Section 2.4 Law of Sines and Cosines

Transcription

1 Section.4 Law of Sines and osines Oblique Triangle A triangle that is not a right triangle, either acute or obtuse. The measures of the three sides and the three angles of a triangle can be found if at least one side and any other two measures are known. The Law of Sines There are many relationships that exist between the sides and angles in a triangle. One such relation is called the law of sines. Given triangle A or, equivalently sin A sin sin a b c sin a sin b A sin c Proof sin A h b sin h a h bsin A (1) h asin () From (1) & () h h bsin A asin bsin A asin ab ab sin A sin a b 8

2 Angle Side - Angle (ASA or AAS) If two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent. Example In triangle A, A ( A ) 180 (30 70) , 70, and a 8.0 cm. Find the length of side c. c sin a sin A c a sin sin A 8 sin sin cm Example Find the missing parts of triangle A if 180 ( ) 180 (3 81.8) 66. a b sin A sin b asin sin A 4.9sin 66. sin cm A 3, c a sin sin A c asin sin A sin 81.8 sin cm,, and a 4.9 cm. 9

3 Example You wish to measure the distance across a River. You determine that = 11.90, A = 31.10, and. Find the distance a across the river. b ft 180 A a b sin A sin a sin 31.1 sin 36 a sin 31.1 sin 36 a ft Example Find distance x if a = 56 ft., x a sin sin A x asin sin A 56sin 5.7 sin ft 5. 7 and A

4 Example A hot-air balloon is flying over a dry lake when the wind stops blowing. The balloon comes to a stop 450 feet above the ground at point D. A jeep following the balloon runs out of gas at point A. The nearest service station is due north of the jeep at point. The bearing of the balloon from the jeep at A is N 13 E, while the bearing of the balloon from the service station at is S 19 E. If the angle of elevation of the balloon from A is 1, how far will the people in the jeep have to walk to reach the service station at point? tan1 D A A D tan1 450 tan1,117 ft A 180 (13 19) 148 Using triangle A A 117 sin148 sin19 A 117sin148 sin19 3, 400 ft 31

5 Ambiguous ase Side Angle Side (SAS) If two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent. Example Find angle in triangle A if a =, b = 6, and A 30 sin sin A b a sin bsin A a Since 6sin 30 1 sin sin 1 is impossible, no such triangle exists. Example Find the missing parts in triangle A if = 35.4, a = 05 ft., and c = 314 ft. sin A asin c 05 sin A sin 1 (0.378) A. A (. 35.4) 1. 4 b csin sin 314sin1.4 sin ft A

6 Example Find the missing parts in triangle A if a = 54 cm, b = 6 cm, and A = 40. sin sin A b a sin bsin A a 6sin sin 1 (0.738) (40 48) 180 (40 13) 9 c asin sin A 54sin 9 sin cm 8 c asin sin A 54sin 8 sin 40 1 cm 33

7 Area of a Triangle (SAS) In any triangle A, the area A is given by the following formulas: A 1 bcsin A A 1 acsin A 1 absin Example Find the area of triangle A if A 440, b 7.3 cm, and a b sin A sin a 7.3 sin 4 40 sin sin 440 a sin cm A 1 acsin 1 (11.7)(7.3)sin cm Example Find the area of triangle A. A 1 acsin sin ft 34

8 Number of Triangles Satisfying the Ambiguous ase (SSA) Let sides a and b and angle A be given in triangle A. (The law of sines can be used to calculate the value of sin.) 1. If applying the law of sines results in an equation having sin > 1, then no triangle satisfies the given conditions.. If sin = 1, then one triangle satisfies the given conditions and = If 0 < sin < 1, then either one or two triangles satisfy the given conditions. a) If sin = k, then let b) Let 180. If 1 in the second triangle. sin 1 k 1 and use A for in the first triangle., then a second triangle exists. In this case, use for 35

9 Law of osines (SAS) a b c bccos A b a c accos c a b abcos Derivation a ( c x) h c cx x h (1) b x h () From (): (1) a c cx b a c b cx (3) cos A x b bcos A x (3) a c b cb cos A 36

10 Example Find the missing parts in triangle A if A = 60, b = 0 in, and c = 30 in. a b c bccos A 0 30 (0)(30) cos a 6 sin bsin A a 0sin sin 1 (0.666) A Example A surveyor wishes to find the distance between two inaccessible points A and on opposite sides of a lake. While standing at point, she finds that A = 59 m, = 43 m, and angle A = Find the distance A. A A A cos cos A 68 37

