8.1 Right Triangle Trigonometry; Applications

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1 SECTION 8.1 Right Tringle Trigonometry; pplitions Right Tringle Trigonometry; pplitions PREPRING FOR THIS SECTION efore getting strted, review the following: Pythgoren Theorem (ppendix, Setion., pp ) Now Work the re You Prepred? prolems on pge 51. Trigonometri Equtions (Setion 7., pp ) OJECTIVES 1 Find the Vlue of Trigonometri Funtions of ute ngles Using Right Tringles (p. 505) Use the Complementry ngle Theorem (p. 507) Solve Right Tringles (p. 507) 4 Solve pplied Prolems (p. 508) Figure 1 1 Find the Vlue of Trigonometri Funtions of ute ngles Using Right Tringles tringle in whih one ngle is right ngle 190 is lled right tringle. Rell tht the side opposite the right ngle is lled the hypotenuse, nd the remining two sides re lled the legs of the tringle. In Figure 1(), we hve leled the hypotenuse s to indite tht its length is units, nd, in like mnner, we hve leled the legs s nd. euse the tringle is right tringle, the Pythgoren Theorem tells us tht In Figure 1(), we lso show the ngle u. The ngle u is n ute ngle: tht is, 0 6 u 6 90 for umesured in degrees nd 0 6 u 6 p for umesured in rdins. Ple uin stndrd position, s shown in Figure 1(). Then the oordintes of the point P re 1,. lso, P is point on the terminl side of utht is on the irle x + y =. (Do you see why?) Hypotenuse + = y O P (, ) x y x Figure Hypotenuse djent to Opposite () Now use the theorem on pge 74 with irle of rdius, x + y =. y referring to the lengths of the sides of the tringle y the nmes hypotenuse 1, opposite 1, nd djent 1, s indited in Figure, we n express the trigonometri funtions of us rtios of the sides of right tringle. sin u = os u = Opposite Hypotenuse = djent Hypotenuse = tn u = Opposite djent = () s u = Hypotenuse Opposite se u = Hypotenuse djent ot u = djent Opposite = = = (1) Notie tht eh trigonometri funtion of the ute ngle uis positive.

2 506 CHPTER 8 pplitions of Trigonometri Funtions EXMPLE 1 Finding the Vlue of Trigonometri Funtions from Right Tringle Find the ext vlue of the six trigonometri funtions of the ngle uin Figure. We see in Figure tht the two given sides of the tringle re Figure = Hypotenuse = 5, = djent = To find the length of the opposite side, we use the Pythgoren Theorem. 5 Opposite 1djent + 1Opposite = 1Hypotenuse + 1Opposite = 5 1Opposite = 5-9 = 16 Opposite = 4 Now tht we know the lengths of the three sides, we use the rtios in equtions (1) to find the vlue of eh of the six trigonometri funtions. sin u = Opposite Hypotenuse = 4 5 s u = Hypotenuse = 5 Opposite 4 os u = djent Hypotenuse = 5 se u = Hypotenuse = 5 djent tn u = Opposite djent = 4 ot u = djent Opposite = 4 Now Work PROLEM 9 The vlues of the trigonometri funtions of n ute ngle re rtios of the lengths of the sides of right tringle. This wy of viewing the trigonometri funtions leds to mny pplitions nd, in ft, ws the point of view used y erly mthemtiins (efore lulus) in studying the sujet of trigonometry. EXMPLE Construting Rin Gutter rin gutter is to e onstruted of luminum sheets 1 inhes wide. See Figure 4(). fter mrking off length of 4 inhes from eh edge, the sides re ent up t n ngle u. See Figure 4(). () Express the re of the opening s funtion of u. [Hint: Let denote the vertil height of the end.] () Grph = 1u. Find the ngle utht mkes lrgest. (This end will llow the most wter to flow through the gutter.) Figure 4 4 in 1 in 4 in () 4 in () Look gin t Figure 4(). The re of the opening is the sum of the res of two ongruent right tringles nd one retngle. Look t Figure 4(), whih shows the tringle on the right in Figure 4() redrwn. We see tht os u = 4 so = 4 os u sin u = 4 so = 4 sin u 4 in 4 in 4 in The re of the tringle is () () 4 re = 1 1se1height = 1 = 1 14 os u14 sin u = 8 sin u os u So the re of the two ongruent tringles is 16 sin u os u. The retngle hs length 4 nd height, so its re is 4 = 414 sin u = 16 sin u

