Section.3 Solving Rigt Triangle Trigonometry Eample In te rigt triangle ABC, A = 40 and c = 1 cm. Find a, b, and B. sin 40 a a c 1 a 1sin 40 7.7cm cos 40 b c b 1 b 1cos40 9.cm A 40 1 b C B a B = 90 - A = 90-40 50 Eample A circle as its center at C and a radius of 18 inces. If triangle ADC is a rigt triangle and A = 35. Find, te distance from A to B. sin35 18 18 18sin35 18 18 18 sin35 18 18 sin35 13 in A 35 B C 18 D 13
Definitions An angle measured from te orizontal up is called an angle of elevation. angle of elevation orizontal An angle measured from te orizontal down is called an angle of depression. orizontal angle of depression Eample Te two equal sides of an isosceles triangle are eac 4 cm. If eac of te two equal angles measures 5, find te lengt of te base and te altitude. sin5 4 4sin5 19 cm cos5 y 4 y 4cos5 A 4 4 5 y C B y 15 cm AB y 30 cm Eample A man climbs 13 meters up te side of a pyramid. Find tat te angle of depression to is starting point is 5.6. How ig off of te ground is e? C 5.6 sin 5.6 13 13 13sin 5.6 169 m 14 A B
Eample From a given point on te ground, te angle of elevation to te top of a tree is 36.7. From a second point, 50 feet back, te angle of elevation to te top of te tree is.. Find te eigt of te tree to te nearest foot. Triangle DCB tan. 50 50 tan. Triangle ACB tan 36.7 tan36.7 tan 36.7 50 tan. tan 36.7 50 tan. tan. tan 36.7 tan. 50 tan. tan 36.7 tan. 50 tan. 50tan. tan 36.7 tan. tan36.7 50 tan. tan 36.7 tan 36.7 tan. 45 ft Te tree is about 45 feet tall. 15
Bearing Definition Te bearing of a line l is te acute angle formed by te nort-sout line and te line l. Te notation used to designate te bearing of a line begins wit N (for nort) or S (for sout), followed by te number of degrees in te angle, and ends wit E (for east) or W (for west). Eample A boat travels on a course of bearing N 5 40 E for distance of 38 miles. How many miles nort and ow many miles east ave te boat traveled? 540 5 40 1 60 5.6667 sin 5.6667 38 38sin5.6667 189 mi y cos 5.6667 38 y 38cos5.6667 144 mi 16
Eample A elicopter is overing over te desert wen it develops mecanical problems and is forced to land. After landing, te pilot radios is position to a pair of radar station located 5 miles apart along a straigt road running nort and sout. Te bearing of te elicopter from one station is N 13 E, and from te oter it is S 19 E. After doing a few trigonometric calculations, one of te stations instructs te pilot to walk due west for 3.5 miles to reac te road. Is tis information correct? In triangle AFC tan13 y ytan13 In triangle BFC y tan19 5 y (5 ) tan19 y y (5 ) tan19 tan13 5tan19 tan19 tan13 5tan19 tan13 tan19 5tan19 (tan13 tan19 ) 5 tan19 tan13 tan19 14.966 ytan13 14.966 tan13 3.5 mi 17
Eercises Section.3 Solving Rigt Triangle Trigonometry 1. In te rigt triangle ABC, a = 9.43 and c = 53.58. Find te remaining side and angles.. In te rigt triangle ABC, a =.73 and b = 3.41. Find te remaining side and angles. 3. Find as indicated in te figure. 9.5 49. 39 ft 4. Te distance from A to D is 3 feet. Use te information in figure to solve, te distance between D and C. B A 38 3 D 54 C 5. If C = 6 and r = 19, find. 6. If ABD = 53, C = 48, and BC = 4, find and ten find. 18