11 Law of osines (SSS) - Three Sides cos A b c a bc cos a c b ac cos a b c ab Example Solve triangle A if a = 34 km, b = 0 km, and c = 18 km b c cos A a bc (0)(18) 0.6 A cos 1 ( 0.6) 17 a b cos c ab (34)(0) 0.91 cos 1 (0.91) A OR sin csin A a 18sin sin 1 (0.48) 5 38

12 Example A plane is flying with an airspeed of 185 miles per hour with heading 10. The wind currents are running at a constant 3 miles per hour at 165 clockwise from due north. Find the true course and ground speed of the plane V W V W V W cos (185)(3) cos135 43,61 V W 10 mph sin sin 3 10 sin 3sin sin 1 (0.1077) The true course is: The speed of the plane with respect to the ground is 10 mph. Example Find the measure of angle in the figure of a roof truss. cos a c b ac (11)(9) cos (11)(9) 33 39

13 Heron s Area Formula (SSS) If a triangle has sides of lengths a, b, and c, with semi-perimeter 1 s a b c Then the area of the triangle is: A ss as bs c Example The distance as the crow flies from Los Angeles to New York is 451 miles, from New York to Montreal is 331 miles, and from Montreal to Los Angeles is 47 miles. What is the area of the triangular region having these three cities as vertices? (Ignore the curvature of Earth.) The semiperimeter s is: 1 s a b c A ss as bs c ,700 mi 40

14 s Section.4 Law of Sines and osines 1. In triangle A, 110, 40 and b 18in. Find the length of side c.. In triangle A, A , Find the missing parts of triangle A, if and c 46 in 34,. Find all the missing parts Solve triangle A if = 55 40, b = 8.94 m, and a = 5.1 m. 5. Solve triangle A if A = 55.3, a =.8 ft., and b = 4.9 ft., and 6. Solve triangle A given A = 43.5, a = 10.7 in., and c = 7. in. a 5.6 cm. 7. If A 6, s, and r 19, find x 8. If a = 13 yd., b = 14 yd., and c = 15 yd., find the largest angle. 9. Solve triangle A if b = 63.4 km, and c = 75. km, A = Solve triangle A if A 4.3, b 1.9 m, and c 15.4m 11. Solve triangle A if a = 83 ft., b = 63 ft., and c = 345 ft. 1. Solve triangle A if a 9.47 ft, b 15.9 ft, and c 1.1ft 13. The diagonals of a parallelogram are 4. cm and 35.4 cm and intersect at an angle of Find the length of the shorter side of the parallelogram 14. A man flying in a hot-air balloon in a straight line at a constant rate of 5 feet per second, while keeping it at a constant altitude. As he approaches the parking lot of a market, he notices that the angle of depression from his balloon to a friend s car in the parking lot is 35. A minute and a half later, after flying directly over this friend s car, he looks back to see his friend getting into the car and observes the angle of depression to be 36. At that time, what is the distance between him and his friend? 15. A satellite is circling above the earth. When the satellite is directly above point, angle A is If the distance between points and D on the circumference of the earth is 910 miles and the radius of the earth is 3,960 miles, how far above the earth is the satellite? 41

15 16. A pilot left Fairbanks in a light plane and flew 100 miles toward Fort in still air on a course with bearing of 18. She then flew due east (bearing 90) for some time drop supplies to a snowbound family. After the drop, her course to return to Fairbanks had bearing of 5. What was her maximum distance from Fairbanks? 17. The dimensions of a land are given in the figure. Find the area of the property in square feet. 18. The angle of elevation of the top of a water tower from point A on the ground is From point, 50.0 feet closer to the tower, the angle of elevation is 1.8. What is the height of the tower? 19. A 40-ft wide house has a roof with a 6-1 pitch (the roof rises 6 ft for a run of 1 ft). The owner plans a 14-ft wide addition that will have a 3-1 pitch to its roof. Find the lengths of A and 0. A hill has an angle of inclination of 36. A study completed by a state s highway commission showed that the placement of a highway requires that 400 ft of the hill, measured horizontally, be removed. The engineers plan to leave a slope alongside the highway with an angle of inclination of 6. Located 750 ft up the hill measured from the base is a tree containing the nest of an endangered hawk. Will this tree be removed in the excavation? 4