3 SECTION 8.1 Right Tringle Trigonometry; pplitions 507 Figure 5 The re of the opening is 4 = re of the two tringles + re of the retngle 1u = 16 sin u os u + 16 sin u = 16 sin u1os u Figure 6 djent to opposite 90 djent to opposite () Figure 5 shows the grph of = 1u. Using MXIMUM, the ngle u tht mkes lrgest is 60. Use the Complementry ngle Theorem Two ute ngles re lled omplementry if their sum is right ngle. euse the sum of the ngles of ny tringle is 180, it follows tht, for right tringle, the two ute ngles re omplementry. Refer now to Figure 6. We hve leled the ngle opposite side s nd the ngle opposite side s. Notie tht side is djent to ngle nd side is djent to ngle. s result, sin = = os os = = sin tn = = ot s = = se se = = s ot = = tn () euse of these reltionships, the funtions sine nd osine, tngent nd otngent, nd sent nd osent re lled ofuntions of eh other. The identities () my e expressed in words s follows: THEOREM Complementry ngle Theorem Cofuntions of omplementry ngles re equl. Exmples of this theorem re given next: Complementry ngles Complementry ngles Complementry ngles sin 0 = os 60 tn 40 = ot 50 se 80 = s 10 ^ ^ ^ ^ ^ ^ Cofuntions Cofuntions Cofuntions EXMPLE Using the Complementry ngle Theorem () () () sin 6 = os190-6 = os 8 tn p 1 = otp - p 5p = ot 1 1 sin 40 + sin 50 = sin 40 + os 40 = 1 sin 50 = os 40 Now Work PROLEM 19 Figure 7 Solve Right Tringles In the disussion tht follows, we will lwys lel right tringle so tht side is opposite ngle, side is opposite ngle, nd side is the hypotenuse, s shown in Figure 7. To solve right tringle mens to find the missing lengths of its sides nd the mesurements of its ngles. We shll follow the prtie of expressing the lengths of the sides rounded to two deiml ples nd expressing ngles in degrees rounded to one deiml ple. (e sure tht your lultor is in degree mode.)

4 508 CHPTER 8 pplitions of Trigonometri Funtions To solve right tringle, we need to know one of the ute ngles or nd side, or else two sides. Then we mke use of the Pythgoren Theorem nd the ft tht the sum of the ngles of tringle is 180. The sum of the ngles nd in right tringle is therefore 90. THEOREM For the right tringle shown in Figure 7, we hve = + + = 90 Figure 8 EXMPLE 4 Solving Right Tringle Use Figure 8. If = nd = 40, find,, nd. Sine = 40 nd + = 90, it follows tht = 50. To find the sides nd, use the fts tht tn 40 = nd os 40 = 40 Now solve for nd. = tn 40 L 1.68 nd = Now Work PROLEM 9 os 40 L.61 EXMPLE 5 Solving Right Tringle Use Figure 9. If = nd =, find,, nd. Sine = nd =, then, y the Pythgoren Theorem, we hve Figure 9 = + = + = = 1 = 1 L.61 To find ngle, use the ft tht NOTE To void round-off errors when using lultor, we will store unrounded vlues in memory for use in susequent lultions. tn = so = tn - 1 Set the mode on your lultor to degrees. Then, rounded to one deiml ple, we find tht = 56.. Sine + = 90, we find tht =.7. Now Work PROLEM 9 4 Solve pplied Prolems* In ddition to developing models using right tringles, we n use right tringle trigonometry to mesure heights nd distnes tht re either wkwrd or impossile to mesure y ordinry mens. When using right tringles to solve these prolems, py ttention to the known mesures. This will indite wht trigonometri funtion to use. For exmple, if we know the mesure of n ngle nd the length of the side djent to the ngle, nd wish to find the length of the opposite side, we would use the tngent funtion. Do you know why? * In pplied prolems, it is importnt tht nswers e reported with oth justifile ury nd pproprite signifint figures. In this hpter we shll ssume tht the prolem dt re urte to the numer of signifint digits resulting in sides eing rounded to two deiml ples nd ngles eing rounded to one deiml ple.