7. If A = 41, BDC = 58, and AB = 8, find, ten. 8. A plane flies 1.7 ours at 10 mp on a bearing of 10. It ten turns and flies 9.6 ours at te same speed on a bearing of 100. How far is te plane from its starting point? 9. Te sadow of a vertical tower is 67.0 ft long wen te angle of elevation of te sun is 36.0. Find te eigt of te tower. 10. Te base of a pyramid is square wit sides 700 ft long, and te eigt of te pyramid is 600 ft. Find te angle of elevation of te edge indicated in te figure to two significant digits. (Hint: Te base of te triangle in te figure is alf te diagonal of te square base of te pyramid.) 11. If a 73-foot flagpole casts a sadow 51 feet long, wat is te angle of elevation of te sun (to te nearest tent of a degree)? 19
1. Suppose eac edge of te cube is 3.00 inces long. Find te measure of te angle formed by diagonals DE and DG. Round your answer to te nearest tent of a degree. 13. A person standing at point A notices tat te angle of elevation to te top of te antenna is 47 30. A second person standing 33.0 feet farter from te antenna tan te person at A finds te angle of elevation to te top of te antenna to be 4 10. How far is te person at A from te base of te antenna? 14. Find as indicated in te figure. 15. Find as indicated in te figure. 16. Te angle of elevation from a point on te ground to te top of a pyramid is 31 40. Te angle of elevation from a point 143 ft farter back to te top of te pyramid is 14 50. Find te eigt of te pyramid. 0
17. In one area, te lowest angle of elevation of te sun in winter is 1 16. Find te minimum distance,, tat a plant needing full sun can be placed from a fence 4.41 ft ig. 18. A sip leaves its port and sails on a bearing of N 30 10 E, at speed 9.4 mp. Anoter sip leaves te same port at te same time and sails on a bearing of S 59 50 E, at speed 17.1 mp. Find te distance between te two sips after rs. 19. Radar stations A and B are on te east-west line, 3.7 km apart. Station A detects a place at C, on a bearing of 61. Station B simultaneously detects te same plane, on a bearing of 331. Find te distance from A to C. 0. Suppose te figure below is eaggerated diagram of a plane flying above te eart. If te plane is 4.55 miles above te eart and te radius of te eart is 3,960 miles, ow far is it from te plane to te orizon? Wat is te measure of angle A? 1
1. Te Ferry weel as a 50 feet diameter and 14 feet above te ground. If is te central angle formed as a rider moves from position P 0 to position P 1, find te rider s eigt above te ground wen is 45. O P P 1 P 0 14 ft. Te lengt of te sadow of a building 34.09 m tall is 37.6 m. Find te angle of te elevation of te sun. 3. San Luis Obispo, California is 1 miles due nort of Grover Beac. If Arroyo Grande is 4.6 miles due east of Grover Beac, wat is te bearing of San Luis Obispo from Arroyo Grande? 4. Te bearing from A to C is S 5 E. Te bearing from A to B is N 84 E. Te bearing from B to C is S 38 W. A plane flying at 50 mp takes.4 ours to go from A to B. Find te distance from A to C. 5. From a window 31.0 ft. above te street, te angle of elevation to te top of te building across te street is 49.0 and te angle of depression to te base of tis building is 15.0. Find te eigt of te building across te street. 6. A man wondering in te desert walks.3 miles in te direction S 31 W. He ten turns 90 and walks 3.5 miles in te direction N 59 W. At tat time, ow far is e from is starting point, and wat is is bearing from is starting point?
