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1 538 HPTER 7 pplitions of Trigonometri Funtions 7. 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. The differene formul for sine is sin1 - =. 3. If u is n ute ngle, solve the eqution sin u =. (p. 476). If u is n ute ngle, solve the eqution os u = 3 (pp ). (pp ) onepts nd Voulry 4. If none of the ngles of tringle is right ngle, the tringle is lled. 5. For tringle with sides,, nd opposite ngles,, g, the Lw of Sines sttes tht. 6. True or Flse: n olique tringle in whih two sides nd n ngle re given lwys results in t lest one tringle. 7. True or Flse: The sum of the ngles of ny tringle equls True or Flse: The miguous se refers to the ft tht, when two sides nd the ngle opposite one of them re given, sometimes the Lw of Sines nnot e used.

2 SETION 7. The Lw of Sines 539 Skill uilding In Prolems 9 16, solve eh tringle In Prolems 17 4, solve eh tringle. 17. = 40, = 0, = 18. = 50, g = 0, = = 70, g = 10, = 5 0. = 70, = 60, = 4 1. = 110, g = 30, = 3. = 10, g = 100, = 3. = 40, = 40, = 4. = 0, g = 70, = 1 In Prolems 5 36, two sides nd n ngle re given. Determine whether the given informtion results in one tringle, two tringles, or no tringle t ll. Solve ny tringle(s) tht results. 5. = 3, =, = = 4, = 3, = = 5, = 3, = =, = 1, = = 4, = 5, = =, = 3, = = 4, = 6, = 0 3. = 3, = 7, = = 4, = 5, = =, = 1, g = =, = 1, g = 100 = 4, = 5, = 40 pplitions nd Extensions 37. Resue t Se ost Gurd Sttion le is loted 150 miles due south of Sttion ker. ship t se sends n SOS ll tht is reeived y eh sttion. The ll to Sttion le indites tht the ship is loted N55 E; the ll to Sttion ker indites tht the ship is loted S60 E. () How fr is eh sttion from the ship? () If heliopter ple of flying 00 miles per hour is dispthed from the sttion nerest to the ship, how long will it tke to reh the ship? 38. Surveying onsult the figure. To find the distne from the house t to the house t, surveyor mesures the ngle to e 40 nd then wlks off distne of 100 feet to nd mesures the ngle to e 50. Wht is the distne from to? 150 mi ker 60 W N S E le 55

3 540 HPTER 7 pplitions of Trigonometri Funtions 39. Finding the Length of Ski Lift onsult the figure.to find the length of the spn of proposed ski lift from to, surveyor mesures the ngle D to e 5 nd then wlks off distne of 1000 feet to nd mesures the ngle to e 15. Wht is the distne from to? 43. Nvigtion n irplne flies from ity to ity, distne of 150 miles, nd then turns through n ngle of 40 nd heds towrd ity, s shown in the figure. 300 mi mi D 40. Finding the Height of Mountin Use the illustrtion in Prolem 39 to find the height D of the mountin t. 41. Finding the Height of n irplne n irrft is spotted y two oservers who re 1000 feet prt. s the irplne psses over the line joining them, eh oserver tkes sighting of the ngle of elevtion to the plne, s indited in the figure. How high is the irplne? ft () If the distne etween ities nd is 300 miles, how fr is it from ity to ity? () Through wht ngle should the pilot turn t ity to return to ity? 44. Time Lost due to Nvigtion Error In ttempting to fly from ity to ity, n irrft followed ourse tht ws 10 in error, s indited in the figure.fter flying distne of 50 miles, the pilot orreted the ourse y turning t point nd flying 70 miles frther. If the onstnt speed of the irrft ws 50 miles per hour, how muh time ws lost due to the error? ft mi mi 4. Finding the Height of the ridge over the Royl Gorge The highest ridge in the world is the ridge over the Royl Gorge of the rknss River in olordo. Sightings to the sme point t wter level diretly under the ridge re tken from eh side of the 880-foot-long ridge, s indited in the figure. How high is the ridge? SOURE: Guinness ook of World Reords. 45. Finding the Len of the Lening Tower of Pis The fmous Lening Tower of Pis ws originlly feet high. * t distne of 13 feet from the se of the tower, the ngle of elevtion to the top of the tower is found to e 60. Find the ngle indited in the figure. lso, find the perpendiulr distne from to. 880 ft h ft ft * In their 1986 report on the frgile seven-entury-old ell tower, sientists in Pis, Itly, sid tht the Lening Tower of Pis hd inresed its fmous len y 1 millimeter, or 0.04 inh. This is out the nnul verge, lthough the tilting hd slowed to out hlf tht muh in the previous yers. (Soure: United Press Interntionl, June 9, 1986.) PIS, ITLY. Septemer 1995.The Lening Tower of Pis hs suddenly shifted, jeoprdizing yers of preservtion work to stilize it, Itlin newsppers sid Sundy.The tower, uilt on shifting susoil etween 1174 nd 1350 s elfry for the nery thedrl, reently moved 0.07 inh in one night. Updte The tower, whih hd een losed to tourists sine 1990, ws reopened in Deemer, 001, fter reinforement of its se.

