Solution. Section 4.1 Law of Sines. Exercise. Exercise in. 9.6 in in. 621 in. In triangle ABC, A 110.

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1 Section 4.1 Law of Sines In triangle ABC, B 110, C 40 and b 18in. Find the length of side c. A 180 ( B C) a b c b sin A sin B sin C sin B a 18 c 18 sin 30 sin110 sin 40 sin110 a 18sin 30 c 18sin 40 sin110 sin in 1.3 in In triangle ABC, A , C 1. 8 and c 46 in. Find all the missing parts. B 180 A C a c b c sin A sin C sin B sin C a 46 b 46 sin110.4 sin sin 1.8 a 46sin110.4 sin 1.8 b 46sin 47.8 sin in 491 in 1

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

3 Solve triangle ABC if A = 55.3, a =.8 ft, and b = 4.9 ft sin B sin A b a sin B 4.9sin B sin B 63.9 and B C 180 A B C C 60.8 c asinc sin A.8sin 60.8 sin ft C C 8.6 c asinc sin A.8sin8.6 sin ft 3

4 Solve triangle ABC given A = 43.5, a = 10.7 in., and c = 7. in sin C sin A c a sin C 7.sin C sin C 7.6 and C B 180 AC B B b asin B sin A 10.7sin108.9 sin in B B 15.9 Is not possible If A 6, s, and r 19 find C s ad r r 19 D 180 AC r r sin D sin A 19 19sin 88 sin 6 19sin sin 6 4

5 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 B, angle A is If the distance between points B 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 C arc length C 910 rad D 180 ( A C) BD divides by radius 180 ( ) 5

91. 4 CA 3960 sin D sin A 3960 3960 sin91.4 sin 75.4 3960 3960sin91.4 sin 75.4 3960sin91.4 3960 sin 75.4 130 mi The dimensions of a land are given in the figure.

6 91. 4 CA 3960 sin D sin A sin91.4 sin sin91.4 sin sin sin mi 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 6

7 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 maimum distance from Fairbanks? From the triangle ABC: ABC ACB BAC The length AC is the maimum distance from Fairbanks: b 100 sin108 sin 45 b 100sin108 sin miles The angle of elevation of the top of a water tower from point A on the ground is From point B, 50.0 feet closer to the tower, the angle of elevation is 1.8. What is the height of the tower? ABC ACB Apply the law of sines in triangle ABC: BC 50 BC 50sin sin19.9 sin1.9 sin1.9 y Using the right triangle: sin 1.8 BC y 513.3sin ft 7

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 AB and BC.

8 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 AB and BC. tan 6 tan tan 3 tan AB 14 sin sin1.59 AB 14sin sin1.59 BC 14 sin sin1.59 BC 14sin sin ft 15.7 ft A C B

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.

9 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 ecavation? ACB ABC Using the law of sines: 400 sin118 sin 6 400sin ft sin 6 A 750 ft ft C 6 B Yes, the tree will have to be ecavated. 9

10 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? ACB 35 BAC After t seconds; The cruise missile distance: 548 t miles 3600 The Projectile distance: 688 t miles 3600 Using the law of sines: 548t 688t sin 145 sin 35 A 688t mi 35 B 548t mi C 1 mi D 548t sin t sin sin sin sin 145 sin sin sin sin sin BAC The angle of elevation of the projectile must be

11 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.) ABD ABC Using the law of sines: 17t 60t sin sin sin sin110 sin 17 sin sin 1 17 sin A 60t ft C B 17t ft D 11

12 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 fo starts running in a straight line from a position 30 ft to the west of the rabbit 6.5 ft/sec. The fo chooses his path so that he will catch the rabbit at point C. In how many seconds will the fo catch the rabbit? BAC t 3.5t sin10 sin B sin10 sin B sin B 3.5sin B sin 1 3.5sin C t 30 sin 8 sin 3 t 30sin 8 3.5sin sec It will take 7.6 sec. to catch the rabbit. B A C 1