7. A 10.5-m fire truck ladder is leaning against a wall. Find te distance d te ladder goes up te wall (above te fire truck) if te ladder makes an angle of 35 9 wit te orizontal. 8. Te angle of elevation from a point 93. ft from te base of a tower to te top of te tower is 38 0. Find te eigt of te tower. 9. A basic curve connecting two straigt sections of road is often circular. In te figure, te points P and S mark te beginning and end of te curve. Let Q be te point of intersection were te two straigt sections of igway leading into te curve would meet if etended. Te radius of te curve is R, and te central angle denotes ow many degrees te curve turns. a) If R = 965 ft. and = 37, find te distance d between P and Q. b) Find an epression in terms of R and for te distance between points M and N. 30. Jane was iking directly toward a long straigt road wen se encountered a swamp. Se turned 65 to te rigt and iked 4 mi in tat direction to reac te road. How far was se form te road wen se encountered te swamp? 3
31. From a igway overpass, 14.3 m above te road, te angle of depression of an oncoming car is measured at 18.3. How far is te car from a point on te igway directly below te observer? 3. A tunnel under a river is 196.8 ft. below te surface at its lowest point. If te angle of depression of te tunnel is 4.96, ten ow far apart on te surface are te entrances to te tunnel? How long is te tunnel? 33. A boat sailing nort sigts a ligtouse to te east at an angle of 3 from te nort. After te boat travels one more kilometer, te angle of te ligtouse from te nort is 36. If te boat continues to sail nort, ten ow close will te boat come to te ligtouse? 34. Te angle of elevation of a pedestrian crosswalk over a busy igway is 8.34, as sown in te drawing. If te distance between te ends of te crosswalk measured on te ground is 34 ft., ten wat is te eigt of te crosswalk at te center? 35. A policewoman as positioned erself 500 ft. from te intersection of two roads. Se as carefully measured te angles of te lines of sigt to points A and B. If a car passes from A to B is 1.75 sec and te speed limit is 55 mp, is te car speeding? (Hint: Find te distance from B to A and use R = D/T) 4
36. From point A te angle of elevation to te top of te building is 30. From point B, 0 meters closer to te building, te angle of elevation is 45. Find te angle of elevation of te building from point C, wic is anoter 0 meters closer to te building. 37. A ot air balloon is rising upward from te eart at a constant rate. An observer 50 m away spots te balloon at an angle of elevation of 4. Two minutes later te angle of elevation of te balloon is 58. At wat rate is te balloon ascending? 38. A skateboarder wises to build a jump ramp tat is inclined at a 19 angle and tat as a maimum eigt of 3.0 inces. Find te orizontal widt of te ramp. 39. For best illumination of a piece of art, a ligting specialist for an art gallery recommends tat a ceiling-mounted ligt be 6 ft from te piece of art and tat te angle of depression of te ligt be 38. How far from a wall sould te ligt be placed so tat te recommendations of te specialist are met? Notice tat te art etends outward 4 inces from te wall. 5
40. A surveyor determines tat te angle of elevation from a transit to te top of a building is 7.8. Te transit is positioned 5.5 feet above ground level and 131 feet from te building. Find te eigt of te building to te nearest tent of a foot. 41. From a point A on a line from te base of te Wasington Monument, te angle of elevation to te top of te monument is 4.0. From a point 100 ft away from A and on te same line, te angle to te top is 37.8. Find te eigt, to te nearest foot, of te Monument. 