Section 2.3 Solving Right Triangle Trigonometry

Transcription

1 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 cm cos 40 b c b 1 b 1cos40 9.cm A 40 1 b C B a B = 90 - A = 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. sin sin sin sin35 13 in A 35 B C 18 D 13

2 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 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 sin m 14 A B

3 Eample From a given point on te ground, te angle of elevation to te top of a tree is 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 tan. Triangle ACB tan 36.7 tan36.7 tan tan. tan tan. tan. tan 36.7 tan. 50 tan. tan 36.7 tan. 50 tan. 50tan. tan 36.7 tan. tan tan. tan 36.7 tan 36.7 tan. 45 ft Te tree is about 45 feet tall. 15

4 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? sin sin mi y cos y 38cos mi 16

5 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 tan ytan tan mi 17

6 Eercises Section.3 Solving Rigt Triangle Trigonometry 1. In te rigt triangle ABC, a = 9.43 and c = Find te remaining side and angles.. In te rigt triangle ABC, a =.73 and b = Find te remaining side and angles. 3. Find as indicated in te figure 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 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 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

8 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 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 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 Te angle of elevation from a point 143 ft farter back to te top of te pyramid is Find te eigt of te pyramid. 0

9 17. In one area, te lowest angle of elevation of te sun in winter is 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 E, at speed 9.4 mp. Anoter sip leaves te same port at te same time and sails on a bearing of S 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

10 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 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 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?

11 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 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

12 31. From a igway overpass, 14.3 m above te road, te angle of depression of an oncoming car is measured at How far is te car from a point on te igway directly below te observer? 3. A tunnel under a river is 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

13 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

14 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 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 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 Ten se turns 90 and walks 5.0 ft to point B, were se measures te angle 6

15 between er pat and a line from B to te base of te pole. Se finds tat angle is Use tis information to find te eigt of te pole. 7

16 Section.3 Solving Rigt Triangle Trigonometry Eercise In te rigt triangle ABC, a = 9.43 and c = Find te remaining side and angles. c a b b c a B b sin A a c A sin A 3.41 b.73 C B 90 A Eercise In te rigt triangle ABC, a =.73 and b = Find te remaining side and angles. c a b c B tan A a b or sin A a c tan 1.73 A sin 1.73 A A c C B = 90 - A =

17 Eercise Find as indicated in te figure. B Triangle DCB tan 49. tan 49. Triangle ACB tan ( 39) tan 9.5 A ft D C tan 49. ( 39) tan 9.5 tan 49. tan tan 9.5 tan 49. tan tan 9.5 tan 49. tan tan tan 9.5 tan 49. tan tan 9.5 tan 49. tan 49.tan 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

18 Eercise If C = 6 and r = 19, find. cos 6 r 19 r 19 ( 19 )cos cos 6 cos 6 19 cos cos cos cos 6 Eercise If ABD = 53, C = 48, and BC = 4, find and ten find. tan tan tan53 47 tan 53 6 Eercise If A = 41, BDC = 58, and AB = 8, find, ten. sin 41 AB 8sin tan tan58 6

19 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 a c a b mi Eercise Te sadow of a vertical tower is 67.0 ft long wen te angle of elevation of te sun is Find te eigt of te tower. tan tan 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 b tan b/ tan b/ b/ 7

20 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 tan 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 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 cos cos W d mi.3 mi E S (57+31) W Bearing S 88 W S 8

21 Eercise A person standing at point A notices tat te angle of elevation to te top of te antenna is 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 How far is te person at A from te base of te antenna? = tan 47.5 tan 47.5 (1) 4 10 = tan (33 ) tan () = (33 ) tan tan tan tan tan tan tan 47.5 tan tan tan 47.5 tan

22 Eercise Find as indicated in te figure. Outside triangle: tan tan Inside triangle: tan 60.4 tan 60.4 Bot triangles ave te same, terefore: tan tan 7.6 tan 7.6 tan 60.4 tan tan 7.6 tan 60.4 tan tan tan 7.6 tan 60.4 tan tan Eercise Find as indicated in te figure. Outside triangle: Inside triangle: tan tan tan53.5 tan53.5 Bot triangles ave te same, terefore: tan tan 1.6 tan 1.6 tan53.5 tan tan 1.6 tan53.5 tan tan tan 1.6 tan 53.5 tan tan tan

23 Eercise Te angle of elevation from a point on te ground to te top of a pyramid is Te angle of elevation from a point 143 ft farter back to te top of te pyramid is Find te eigt of te pyramid and tan tan tan tan Bot triangles ave te same, terefore: tan tan tan tan tan tan tan tan tan tan tan tan tan tan tan tan tan tan tan Eercise In one area, te lowest angle of elevation of te sun in winter is Find te minimum distance,, tat a plant needing full sun can be placed from a fence 4.41 ft ig tan tan ft 31

24 Eercise A sip leaves its port and sails on a bearing of N E, at speed 9.4 mp. Anoter sip leaves te same port at te same time and sails on a bearing of S E, at speed 17.1 mp. Find te distance between te two sips after rs After ours: s 9.4 mi. r r 17.1 mi s. r 34. r tan tan 30. s 1 y tan 59.8 y 34. tan 59.8 s a y 58.8tan tan miles 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? Te plane is 190 miles from te orizon. sin A A sin 1 (0.9989)

25 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 cos ft P 0 P 1 O and P radius 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 tan ft 43 ft Eercise Te lengt of te sadow of a building m tall is 37.6 m. Find te angle of te elevation of te sun. tan B B tan Te angle of elevation is

26 =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 tan 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 ABC C c rate time mi. sin 46 b b c 600 b 600sin mi 46 W 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 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 tan 49 tan ft d y 34

27 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 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 R CPN NPQ 90 CPN PN cos( NPQ) d PN d cos18.5 cos18.5 cos CN R CN Rcos R CQ CM NM NM R CM NM R Rcos NM 1 R 1cos 35

28 Eercise Te angle of elevation from a point 93. ft from te base of a tower to te top of te tower is Find te eigt of te tower. tan tan 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 4cos miles Eercise From a igway overpass, 14.3 m above te road, te angle of depression of an oncoming car is measured at How far is te car from a point on te igway directly below te observer? tan(71.7 ) tan(71.7 ) 43. m Eercise A tunnel under a river is 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

29 tan 4.96 d 4533 ft sin y y 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? tan tan ft 37

30 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 500tan tan15.4 b a ba 500tan a 500tan15.4 b 500tan tan ft 1 mi 580 ft mi Te speed is: Te car is not speeding mi 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 tan tan 45 0 tan 45 0 (1) tan C C tan

31 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 tan tan58 50tan58 50 It took minutes to get from rate 1 50tan58 50tan 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. tan in tan 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 6cos feet distance 4.7 ft 1 in 1 ft 4 in 60.7 in distance ft 39 4 in Distance = + 4 inces 38 6 ft = 7 in

32 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 y tan 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 Find te eigt, to te nearest foot, of te Monument. Triangle ACB: Triangle DCB: tan tan tan 4 tan 4 tan tan37.8 tan 4 tan tan 37.8 tan 4 tan tan 37.8 tan 4 tan tan tan37.8 tan 4 tan 37.8 tan tan 37.8 tan 4 tan 4 tan ft A 100 ft D B C 40

33 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 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 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 d cot cm b) d cot Te cange is: cm cm 41

34 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 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 Use tis information to find te eigt of te pole. tan tan ft tan tan ft 4