4 SETION 7. The Lw of Sines rnkshfts on rs On ertin utomoile, the rnkshft is 3 inhes long nd the onneting rod is 9 inhes long (see the figure).t the time when the ngle OP is 15, how fr is the piston (P) from the enter (O) of the rnkshft? 3 in. O 15 P 47. onstruting Highwy U.S. 41, 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? Oen mi Pelin y in. lm y 1 8 mi 1 8 mi Highwy U.S lulting Distnes t Se The nvigtor of ship t se spots two lighthouses tht she knows to e 3 miles prt long stright seshore. She determines tht the ngles formed etween two line-of-sight oservtions of the lighthouses nd the line from the ship diretly to shore re 15 nd 35. See the illustrtion. () How fr is the ship from lighthouse? () How fr is the ship from lighthouse? () How fr is the ship from shore? 49. Designing n wning n wning tht overs sliding glss door tht is 88 inhes tll forms n ngle of 50 with the wll. The purpose of the wning is to prevent sunlight from entering the house when the ngle of elevtion of the sun is more thn 65. See the figure. Find the length L of the wning. 50. Finding Distnes forest rnger is wlking on pth inlined t 5 to the horizontl diretly towrd 100-foot-tll fire oservtion tower. The ngle of elevtion from the pth to the top of the tower is 40. How fr is the rnger from the tower t this time? 51. Gret Pyrmid of heops One of the originl Seven Wonders of the World, the Gret Pyrmid of heops ws uilt out 580. Its originl height ws 480 feet 11 inhes, ut due to the loss of its topmost stones, it is now shorter. Find the urrent height of the Gret Pyrmid, using the informtion given in the illustrtion. SOURE: Guinness ook of World Reords L 65 pth horizontl 40 5 Step 3 mi Oen 00 ft 5. Determining the Height of n irrft Two sensors re sped 700 feet prt long the pproh to smll irport. When n irrft is nering the irport, the ngle of elevtion from the first sensor to the irrft is 0, nd from the seond sensor to the irrft it is 15. Determine how high the irrft is t this time.

5 54 HPTER 7 pplitions of Trigonometri Funtions 53. Merury The distne from the Sun to Erth is pproximtely 149,600,000 kilometers (km). The distne from the Sun to Merury is pproximtely 57,910,000 km. The elongtion ngle is the ngle formed etween the line of sight from Erth to the Sun nd the line of sight from Erth to Merury. See the figure. Suppose tht the elongtion ngle for Merury is 15. Use this informtion to find the possile distnes etween Erth nd Merury. Erth Merury Merury 54. Venus The distne from the Sun to Erth is pproximtely 149,600,000 km.the distne from the Sun to Venus is pproximtely 108,00,000 km. The elongtion ngle is the ngle formed etween the line of sight from Erth to the Sun nd the line of sight from Erth to Venus. Suppose tht the elongtion ngle for Venus is 10. Use this informtion to find the possile distnes etween Erth nd Venus. 55. Lndsping Pt needs to determine the height of tree efore utting it down to e sure tht it will not fll on nery fene. The ngle of elevtion of the tree from one position on flt pth from the tree is 30, nd from seond position 40 feet frther long this pth it is 0. Wht is the height of the tree? 56. onstrution loding rmp 10 feet long tht mkes n ngle of 18 with the horizontl is to e repled y one tht mkes n ngle of 1 with the horizontl. How long is the new rmp? 57. Finding the Height of Heliopter Two oservers simultneously mesure the ngle of elevtion of heliopter. One ngle is mesured s 5, the other s 40 (see the figure). If the oservers re 100 feet prt nd the heliopter lies over the line joining them, how high is the heliopter? Sun Derive it. [Hint: Use the Lw of Sines nd then Sum-to-Produt Formul. Notie tht this formul involves ll six prts of tringle. s result, it is sometimes used to hek the solution of tringle.] 59. Mollweide s Formul nother form of Mollweide s Formul is Derive it For ny tringle, derive the formul [Hint: Use the ft tht sin = sin g. ] 61. Lw of Tngents For ny tringle, derive the Lw of Tngents. [Hint: Use Mollweide s Formul.] 6. irumsriing Tringle Show tht sin where r is the rdius of the irle irumsriing the tringle whose sides re,, nd, s shown in the figure. = = = sin [Hint: Drw the dimeter. Then = ngle = ngle, nd ngle =90. ] os d sin 1 g sin d os 1 g = os g + os tn d - + = tn d = sin g = 1 r Mollweide s Formul For ny tringle, Mollweide s Formul (nmed fter Krl Mollweide, ) sttes tht 40 Disussion nd Writing 63. Mke up three prolems involving olique tringles. One should result in one tringle, the seond in two tringles, nd the third in no tringle. 64. Wht do you do first if you re sked to solve tringle nd re given one side nd two ngles? 65. Wht do you do first if you re sked to solve tringle nd re given two sides nd the ngle opposite one of them? re You Prepred? nswers 1. sin os - os sin No solution

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