4. A metod tat surveyors use to determine a small distance d between two points P and Q is called te subtense bar metod. Te subtense bar wit lengt b is centered at Q and situated perpendicular to te line of sigt between P and Q. Angle is measured, ten te distance d can be determined. a) Find d wit 1 3 1 and b.000 cm b) Angle usually cannot be measured more accurately tan to te nearest 1. How muc cange would tere be in te value of d if were measured 1 larger? 43. A diagram tat sows ow Diane estimates te eigt of a flagpole. Se can't measure te distance between erself and te flagpole directly because tere is a fence in te way. So se stands at point A facing te pole and finds te angle of elevation from point A to te top of te pole to be 61.7. Ten se turns 90 and walks 5.0 ft to point B, were se measures te angle 6
between er pat and a line from B to te base of te pole. Se finds tat angle is 54.5. Use tis information to find te eigt of te pole. 7
Section.3 Solving Rigt Triangle Trigonometry Eercise In te rigt triangle ABC, a = 9.43 and c = 53.58. Find te remaining side and angles. c a b b c a B b sin A a c 53.58 9.43 44.77 9.43 53.58 A sin 1 9.43 53.58 33.3 A 3.41 b.73 C B 90 A 90 33.3 56.68 Eercise In te rigt triangle ABC, a =.73 and b = 3.41. Find te remaining side and angles. c a b c.73 3.41 4.37 B tan A a b.73 3.41 or sin A a c.73 4.37 tan 1.73 A sin 1.73 A 3.41 4.37 38.7 38.7 A c 3.41.73 C B = 90 - A = 90-38.7 4
Eercise Find as indicated in te figure. B Triangle DCB tan 49. tan 49. Triangle ACB tan 9.5 39 ( 39) tan 9.5 A 9.5 49. 39 ft D C tan 49. ( 39) tan 9.5 tan 49. tan 9.5 39 tan 9.5 tan 49. tan 9.5 39 tan 9.5 tan 49. tan 9.5 39 tan 9.5 39tan 9.5 tan 49. tan 9.5 39tan 9.5 tan 49. tan 49.tan 9.5 433.5 ft Eercise Te distance from A to D is 3 feet. Use te figure to solve, te distance between D and C. Triangle DCB tan54 tan 54 B Triangle ACB ( 3) tan38 tan38 3 tan54 ( 3) tan 38 A 38 3 D 54 C tan54 tan 38 3 tan 38 tan54 tan 38 3 tan 38 (tan54 tan38 ) 3tan38 3 tan38 tan54 tan38 4 ft 5
Eercise If C = 6 and r = 19, find. cos 6 r 19 r 19 ( 19 )cos 6 19 19cos 6 cos 6 19 cos 6 19 19cos 6 19 19cos 6. 14 cos 6 Eercise If ABD = 53, C = 48, and BC = 4, find and ten find. tan 48 4 4tan48 46.65 47 tan53 47 tan 53 6 Eercise If A = 41, BDC = 58, and AB = 8, find, ten. sin 41 AB 8sin 41 18 tan58 18 11 tan58 6
Eercise A plane flies 1.7 ours at 10 mp on a bearing of 10. It ten turns and flies 9.6 ours at te same speed on a bearing of 100. How far is te plane from its starting point? b 10 mi 1.7rs 04 mi r a 10 mi 9.6rs 115 mi r Te triangle is rigt triangle. b 10 80 a c a b 115 04 1170 mi Eercise Te sadow of a vertical tower is 67.0 ft long wen te angle of elevation of te sun is 36.0. Find te eigt of te tower. tan36 67 67 tan 36 48.7 ft Eercise Te base of a pyramid is square wit sides 700 ft. long, and te eigt of te pyramid is 600 ft. Find te angle of elevation of te edge indicated in te figure to two significant digits. (Hint: Te base of te triangle in te figure is alf te diagonal of te square base of te pyramid.) b 700 700 b 700 700 tan 600 600 600 1 b/ 700 700 7 tan 1 1 50.48 7 b/ b/ 7
Eercise If a 73-foot flagpole casts a sadow 51 feet long, wat is te angle of elevation of te sun (to te nearest tent of a degree)? tan 73 51 tan 1 73 55.1 51 Eercise Suppose eac edge of te cube is 3.00 inces long. Find te measure of te angle formed by diagonals DE and DG. Round your answer to te nearest tent of a degree. DG 3 3 3 tan( EDG) EG 3 GD EDG tan 1 EDG 45 3 Eercise A man wondering in te desert walks.3 miles in te direction S 31 W. He ten turns 90 and walks 3.5 miles in te direction N 59 W. At tat time, ow far is e from is starting point, and wat is is bearing from is starting point? N d.3 3.5 4. cos.3 4..55 cos 1 0.55 57 W d 31 3.5 mi.3 mi E S (57+31) W Bearing S 88 W S 8