5 SECTION 8.1 Right Tringle Trigonometry; pplitions 509 EXMPLE 6 Finding the Width of River surveyor n mesure the width of river y setting up trnsit* t point C on one side of the river nd tking sighting of point on the other side. Refer to Figure 10. fter turning through n ngle of 90 t C, the surveyor wlks distne of 00 meters to point. Using the trnsit t, the ngle u is mesured nd found to e 0. Wht is the width of the river rounded to the nerest meter? Figure 10 C 00m 0 We seek the length of side. We know nd u. So we use the ft tht is opposite und is djent to und write tn u = whih leds to tn 0 = 00 = 00 tn 0 L 7.79 meters The width of the river is 7 meters, rounded to the nerest meter. Now Work PROLEM 49 EXMPLE 7 Finding the Inlintion of Mountin Tril stright tril leds from the lpine Hotel, elevtion 8000 feet, to seni overlook, elevtion 11,100 feet. The length of the tril is 14,100 feet. Wht is the inlintion (grde) of the tril? Tht is, wht is the ngle in Figure 11? Figure 11 Hotel Elevtion 8000 ft Tril 14,100 ft Overlook elevtion 11,100 ft 100 ft s we n see in Figure 11, we know the length of the side opposite ngle is 11, = 100 feet nd the length of the hypotenuse is 14,100 feet. The ngle oeys the eqution Using lultor, The inlintion (grde) of the tril is pproximtely 1.7. Now Work PROLEM 55 sin = ,100 = sin ,100 L 1.7 Vertil heights n sometimes e mesured using either the ngle of elevtion or the ngle of depression. If person is looking up t n ojet, the ute ngle mesured from the horizontl to line of sight to the ojet is lled the ngle of elevtion. See Figure 1(). * n instrument used in surveying to mesure ngles.

6 510 CHPTER 8 pplitions of Trigonometri Funtions Figure 1 Ojet ngle of depression Horizontl Line of sight Line of sight ngle of elevtion Horizontl Ojet () () If person is stnding on liff looking down t n ojet, the ute ngle mde y the line of sight to the ojet nd the horizontl is lled the ngle of depression. See Figure 1(). EXMPLE 8 Finding the Height of Cloud Meteorologists find the height of loud using n instrument lled eilometer. eilometer onsists of light projetor tht direts vertil light em up to the loud se nd light detetor tht sns the loud to detet the light em. See Figure 1(). On Deemer 8, 010, t Midwy irport in Chigo, eilometer ws employed to find the height of the loud over. It ws set up with its light detetor 00 feet from its light projetor. If the ngle of elevtion from the light detetor to the se of the loud ws 75, wht ws the height of the loud over? Figure 1 Illuminted spot on se of louds Vertil light em Cloud height h h Light detetor se () Light projetor ft () Figure 1() illustrtes the sitution. To find the height h, use the ft tht tn 75 = h so 00, The eiling (height to the se of the loud over) ws pproximtely 110 feet. Now Work PROLEM 51 h = 00 tn 75 L 110 feet The ide ehind Exmple 8 n lso e used to find the height of n ojet with se tht is not essile to the horizontl. EXMPLE 9 Finding the Height of Sttue on uilding dorning the top of the ord of Trde uilding in Chigo is sttue of Ceres, the Romn goddess of whet. From street level, two oservtions re tken 400 feet from the enter of the uilding. The ngle of elevtion to the se of the sttue is

