. Trigonometric Functions of ny ngle Trigonometric Functions of Qudrntl ngles Signs of Trigonometric Functions The Reference ngle Trigonometric Functions of ny ngle PREPRE FOR THIS SECTION Prepre for this section by completing the following exercises. The nswers cn be found on pge 4. PS. Find the reciprocl of - [P.] PS. Find the reciprocl of [P.] 4.. PS. Evlute: 0-80 [P.] PS4. Simplify: p - 9p [P.] y P(x, y) r y x O Figure.4 x PS. Simplify: [P.] PS6. Simplify: - +- [P.] p -p Trigonometric Functions of ny ngle The pplictions of trigonometry would be quite limited if ll ngles hd to be cute ngles. Fortuntely, this is not the cse. In this section we extend the definition of trigonometric function to include ny ngle. Consider ngle u in Figure.4 in stndrd position nd point Px, yon the terminl side of the ngle. We define the trigonometric functions of ny ngle ccording to the following definitions. Definitions of the Trigonometric Functions of ny ngle Let Px, y be ny point, except the origin, on the terminl side of n ngle u in stndrd position. Let r = do, P, the distnce from the origin to P. The six trigonometric functions of u re sinu = y r cscu = r y, y Z 0 cosu = x r secu = r x, x Z 0 tnu = y x, x Z 0 cotu = x y, y Z 0 where r = x + y.