Eercise A person standing at point A notices tat te angle of elevation to te top of te antenna is 47 30. A second person standing 33.0 feet farter from te antenna tan te person at A finds te angle of elevation to te top of te antenna to be 4 10. How far is te person at A from te base of te antenna? 47 30 = 47 30 1 47. 5 60 tan 47.5 tan 47.5 (1) 4 10 = 4 10 1 4. 167 60 tan 4.167 33 (33 ) tan 4.167 () = (33 ) tan 4.167 tan 47.5 33 tan 4.167 tan 4.167 tan 47. 5 9.88 0.906 1.09 33 tan 4.167 tan 47.5 tan 4. 167 9.88 1.09. 906 33tan 4.167 tan 47.5 tan 4.167 9.88 161 0.18 9.88 0.184 9
Eercise Find as indicated in te figure. Outside triangle: tan 7.6 371 tan 7.6 371 Inside triangle: tan 60.4 tan 60.4 Bot triangles ave te same, terefore: tan 60.4 371tan 7.6 tan 7.6 tan 60.4 tan 7.6 371tan 7.6 tan 60.4 tan 7.6 371tan 7.6 371tan 7.6 tan 60.4 tan 7.6 157 tan 60.4 76 Eercise Find as indicated in te figure. Outside triangle: Inside triangle: tan 1.6 449 tan 1.6 449 tan53.5 tan53.5 Bot triangles ave te same, terefore: tan53.5 449tan 1.6 tan 1.6 tan53.5 tan 1.6 449tan 1.6 tan53.5 tan 1.6 449tan 1.6 449tan 1.6 tan 53.5 tan 1.6 186 tan53.5 186tan53.5 5 30
Eercise Te angle of elevation from a point on te ground to te top of a pyramid is 31 40. Te angle of elevation from a point 143 ft farter back to te top of te pyramid is 14 50. Find te eigt of te pyramid. 1450 14 50 14.833 and 3140 31 40 31.667 60 60 tan14.833 143 tan14.833 143 tan 31.667 tan 31.667 Bot triangles ave te same, terefore: tan 31.667 143 tan14.833 tan 31.667 143tan14.833 tan14.833 tan 31.667 tan14.833 143tan14.833 tan 31.667 tan14.833 143tan14.833 143tan14.833 tan 31.667 tan14.833 tan31.667 143tan14.833 tan 31.667 tan 31.667 tan14.833 66 Eercise In one area, te lowest angle of elevation of te sun in winter is 1 16. Find te minimum distance,, tat a plant needing full sun can be placed from a fence 4.41 ft ig. 4.41 tan 1 16 4.41 tan 16 1 60 11.33 ft 31
Eercise A sip leaves its port and sails on a bearing of N 30 10 E, at speed 9.4 mp. Anoter sip leaves te same port at te same time and sails on a bearing of S 59 50 E, at speed 17.1 mp. Find te distance between te two sips after rs. 3010 30 10 30.16667 60 5950 59 50 59.8333 60 After ours: s 9.4 mi. r 58.8 1 r 17.1 mi s. r 34. r tan 30. 58.8 tan 30. s 1 y tan 59.8 y 34. tan 59.8 s a y 58.8tan 30. 34. tan 59.8 93 miles 30. 59.8 s 1 s y Eercise Suppose te figure below is eaggerated diagram of a plane flying above te eart. If te plane is 4.55 miles above te eart and te radius of te eart is 3,960 miles, ow far is it from te plane to te orizon? Wat is te measure of angle A? 3960 3964.55 3964.55 3960 3964.55 3960 190 Te plane is 190 miles from te orizon. sin A 3960 3964.55 0.9989 A sin 1 (0.9989) 87. 3 3
Eercise Te Ferry weel as a 50 feet diameter and 14 feet above te ground. If is te central angle formed as a rider moves from position to position, find te rider s eigt above te ground wen is 45. Distance between cos OP OP 1 cos 45 OP 15 OP 15cos45 PP 14 0 OP OP 14 0 15 15cos45 14 51 ft P 0 P 1 O and P radius 50 15 ft 0 P 0 P O 14 ft P 1 Eercise If a 75-foot flagpole casts a sadow 43 ft long, to te nearest 10 minutes wat is te angle of elevation of te sum from te tip of te sadow? tan 75 43 tan 1 75 43 60.17 60 0.17 60 10 60 1 75 ft 43 ft Eercise Te lengt of te sadow of a building 34.09 m tall is 37.6 m. Find te angle of te elevation of te sun. tan B 34.09 37.6 B tan 1 34.09 37.6 4.18 Te angle of elevation is 4.18 33