7 SECTION 8.1 Right Tringle Trigonometry; pplitions 511 found to e 55.1 nd the ngle of elevtion to the top of the sttue is See Figure 14(). Wht is the height of the sttue? Figure ft () ft () ft Figure 14() shows two tringles tht replite Figure 14(). The height of the sttue of Ceres will e -. To find nd, refer to Figure 14(). tn 55.1 = 400 tn 56.5 = 400 = 400 tn 55.1 L 57.9 = 400 tn 56.5 L 604. The height of the sttue is pproximtely = 0.94 feet L 1 feet. Now Work PROLEM 71 EXMPLE 10 The Gi s Hill Lighthouse, Southmpton, ermud In opertion sine 1846, the Gi s Hill Lighthouse stnds 117 feet high on hill 45 feet high, so its em of light is 6 feet ove se level. rohure sttes tht the light n e seen on the horizon out 6 miles from the lighthouse. Verify the ury of this sttement. Figure 15 s 6 ft Figure 15 illustrtes the sitution. The entrl ngle u, positioned t the enter of Erth, rdius 960 miles, oeys the eqution os u = 960 L mile = 580 feet Solving for u, we find 960 mi 960 mi u L L 0. θ The rohure does not indite whether the distne is mesured in nutil miles or sttute miles. Let s lulte oth distnes. The distne s in nutil miles (refer to Prolem 114, p. 6) is the mesure of the ngle uin minutes, so s L 0. nutil miles. The distne s in sttute miles is given y the formul s = ru, where u is mesured in rdins. Then, sine u L 0. L0.715 L rdin 1 1 = 60 1 = p 180 rdin

8 51 CHPTER 8 pplitions of Trigonometri Funtions Figure 16 N N0 E we find tht s = ru L L. miles N70 W P W P S50 W 0 70 O 50 0 S P 4 P 1 E In either se, it would seem tht the rohure overstted the distne somewht. In nvigtion nd surveying, the diretion or ering from point O to point P equls the ute ngle uetween the ry OP nd the vertil line through O, the north south line. Figure 16 illustrtes some erings. Notie tht the ering from O to P 1 is denoted y the symolism N0 E, inditing tht the ering is 0 est of north. In writing the ering from O to P, the diretion north or south lwys ppers first, followed y n ute ngle, followed y est or west. In Figure 16, the ering from O to is S50 W, nd from O to it is N70 W. P P EXMPLE 11 EXMPLE 1 Finding the ering of n Ojet In Figure 16, wht is the ering from O to n ojet t P 4? The ute ngle etween the ry OP 4 nd the north south line through O is given s 0. The ering from O to P 4 is S0 E. Finding the ering of n irplne oeing 777 irrft tkes off from O Hre irport on runwy LEFT, whih hs ering of N0 E.* fter flying for 1 mile, the pilot of the irrft requests permission to turn 90 nd hed towrd the northwest. The request is grnted. fter the plne goes miles in this diretion, wht ering should the ontrol tower use to lote the irrft? Figure 17 Q N Figure 17 illustrtes the sitution. fter flying 1 mile from the irport O (the ontrol tower), the irrft is t P.fter turning 90 towrd the northwest nd flying miles, the irrft is t the point Q. In tringle OPQ, the ngle uoeys the eqution 0 P tn u = 1 = so u = tn- 1 L 6.4 W θ O 1 Runwy LEFT E The ute ngle etween north nd the ry OQ is = 4.4. The ering of the irrft from O to Q is N4.4 W. Now Work PROLEM 6 S 8.1 ssess Your Understnding re You Prepred? nswers re given t the end of these exerises. If you get wrong nswer, red the pges listed in red. 1. In right tringle, if the length of the hypotenuse is 5 nd the length of one of the other sides is, wht is the length of the third side? (pp ). If u is n ute ngle, solve the eqution tn u = 1 Express. your nswer in degrees, rounded to one deiml ple. (pp ). If u is n ute ngle, solve the eqution (pp ) sin u = 1. * In ir nvigtion, the term zimuth denotes the positive ngle mesured lokwise from the north (N) to ry OP. In Figure 16, the zimuth from O to P 1 is 0 ; the zimuth from O to P is 0 ; the zimuth from O to P is 90. In nming runwys, the units digit is left off the zimuth. Runwy LEFT mens the left runwy with diretion of zimuth 0 (ering N0 E). Runwy is the runwy with zimuth 0 nd ering S50 W.