16 1. A cruise missile is traveling straight across the desert at 548 mph at an altitude of 1 mile. A gunner spots the missile coming in his direction and fires a projectile at the missile when the angle of elevation of the missile is 35. If the speed of the projectile is 688 mph, then for what angle of elevation of the gun will the projectile hit the missile?. When the ball is snapped, Smith starts running at a 50 angle to the line of scrimmage. At the moment when Smith is at a 60 angle from Jones, Smith is running at 17 ft/sec and Jones passes the ball at 60 ft/sec to Smith. However, to complete the pass, Jones must lead Smith by the angle. Find (find in his head. Note that can be found without knowing any distances.) 43

17 3. A rabbit starts running from point A in a straight line in the direction 30 from the north at 3.5 ft/sec. At the same time a fox starts running in a straight line from a position 30 ft to the west of the rabbit 6.5 ft/sec. The fox chooses his path so that he will catch the rabbit at point. In how many seconds will the fox catch the rabbit? 4. An engineer wants to position three pipes at the vertices of a triangle. If the pipes A,, and have radii in, 3 in, and 4 in, respectively, then what are the measures of the angles of the triangle A? 5. A solar panel with a width of 1. m is positioned on a flat roof. What is the angle of elevation of the solar panel? 6. Andrea and Steve left the airport at the same time. Andrea flew at 180 mph on a course with bearing 80, and Steve flew at 40 mph on a course with bearing 10. How far apart were they after 3 hr.? 44

18 7. A submarine sights a moving target at a distance of 80 m. A torpedo is fired 9 ahead of the target, and travels 94 m in a straight line to hit the target. How far has the target moved from the time the torpedo is fired to the time of the hit? 8. A tunnel is planned through a mountain to connect points A and on two existing roads. If the angle between the roads at point is 8, what is the distance from point A to? Find A and A to the nearest tenth of a degree. 9. A 6-ft antenna is installed at the top of a roof. A guy wire is to be attached to the top of the antenna and to a point 10 ft down the roof. If the angle of elevation of the roof is 8, then what length guy wire is needed? 30. On June 30, 1861, omet Tebutt, one of the greatest comets, was visible even before sunset. One of the factors that causes a comet to be extra bright is a small scattering angle. When omet Tebutt was at its brightest, it was a.u. from the earth, a.u. from the sun, and the earth was a.u. from the sun. Find the phase angle and the scattering angle for omet Tebutt on June 30, (One astronomical unit (a.u) is the average distance between the earth and the sub.) 45

19 31. A human arm consists of an upper arm of 30 cm and a lower arm of 30 cm. To move the hand to the point (36, 8), the human brain chooses angle to the nearest tenth of a degree. and 1 3. A forest ranger is 150 ft above the ground in a fire tower when she spots an angry grizzly bear east of the tower with an angle of depression of 10. Southeast of the tower she spots a hiker with an angle of depression of 15. Find the distance between the hiker and the angry bear. 33. Two ranger stations are on an east-west line 110 mi apart. A forest fire is located on a bearing N 4 E from the western station at A and a bearing of N 15 E from the eastern station at. How far is the fire from the western station? 46

20 Section.4 Law of Sines and osines In triangle A, 110, 40 and b 18in. Find the length of side c. A 180 ( ) a b sin A sin a 18 sin 30 sin110 a 18sin 30 sin in c b sin sin c 18 sin 40 sin110 c 18sin 40 sin in In triangle A, A , 180 A a c sin A sin a 46 sin110.4 sin 1.8 a 46sin110.4 sin and c 46 in b c sin sin b sin 1.8 b 46sin 47.8 sin in 491 in. Find all the missing parts. 43

21 Find the missing parts of triangle A if A 180 ( ) 180 (34 8) , 8, and a 5.6 cm. b a sin sin A b asin sin A 5.6sin34 sin64 3.5cm c a sin sin A c asin sin A 5.6sin8 sin64 6. cm Solve triangle A if = 55 40, b = 8.94 m, and a = 5.1 m. sin A sin a b sin 5540 sin A sin sin A Since sin A > 1 is impossible, no such triangle exists. 44

22 Solve triangle A if A = 55.3, a =.8 ft, and b = 4.9 ft sin sin A b a sin 4.9sin sin and A c asin sin A.8sin 60.8 sin ft c asin sin A.8sin8.6 sin ft 45