=31 Eercise San Luis Obispo, California is 1 miles due nort of Grover Beac. If Arroyo Grande is 4.6 miles due east of Grover Beac, wat is te bearing of San Luis Obispo from Arroyo Grande? tan 4.6 0.3833 1 tan 1 0.3833 1 Te bearing of San Luis Obispo from Arroyo Grande is N 1 W Eercise Te bearing from A to C is S 5 E. Te bearing from A to B is N 84 E. Te bearing from B to C is S 38 W. A plane flying at 50 mp takes.4 ours to go from A to B. Find te distance from A to C. ABD 180 84 96 ABC 180 96 38 C 180 46 44 90 c rate time 50.4 600 mi. sin 46 b b c 600 b 600sin 46 430 mi 46 W 1 4. 6 N S E Eercise From a window 31.0 ft. above te street, te angle of elevation to te top of te building across te street is 49.0 and te angle of depression to te base of tis building is 15.0. Find te eigt of te building across te street. tan15 31 d 31 d tan15 y tan 49 y 31 tan 49 d tan15 y 31 31 tan 49 tan15 164 ft d y 34
Eercise A 10.5-m fire truck ladder is leaning against a wall. Find te distance d te ladder goes up te wall (above te fire truck) if te ladder makes an angle of 35 9 wit te orizontal. d 10.5 sin 35 9 d 10.5sin 35 9 60 d 6.1 m Eercise A basic curve connecting two straigt sections of road is often circular. In te figure, te points P and S mark te beginning and end of te curve. Let Q be te point of intersection were te two straigt sections of igway leading into te curve would meet if etended. Te radius of te curve is R, and te central angle denotes ow many degrees te curve turns. a) If R = 965 ft. and = 37, find te distance d between P and Q. b) Find an epression in terms of R and for te distance between points M and N. a) b) sin PN PN 965sin 37 306. R CPN 90 71.5 NPQ 90 CPN 90 71.5 18.5 PN cos( NPQ) d PN d 306. 3.9 cos18.5 cos18.5 cos CN R CN Rcos R CQ CM NM NM R CM NM R Rcos NM 1 R 1cos 35
Eercise Te angle of elevation from a point 93. ft from te base of a tower to te top of te tower is 38 0. Find te eigt of te tower. tan 38 0 93. 93.tan 38 0 73.7 Eercise Jane was iking directly toward a long straigt road wen se encountered a swamp. Se turned 65 to te rigt and iked 4 mi in tat direction to reac te road. How far was se forms te road wen se encountered te swamp? cos65d 4 d 4cos65 1.7 miles Eercise From a igway overpass, 14.3 m above te road, te angle of depression of an oncoming car is measured at 18.3. How far is te car from a point on te igway directly below te observer? 90 18.3 71.7 tan(71.7 ) 14.3 14.3tan(71.7 ) 43. m Eercise A tunnel under a river is 196.8 ft. below te surface at its lowest point. If te angle of depression of te tunnel is 4.96, ten ow far apart on te surface are te entrances to te tunnel? How long is te tunnel? tan 4.96 196.8 36
196.8 66.75 tan 4.96 d 4533 ft sin 4.96196.8 y y 196.8 75.3 sin 4.96 Te tunnel lengt: y 4551ft Eercise A boat sailing nort sigts a ligtouse to te east at an angle of 3 from te nort. After te boat travels one more kilometer, te angle of te ligtouse from te nort is 36. If te boat continues to sail nort, ten ow close will te boat come to te ligtouse? tan36 y tan36 y tan3 y 1 tan3 y 1 ytan36 y1 tan3 ytan36 ytan3 tan3 ytan36 ytan3 tan3 y tan36 tan3 tan3 y tan3 tan36 tan3 y tan36 tan3 tan36 tan36 tan3 4.5 km Te closest will te boat come to te ligtouse is 4.5 km. Eercise Te angle of elevation of a pedestrian crosswalk over a busy igway is 8.34, as sown in te drawing. If te distance between te ends of te crosswalk measured on te ground is 34 ft., ten wat is te eigt of te crosswalk at te center? tan8.34 171 171tan8.34 5. 1ft 37
Eercise A policewoman as positioned erself 500 ft. from te intersection of two roads. Se as carefully measured te angles of te lines of sigt to points A and B. If a car passes from A to B is 1.75 sec and te speed limit is 55 mp, is te car speeding? (Hint: Find te distance from B to A and use R = D/T) tan1.3 b b 500tan1.3 500 tan15.4 b a ba 500tan15.4 500 a 500tan15.4 b 500tan15.4 500tan1.3 8.7 ft 1 mi 580 ft 0.0054356 mi Te speed is: Te car is not speeding. 0. 0054356 mi 1 3600 sec 11. mp 1.75 sec 1 r Eercise From point A te angle of elevation to te top of te building is 30. From point B, 0 meters closer to te building, te angle of elevation is 45. Find te angle of elevation of te building from point C, wic is anoter 0 meters closer to te building. Let be te distance between C and te building. 1 3 tan 30 40 tan 30 40 40 tan 45 0 tan 45 0 (1) 0 40 0 13 40 0 3 3 3 0 3 40 1 3 0 3 40 0 340 7.3 1 3 1 3 40 7.3 7.3 tan C 7.3 7.3 C 1 7.3 tan 75 7.3 38