9 SECTION 8.1 Right Tringle Trigonometry; pplitions 51 Conepts nd Voulry 4. True or Flse sin 5 = os True or Flse In right tringle, if two sides re known, we 5. True or Flse In right tringle, one of the ngles is 90 n solve the tringle. nd the sum of the other two ngles is When you look up t n ojet, the ute ngle mesured from the horizontl to line-of-sight oservtion of the ojet is lled the. Skill uilding In Prolems 9 18, find the ext vlue of the six trigonometri funtions of the ngle uin eh figure True or Flse In right tringle, if we know the two ute ngles, we n solve the tringle θ In Prolems 19 8, find the ext vlue of eh expression. Do not use lultor. os 10 os sin 8 - os 5 0. tn 1 - ot sin 80 sin 50 os os 0 - os tn 5 - s tn 0-6. ot 40 - os 0 7. os 5 sin 55 + sin 5 os se 5 s 55 - tn 5 ot 55 sin 50 sin 40 In Prolems 9 4, use the right tringle shown elow. Then, using the given informtion, solve the tringle. 9. = 5, = 0 ; find,, nd 0. = 4, = 10 ; find,, nd 1. = 6, = 40 ; find,, nd. = 7, = 50 ; find,, nd. = 4, = 10 ; find,, nd 4. = 6, = 0 ; find,, nd 5. = 5, = 5 ; find,, nd 6. = 6, = 40 ; find,, nd 7. = 9, = 0 ; find,, nd 8. = 10, = 40 ; find,, nd 9. = 5, = ; find,, nd 40. =, = 8; find,, nd 41. =, = 5; find,, nd 4. = 4, = 6; find,, nd pplitions nd Extensions 4. Geometry The hypotenuse of right tringle is 5 inhes. If one leg is inhes, find the degree mesure of eh ngle. 44. Geometry The hypotenuse of right tringle is feet. If one leg is 1 foot, find the degree mesure of eh ngle. 45. Geometry right tringle hs hypotenuse of length 8 inhes. If one ngle is 5, find the length of eh leg. 46. Geometry right tringle hs hypotenuse of length 10 entimeters. If one ngle is 40, find the length of eh leg. 47. Geometry right tringle ontins 5 ngle. () If one leg is of length 5 inhes, wht is the length of the hypotenuse? () There re two nswers. How is this possile? p 48. Geometry right tringle ontins n ngle of rdin. 8 () If one leg is of length meters, wht is the length of the hypotenuse? () There re two nswers. How is this possile? 49. Finding the Width of Gorge Find the distne from to C ross the gorge illustrted in the figure. C ft

10 514 CHPTER 8 pplitions of Trigonometri Funtions 50. Finding the Distne ross Pond Find the distne from to C ross the pond illustrted in the figure. 57. Finding the Speed of Truk stte trooper is hidden 0 feet from highwy. One seond fter truk psses, the ngle uetween the highwy nd the line of oservtion from the ptrol r to the truk is mesured. See the illustrtion. C 100 ft The Eiffel Tower The tllest tower uilt efore the er of television msts, the Eiffel Tower ws ompleted on Mrh 1, Find the height of the Eiffel Tower (efore television mst ws dded to the top) using the informtion given in the illustrtion. 0 ft PD 1 se ft 5. Finding the Distne of Ship from Shore person in smll ot, offshore from vertil liff known to e 100 feet in height, tkes sighting of the top of the liff. If the ngle of elevtion is found to e 5, how fr offshore is the ot? 5. Finding the Distne to Plteu Suppose tht you re heded towrd plteu 50 meters high. If the ngle of elevtion to the top of the plteu is 0, how fr re you from the se of the plteu? 