13 Section 4. - Law of Cosines If a = 13 yd, b = 14 yd, and c = 15 yd, find the largest angle. C cos 1 a b c ab cos (13)(14) 67 Solve triangle ABC if b = 63.4 km, and c = 75. km, A = a b c bccos A cos a 1.9 km sin B bsin A a 63.4sin B sin sin C 180 A B

14 Solve triangle ABC if a = 83 ft, b = 63 ft, and c = 345 ft C cos 1 a b c ab cos (83)(63) sin B bsin C c 63sin 345 B sin 1 63sin A Solve triangle ABC if A 4.3, b 1.9 m, and c 15.4m a b c bccos A (1.9)(15.4)cos a m sin B bsin A 1.9sin 4.3 a B sin 1 1.9sin C

15 Solve triangle ABC if a 9.47 ft, b 15.9 ft, and c 1.1ft C cos 1 a b c ab cos (9.47)(15.9) sin B bsin C c 15.9sin B 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 (17.7)(1.1) cos cm 15

16 An engineer wants to position three pipes at the vertices of a triangle. If the pipes A, B, and C have radii in, 3 in, and 4 in, respectively, then what are the measures of the angles of the triangle ABC? AC 6 AB 5 BC 7 A cos (5)(6) 6 7 sin B sin 78.5 sin B 6sin B sin 1 6sin C 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 cos cos miles

A submarine sights a moving target at a distance of 80 m.

17 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 m 17

By the cosine law, AB 5.3 7.6 (5.3)(7.6) cos 8 3.8 1 3.8 5.3 7.6 CBA cos 11 (3.8)(5.3) BAC 180 11 8 40 A 6-ft antenna is installed at the top of a roof.

18 A tunnel is planned through a mountain to connect points A and B on two eisting roads. If the angle between the roads at point C is 8, what is the distance from point A to B? Find CBA and CAB to the nearest tenth of a degree. By the cosine law, AB (5.3)(7.6) cos CBA cos 11 (3.8)(5.3) BAC 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? By the cosine law, (6)(10)cos ft

On June 30, 1861, Comet Tebutt, one of the greatest comets, was visible even before sunset. One of the factors that causes a comet to be etra bright is a small scattering angle.

19 On June 30, 1861, Comet Tebutt, one of the greatest comets, was visible even before sunset. One of the factors that causes a comet to be etra bright is a small scattering angle. When Comet 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 Comet Tebutt on June 30, (One astronomical unit (a.u) is the average distance between the earth and the sub.) By 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.

20 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 AC 30 8 BC 36 AB AC CB By the cosine law: A 1 ADB cos AD DB AB ADDB C B ADB cos ADB D CAB BC AC tan 36 CAB tan sin DAB sin 89.4 sin DAB 30sin DAB sin 1 30sin CAB DAB 1 a 0

58.57 45.3 13.5 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.

21 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. BEC ECD 10 From triangle EBC: tan BE BE tan10 BAC ACF 15 From triangle ABC: tan AB C B 45 A F D AB tan15 AB BE AB BE cos cos ft 1

22 Section 4.3 Vectors and Dot Product An arrow is shot into the air so that its horizontal velocity is 5 feet per second and its vertical is 15 feet per second. Find the velocity of the arrow. The magnitude of the velocity is: V ft / sec V y tan V tan A boat travels 7 miles on a course of bearing N 7 E and then changes its course to travel 37 miles at N 55 E. How far north and how far east has the boat traveled on this 109-mile trip? Total distance traveled east = U V N N = 7 cos63 37cos 35 = 63 mi V y 55 V 37 mi Total distance traveled North = U y Vy = 7 sin63 37sin 35 U y 7 U 7 mi = 85 mi 63 U V E