Eercise A ot air balloon is rising upward from te eart at a constant rate. An observer 50 m away spots te balloon at an angle of elevation of 4. Two minutes later te angle of elevation of te balloon is 58. At wat rate is te balloon ascending? tan 4 1 50tan 4 50 1 tan58 50tan58 50 It took minutes to get from rate 1 50tan58 50tan 4 144.4 m / min to 1 Eercise A skateboarder wises to build a jump ramp tat is inclined at a 19 angle and tat as a maimum eigt of 3.0 inces. Find te orizontal widt of te ramp. tan193 3 9.9 in tan19 19 3 Eercise For best illumination of a piece of art, a ligting specialist for an art gallery recommends tat a ceilingmounted ligt be 6 ft from te piece of art and tat te angle of depression of te ligt be 38. How far from a wall sould te ligt be placed so tat te recommendations of te specialist are met? Notice tat te art etends outward 4 inces from te wall. cos38 6 6cos38 4.7 feet distance 4.7 ft 1 in 1 ft 4 in 60.7 in distance 60.7 1 5.1 ft 39 4 in Distance = + 4 inces 38 6 ft = 7 in
Eercise A surveyor determines tat te angle of elevation from a transit to te top of a building is 7.8. Te transit is positioned 5.5 feet above ground level and 131 feet from te building. Find te eigt of te building to te nearest tent of a foot. tan 7.8 y 131 y 131tan 7.8 y5.5 131tan 7.8 5.5 74.6 ft 7.8 y 131 ft 5.5 ft Eercise From a point A on a line from te base of te Wasington Monument, te angle of elevation to te top of te monument is 4.0. From a point 100 ft away from A and on te same line, te angle to te top is 37.8. Find te eigt, to te nearest foot, of te Monument. Triangle ACB: Triangle DCB: tan 37.8 100 tan 37.8 100 tan 4 tan 4 tan 4 100 tan37.8 tan 4 tan 37.8 100 tan 37.8 tan 4 tan 37.8 100 tan 37.8 tan 4 tan 37.8 100 tan 37.8 100tan37.8 tan 4 tan 37.8 tan 4 100 tan 37.8 tan 4 tan 4 tan 37.8 560 ft A 100 ft 37.8 4 D B C 40
Eercise Radar stations A and B are on te east-west line, 3.7 km apart. Station A detects a place at C, on a bearing of 61. Station B simultaneously detects te same plane, on a bearing of 331. Find te distance from A to C. A 90 61 9 cos 9 b 3.7 b 3.7cos9 3. km Eercise A metod tat surveyors use to determine a small distance d between two points P and Q is called te subtense bar metod. Te subtense bar wit lengt b is centered at Q and situated perpendicular to te line of sigt between P and Q. Angle is measured, ten te distance d can be determined. a) Find d wit 1 3 1 and b.000 cm b) Angle usually cannot be measured more accurately tan to te nearest 1. How muc cange a) would tere be in te value of d if were measured 1 larger? cot d b / d b cot 1 3 1 d 1 + 3 1 60 3600 1.38667 cot 1.38667 8.6341 cm b) 1 3 1 1 1 3 13 1.386944 d cot 1.386944 Te cange is: 8.6341 8.6175 8.617558 cm 0.0166 cm 41
Eercise A diagram tat sows ow Diane estimates te eigt of a flagpole. Se can't measure te distance between erself and te flagpole directly because tere is a fence in te way. So se stands at point A facing te pole and finds te angle of elevation from point A to te top of te pole to be 61.7. Ten se turns 90 and walks 5.0 ft to point B, were se measures te angle between er pat and a line from B to te base of te pole. Se finds tat angle is 54.5. Use tis information to find te eigt of te pole. tan 54.5 5.0 5.0 tan54. 5 35.0487 ft tan 61.7 35.0487 35.0487 tan 61.7 65.1 ft 4