54. Finding the Reh of Ldder -foot extension ldder lening ginst uilding mkes 70 ngle with the ground. How fr up the uilding does the ldder touh? 55. Finding the ngle of Elevtion of the Sun t 10 M on pril 6, 009, uilding 00 feet high sts shdow 50 feet long. Wht ws the ngle of elevtion of the Sun? 56. Direting Lser em lser em is to e direted through smll hole in the enter of irle of rdius 10 feet. The origin of the em is 5 feet from the irle (see the figure). t wht ngle of elevtion should the em e imed to ensure tht it goes through the hole? () If the ngle mesures 15, how fst is the truk trveling? Express the nswer in feet per seond nd in miles per hour. () If the ngle mesures 0, how fst is the truk trveling? Express the nswer in feet per seond nd in miles per hour. () If the speed limit is 55 miles per hour nd speeding tiket is issued for speeds of 5 miles per hour or more over the limit, for wht ngles should the trooper issue tiket? 58. Seurity seurity mer in neighorhood nk is mounted on wll 9 feet ove the floor. Wht ngle of depression should e used if the mer is to e direted to spot 6 feet ove the floor nd 1 feet from the wll? 59. Prllx One method of mesuring the distne from Erth to str is the prllx method. The ide ehind omputing this distne is to mesure the ngle formed etween the Erth nd the str t two different points in time. Typilly, the mesurements re tken so tht the side opposite the ngle is s lrge s possile. Therefore, the optiml pproh is to mesure the ngle when Erth is on opposite sides of the Sun, s shown in the figure. Erth's orit Erth t time 1 Sun Erth t time Prllx Str 10 ft Lser? 5 ft () Proxim Centuri is 4. light-yers from Erth. If 1 light-yer is out 5.9 trillion miles, how mny miles is Proxim Centuri from Erth? () The men distne from Erth to the Sun is 9,000,000 miles. Wht is the prllx of Proxim Centuri?

11 SECTION 8.1 Right Tringle Trigonometry; pplitions Prllx See Prolem Cygni, sometimes lled essel s Str (fter Friedrih essel, who mesured the distne from Erth to the str in 188), is str in the onstelltion Cygnus. () 61 Cygni is light-yers from Erth. If 1 light-yer is out 5.9 trillion miles, how mny miles is 61 Cygni from Erth? () The men distne from Erth to the Sun is 9,000,000 miles. Wht is the prllx of 61 Cygni? 61. Wshington Monument The ngle of elevtion of the Sun is 5.1 t the instnt the shdow st y the Wshington Monument is 789 feet long. Use this informtion to lulte the height of the monument. 6. Finding the Length of Mountin Tril stright tril with n inlintion of 17 leds from hotel t n elevtion of 9000 feet to mountin lke t n elevtion of 11,00 feet. Wht is the length of the tril? 6. Finding the ering of n irrft DC-9 irrft leves Midwy irport from runwy 4 RIGHT, whose ering is 1 N40 E. fter flying for the pilot requests permission mile, to turn 90 nd hed towrd the southest.the permission is grnted. fter the irplne goes 1 mile in this diretion, wht ering should the ontrol tower use to lote the irrft? 64. Finding the ering of Ship ship leves the port of Mimi with ering of S80 E nd speed of 15 knots. fter 1 hour, the ship turns 90 towrd the south.fter hours, mintining the sme speed, wht is the ering to the ship from port? 65. Nigr Flls Inline Rilwy Situted etween Portge Rod nd the Nigr Prkwy diretly ross from the Cndin Horseshoe Flls, the Flls Inline Rilwy is funiulr tht rries pssengers up n emnkment to Tle Rok Oservtion Point. If the length of the trk is 51.8 meters nd the ngle of inlintion is 6, determine the height of the emnkment. Soure: Willis Tower Willis Tower in Chigo is the third tllest uilding in the world nd is topped y high ntenn. surveyor on the ground mkes the following mesurement: 