23 Solve triangle A given A = 43.5, a = 10.7 in., and c = 7. in sin sin A c a sin 7.sin sin and A b asin sin A 10.7sin108.9 sin in Is not possible If A 6, s, and r 19 find x s ad r r 19 D 180 A r x r sin D sin A 19 x 19sin 88 sin 6 x 19sin sin 6 If a = 13 yd, b = 14 yd, and c = 15 yd, find the largest angle. 1 1 cos a b c cos ab (13) ( 14) 46

24 Solve triangle A if b = 63.4 km, and c = 75. km, A = a b c bccos A cos a 1.9 km sin bsin A a 63.4sin sin sin A Solve triangle A if a = 83 ft, b = 63 ft, and c = 345 ft cos 1 a b c ab cos (83)(63) sin bsin c 63sin 345 sin 1 63sin A

25 Solve triangle A if a b c bccos A A 4.3, b 1.9 m, and c 15.4m (1.9)(15.4) cos a m sin bsin A 1.9sin 4.3 a sin 1 1.9sin Solve triangle A if cos 1 a b c ab a 9.47 ft, b 15.9 ft, and c 1.1ft cos (9.47)(15.9) sin bsin c 15.9sin sin sin A The diagonals of a parallelogram are 4. cm and 35.4 cm and intersect at an angle of Find the length of the shorter side of the parallelogram x (17.7)(1.1)cos x 16.8 cm 48

26 A man flying in a hot-air balloon in a straight line at a constant rate of 5 feet per second, while keeping it at a constant altitude. As he approaches the parking lot of a market, he notices that the angle of depression from his balloon to a friend s car in the parking lot is 35. A minute and a half later, after flying directly over this friend s car, he looks back to see his friend getting into the car and observes the angle of depression to be 36. At that time, what is the distance between him and his friend? car d 450 sin 35 sin109 d 450sin35 sin ft 5 ft/sec car d A satellite is circling above the earth. When the satellite is directly above point, angle A is If the distance between points and D on the circumference of the earth is 910 miles and the radius of the earth is 3,960 miles, how far above the earth is the satellite? s r arc length D divides by radius 910 rad D 180 ( A ) 180 ( ) A 3960 sin D sin A x sin91.4 sin 75.4 x sin91.4 sin 75.4 x 3960sin sin 75.4 x 130 mi 49

27 A pilot left Fairbanks in a light plane and flew 100 miles toward Fort in still air on a course with bearing of 18. She then flew due east (bearing 90) for some time drop supplies to a snowbound family. After the drop, her course to return to Fairbanks had bearing of 5. What was her maximum distance from Fairbanks? From the triangle A: A A A The length A is the maximum distance from Fairbanks: b 100 sin108 sin 45 b 100sin108 sin miles The dimensions of a land are given in the figure. Find the area of the property in square feet. 1 A sin ft 1 A sin ft The total area = A A ,748.8 ft 1 The angle of elevation of the top of a water tower from point A on the ground is From point, 50.0 feet closer to the tower, the angle of elevation is 1.8. What is the height of the tower? A A Apply the law of sines in triangle A: 50 sin19.9 sin1.9 50sin sin1.9 50

28 Using the right triangle: sin 1.8 y 513.3sin ft y A 40-ft wide house has a roof with a 6-1 pitch (the roof rises 6 ft for a run of 1 ft). The owner plans a 14-ft wide addition that will have a 3-1 pitch to its roof. Find the lengths of A tan 6 tan tan 3 tan A 14 sin sin1.59 A 14sin sin sin sin sin sin ft 15.7 ft A and A hill has an angle of inclination of 36. A study completed by a state s highway commission showed that the placement of a highway requires that 400 ft of the hill, measured horizontally, be removed. The engineers plan to leave a slope alongside the highway with an angle of inclination of 6. Located 750 ft up the hill measured from the base is a tree containing the nest of an endangered hawk. Will this tree be removed in the excavation? A A Using the law of sines: x 400 sin118 sin 6 A x 750 ft ft 6 51

29 x 400sin ft sin 6 Yes, the tree will have to be excavated. A cruise missile is traveling straight across the desert at 548 mph at an altitude of 1 mile. A gunner spots the missile coming in his direction and fires a projectile at the missile when the angle of elevation of the missile is 35. If the speed of the projectile is 688 mph, then for what angle of elevation of the gun will the projectile hit the missile? A 35 A After t seconds; The cruise missile distance: The Projectile distance: t t miles miles Using the law of sines: 548t 688t sin 145 sin t sin t sin sin sin sin 145 sin sin sin sin sin A 688t mi t mi 1 mi D A The angle of elevation of the projectile must be 6. 5