23 Let u =, 1 and v = 4, 3. Find the following. a) u + v b) u c) 4u 3v a) u + v =, 1 + 4, 3 =, 4 b) u = -, 1 = 4, - c) 4u 3v = 4, 1-34, 3 = -8, 4-1, 9 = -0, -5 Given: V 13.8, 4., find the magnitudes of the horizontal and vertical vector components of V, V and V y, respectively V V cos 13.8cos V y V sin 13.8sin Find the angle θ between the two vectors u = 3, 4 and v =, 1. cos 1 uv. u. v 1 (3)() (4)(1) cos

24 A bullet is fired into the air with an initial velocity of 1,800 feet per second at an angle of 60 from the horizontal. Find the magnitude of the horizontal and vertical vector component as of the velocity vector. V Vy 1800cos ft / sec 1800sin ft / sec A bullet is fired into the air with an initial velocity of 1,00 feet per second at an angle of 45 from the horizontal. a) Find the magnitude of the horizontal and vertical vector component as of the velocity vector. b) Find the horizontal distance traveled by the bullet in 3 seconds. (Neglect the resistance of air on the bullet). a) V 100cos ft /sec Vy 100sin ft / sec b) 850*3, 550 ft V. t 850*3,550 ft A ship travels 130 km on a bearing of S 4 E. How far east and how far south has it traveled? V 130cos km east V 130sin 4 87 km south 4

25 An arrow is shot into the air with so that its horizontal velocity is 15.0 ft./sec and its vertical velocity is 5.0 ft./sec. Find the velocity of the arrow? V V V y V 15 5 V 9. ft tan 5 15 tan 1 60 /sec 5 15 A plane travels 170 miles on a bearing of N 18 E and then changes its course to N 49 E and travels another 10 miles. Find the total distance traveled north and the total distance traveled east. Total distance traveled east = Total distance traveled North = U V = 170 cos7 10cos 41 = mi = 140 mi - East U y Vy = 170 sin7 10sin 41 = 40 mi - South V y U y N 18 7 U U N 49 V V E 5

A boat is crossing a river that run due north. The boat is pointed due east and is moving through the water at 1 miles per hour. If the current of the river is a constant 5.

26 A boat is crossing a river that run due north. The boat is pointed due east and is moving through the water at 1 miles per hour. If the current of the river is a constant 5.1 miles per hour, find the actual course of the boat through the water to two significant digits. tan N tan V 1 mi/hr 5.1 mi/hr E V (using Pythagorean Theorem) 13 Two forces of 15 and Newtons act on a point in the plane. (A Newton is a unit of force that equals.5 lb.) If the angle between the forces is 100, find the magnitude of the resultant vector. P Q R S 360 P Q 360 Q 360 (100 ) Q 160 Q 80 V 15 (15)() cos V 4.4 N 6

Find the magnitude of the equilibrant of forces of 48 N and 60 N acting on a point A, if the angle between the forces is 50. Then find the angle between the equilibrant and the 48-newton force.

27 Find the magnitude of the equilibrant of forces of 48 N and 60 N acting on a point A, if the angle between the forces is 50. Then find the angle between the equilibrant and the 48-newton force. The equilibrant is v. A B 360 B 360 (50 ) B 60 B 130 V (48)(60) cos130 V 98 N sin BAC sin sin BAC 60sin BAC sin 1 60sin Find the force required to keep a 50-lb wagon from sliding down a ramp inclined at 0 to the horizontal. (Assume there is no friction.) Vector BC represents the force with which the weight pushes against the ramp. Vector BF represents the force that would pull the weight up the ramp. The vertical force BA represents the force of gravity. AC sin 0 AC 50sin A force of approimately 17 lbs. will keep the wagon from sliding down the ramp. 7

28 A force of 16.0 lbs. is required to hold a 40.0 lbs. lawn mower on an incline. What angle does the incline make with the horizontal? Vector BE represents the force required to hold the mower on the incline. sin B B sin The hill makes an angle of about 3.6 with the horizontal. A ship leaves port on a bearing of 8.0 and travels 8.0 mi. The ship then turns due east and travels 4.30 mi. How far is the ship from port? What is its bearing from port? NAP PAE PE cos PE 10.9 The ship is about 10.9 miles from port. The bearing is given by: sin APE sin sin APE 4.3sin APE sin 1 4.3sin The bearing:

Two prospectors are pulling on ropes attached around the neck of a donkey that does not want to move. One prospector pulls with a force of 55 lb, and the other pulls with a force of 75 lb.