1. The ngle of elevtion from his position to the top of the uilding is 4.. The distne from his position to the top of the uilding is 59 feet.. The distne from his position to the top of the ntenn is 74 feet. () How fr wy from the (se of the) uilding is the surveyor loted? () How tll is the uilding? () Wht is the ngle of elevtion from the surveyor to the top of the ntenn? (d) How tll is the ntenn? Soure: Construting Highwy highwy whose primry diretions re north south is eing onstruted long the west ost of Florid. Ner Nples, y ostruts the stright pth of the rod. Sine the ost of ridge is prohiitive, engineers deide to go round the y. The illustrtion shows the pth tht they deide on nd the mesurements tken. Wht is the length of highwy needed to go round the y? 68. Photogrphy mer is mounted on tripod 4 feet high t distne of 10 feet from George, who is 6 feet tll. See the illustrtion. If the mer lens hs ngles of depression nd elevtion of 0, will George s feet nd hed e seen y the lens? If not, how fr k will the mer need to e moved to inlude George s feet nd hed? 4' ' 69. Finding the Distne etween Two Ojets limp, suspended in the ir t height of 500 feet, lies diretly over line from Soldier Field to the dler Plnetrium on Lke Mihign (see the figure). If the ngle of depression from the limp to the stdium is nd from the limp to the plnetrium is, find the distne etween Soldier Field nd the dler Plnetrium. Soldier Field US 41 1 mi. 140 mi. 500 ft 70. Hot-ir lloon While tking ride in hot-ir lloon in Np Vlley, Frniso wonders how high he is. To find out, he hooses lndmrk tht is to the est of the lloon nd mesures the ngle of depression to e 54. few minutes lter, fter trveling 100 feet est, the ngle of depression to the sme lndmrk is determined to e 61. Use this informtion to determine the height of the lloon. 71. Mt. Rushmore To mesure the height of Linoln s riture on Mt. Rushmore, two sightings 800 feet from the se of the mountin re tken. If the ngle of elevtion to the ottom of Linoln s fe is nd the ngle of elevtion to the top is 5, wht is the height of Linoln s fe? 10 6' dler Plnetrium Lke Mihign

12 516 CHPTER 8 pplitions of Trigonometri Funtions 7. The CN Tower The CN Tower, loted in Toronto, Cnd, is the tllest struture in the meris. While visiting Toronto, tourist wondered wht the height of the tower ove the top of the Sky Pod is. While stnding 4000 feet from the tower, she mesured the ngle to the top of the Sky Pod to e 0.1. t this sme distne, the ngle of elevtion to the top of the tower ws found to e 4.4. Use this informtion to determine the height of the tower ove the Sky Pod. New em? 46 ft 0 ft 40 ft 10 ft 0 ft 76. Shooting Free Throws in sketll The eyes of sketll plyer re 6 feet ove the floor. The plyer is t the free-throw line, whih is 15 feet from the enter of the sket rim (see the figure). Wht is the ngle of elevtion from the plyer s eyes to the enter of the rim? [Hint: The rim is 10 feet ove the floor.] 7. Chigo Skysrpers The ngle of inlintion from the se of the John Hnok Center to the top of the min struture of the Willis Tower is pproximtely 10.. If the min struture of the Willis Tower is 1451 feet tll, how fr prt re the two skysrpers? ssume the ses of the two uildings re t the sme elevtion. Soure: Estimting the Width of the Mississippi River tourist t the top of the Gtewy rh (height, 60 feet) in St. Louis, Missouri, oserves ot moored on the Illinois side of the Mississippi River 070 feet diretly ross from the rh. She lso oserves ot moored on the Missouri side diretly ross from the first ot (see digrm). Given tht = ot ,estimte the width of the Mississippi River t 55 the St. Louis riverfront. Soure: U.S. rmy Corps of Engineers 77. Geometry Find the vlue of the ngle u in degrees rounded to the nerest tenth of degree. 