30 When the ball is snapped, Smith starts running at a 50 angle to the line of scrimmage. At the moment when Smith is at a 60 angle from Jones, Smith is running at 17 ft/sec and Jones passes the ball at 60 ft/sec to Smith. However, to complete the pass, Jones must lead Smith by the angle. Find (find in his head. Note that can be found without knowing any distances.) AD A Using the law of sines: 17t 60t sin sin sin sin110 sin 17 sin sin 1 17 sin A 60t ft t ft D A rabbit starts running from point A in a straight line in the direction 30 from the north at 3.5 ft/sec. At the same time a fox starts running in a straight line from a position 30 ft to the west of the rabbit 6.5 ft/sec. The fox chooses his path so that he will catch the rabbit at point. In how many seconds will the fox catch the rabbit? A t 3.5t sin10 sin sin10 sin sin 3.5sin sin 1 3.5sin A 53

31 t 30 sin 8 sin 3 t 30sin 8 3.5sin sec It will take 7.6 sec. to catch the rabbit. An engineer wants to position three pipes at the vertices of a triangle. If the pipes A,, and have radii in, 3 in, and 4 in, respectively, then what are the measures of the angles of the triangle A? A 6 A 5 7 A cos (5)(6) 6 7 sin sin 78.5 sin 6sin sin 1 6sin Andrea and Steve left the airport at the same time. Andrea flew at 180 mph on a course with bearing 80, and Steve flew at 40 mph on a course with bearing 10. How far apart were they after 3 hr.? After 3 hrs., Steve flew: 3(40) = 70 mph Andrea flew: 3(180) = 540 mph x cos 10 x cos miles x

32 A solar panel with a width of 1. m is positioned on a flat roof. What is the angle of elevation of the solar panel? cos (1.)(1.) or s 0.4 r A submarine sights a moving target at a distance of 80 m. A torpedo is fired 9 ahead of the target, and travels 94 m in a straight line to hit the target. How far has the target moved from the time the torpedo is fired to the time of the hit? cos 9 m x A tunnel is planned through a mountain to connect points A and on two existing roads. If the angle between the roads at point is 8, what is the distance from point A to? Find A and A to the nearest tenth of a degree. y the cosine law, A (5.3)(7.6) cos A cos 11 (3.8)(5.3) A

33 A 6-ft antenna is installed at the top of a roof. A guy wire is to be attached to the top of the antenna and to a point 10 ft down the roof. If the angle of elevation of the roof is 8, then what length guy wire is needed? x 6 y the cosine law, x (6)(10)cos ft 8 10 On June 30, 1861, omet Tebutt, one of the greatest comets, was visible even before sunset. One of the factors that causes a comet to be extra bright is a small scattering angle. When omet Tebutt was at its brightest, it was a.u. from the earth, a.u. from the sun, and the earth was a.u. from the sun. Find the phase angle and the scattering angle for omet Tebutt on June 30, (One astronomical unit (a.u) is the average distance between the earth and the sub.) y the cosine law: cos 156 (0.133)(0.89 1) A human arm consists of an upper arm of 30 cm and a lower arm of 30 cm. To move the hand to the point (36, 8), the human brain chooses angle and to the nearest tenth of a degree. 1 A A 30 8 A A y the cosine law: 36 D a 56

34 1 AD cos AD D A ADD AD cos A A tan 36 A tan AD sin DA sin 89.4 sin DA 30sin DA sin 1 30sin A DA A forest ranger is 150 ft above the ground in a fire tower when she spots an angry grizzly bear east of the tower with an angle of depression of 10. Southeast of the tower she spots a hiker with an angle of depression of 15. Find the distance between the hiker and the angry bear. E ED 10 From triangle E: E tan10 A AF 15 From triangle A: tan A A tan15 tan10150 E x A E A E cos cos ft 45 A F x D 57

35 Two ranger stations are on an east-west line 110 mi apart. A forest fire is located on a bearing N 4 E from the western station at A and a bearing of N 15 E from the eastern station at. How far is the fire from the western station? A A b c sin sin b 110 sin105 sin 7 b 110sin105 sin 7 b 34 mi The fire is about 34 miles from the western station. 58