10 ft? 15 ft 4 6 ft 60 ft ot MO 070 ft ot 75. Finding the Pith of Roof rpenter is prepring to put roof on grge tht is 0 feet y 40 feet y 0 feet. steel support em 46 feet in length is positioned in the enter of the grge. To support the roof, nother em will e tthed to the top of the enter em (see the figure). t wht ngle of elevtion is the new em? In other words, wht is the pith of the roof? IL 78. Surveillne Stellites surveillne stellite irles Erth t height of h miles ove the surfe. Suppose tht d is the distne, in miles, on the surfe of Erth tht n e oserved from the stellite. See the illustrtion on the following pge. () Find n eqution tht reltes the entrl ngle uto the height h. () Find n eqution tht reltes the oservle distne d nd u. () Find n eqution tht reltes d nd h. (d) If d is to e 500 miles, how high must the stellite orit ove Erth?

13 SECTION 8. The Lw of Sines 517 (e) If the stellite orits t height of 00 miles, wht distne d on the surfe n e oserved? 960 d θ h 960 ll ded enter. He knows from physis tht the white ll will ome off ushion t the sme ngle s it hits ushion. Where on the top ushion should he hit the white ll? 80. The Freedom Tower The Freedom Tower is to e the enterpiee of the reuilding of the World Trde Center in New York City.The tower will e 1776 feet tll (not inluding rodst ntenn). The ngle of elevtion from the se of n offie uilding to the top of the tower is 4. The ngle of elevtion from the helipd on the roof of the offie uilding to the top of the tower is Clulting Pool Shots pool plyer loted t X wnts to shoot the white ll off the top ushion nd hit the red 1 ft 5 ft ft ft 1.8 ft ft X Explining Conepts: Disussion nd Writing.7 ft 81. Explin how you would mesure the width of the Grnd Cnyon from point on its ridge. 8. Explin how you would mesure the height of TV tower tht is on the roof of tll uilding. 8. The Gi s Hill Lighthouse. Southmpton, ermud In opertion sine 1846, the Gi s Hill Lighthouse stnds () How fr wy is the offie uilding from the Freedom Tower? ssume the side of the tower is vertil. Round to the nerest foot. () How tll is the offie uilding? Round to the nerest foot. 117 feet high on hill 45 feet high, so its em of light is 6 feet ove se level. rohure sttes tht ships 40 miles wy n see the light nd plnes flying t 10,000 feet n see it 10 miles wy. Verify the ury of these sttements. Wht ssumption did the rohure mke out the height of the ship? re You Prepred? nswers The Lw of Sines PREPRING FOR THIS SECTION efore getting strted, review the following: Trigonometri Equtions (Setion 7., pp ) Differene Formul for the Sine Funtion (Setion 7.5, p. 475) Now Work the re You Prepred? prolems on pge 54. OJECTIVES 1 Solve S or S Tringles (p. 518) Solve SS Tringles (p. 519) Solve pplied Prolems (p. 5) Geometry Essentils (ppendix, Setion., pp ) If none of the ngles of tringle is right ngle, the tringle is lled olique. n olique tringle will hve either three ute ngles or two ute ngles nd one otuse ngle (n ngle etween 90 nd 180 ). See Figure 18.

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