29 Two prospectors are pulling on ropes attached around the neck of a donkey that does not want to move. One prospector pulls with a force of 55 lb, and the other pulls with a force of 75 lb. If the angle between the ropes is 5, then how much force must the donkey use in order to stay put? (The donkey knows the proper direction in which to apply his force.) 1 OCB OAB 360 (5 ) 155 By the cosine law, the magnitude of the resultant force is: OB OC CB ( OC)( OB)cos OCB (75)(55) cos 155 O C 55 A B 17 lb So the donkey must pull a force of 17 pounds in the direction opposite that of the resultant s. A solid steel ball is placed on a 10 incline. If a force of 3. lb in the direction of the incline is required to keep the ball in place, then what is the weight of the ball? 10 w f sin10 w lb w sin10 9

30 Find the amount of force required for a winch to pull a 3000-lb car up a ramp that is inclined at f f sin 0 f 3000sin lb 3000 If the amount of force required to push a block of ice up an ice-covered driveway that is inclined at 5 is 100lb, then what is the weight of the block? 5 w sin w 100 w sin lb If superman eerts 1000 lb of force to prevent a 5000-lb boulder from rolling down a hill and crushing a bus full of children, then what is the angle of inclination of the hill? w 1000 sin 1000 sin

31 If Sisyphus eerts a 500-lb force in rolling his 4000-lb spherical boulder uphill, then what is the angle of inclination of the hill? sin 500 sin A plane is headed due east with an air speed of 40 mph. The wind is from the north at 30 mph. Find the bearing for the course and the ground speed of the plane The bearing is: tan The ground speed can be found by using Pythagorean Theorem: mph A plane is headed due west with an air speed of 300 mph. The wind is from the north at 80 mph. Find the bearing for the course and the ground speed of the plane tan The bearing is: The ground speed can be found by using Pythagorean Theorem: mph 31

32 An ultralight is flying northeast at 50 mph. The wind is from the north at 0 mph. Find the bearing for the course and the ground speed of the ultralight Using the cosine law, the ground speed is: 50 0 (50)(0) cos mph sin sin 45 sin 0sin sin 1 0sin The bearing is: A superlight is flying northwest at 75 mph. The wind is from the south at 40 mph. Find the bearing for the course and the ground speed of the superlight Using the cosine law, the ground speed is: (75)(40) cos mph sin sin135 sin 40sin sin 1 40sin The bearing is:

33 An airplane is heading on a bearing of 10 with an air speed of 480 mph. If the wind is out of the northeast (bearing 5) at 58 mph, then what are the bearing of the course and the ground speed of the airplane? : the airplane s vector course and ground speed From the parallelogram (from the vectors): By the cosine law: (480)(58) cos mph sin sin sin 58sin sin 1 58sin The bearing of the plane: In Roman mythology, Sisyphus revealed a secret of Zeus and thus incurred the god s wrath. As punishment, Zeus banished him to Hades, where he was doomed for eternity to roll a rock uphill, only to have it roll back on him. If Sisyphus stands in front of a 4000-lb spherical rock on a 0 incline, then what force applied in the direction of the incline would keep the rock from rolling down the incline? 33

cos 70 f 4000 f 4000cos70 1368 lb 0 f 70 4000 A trigonometry student wants to cross a river that is 0. mi wide and has a current of 1 mph. The boat goes 3 mph in still water.

34 cos 70 f 4000 f 4000cos lb 0 f A trigonometry student wants to cross a river that is 0. mi wide and has a current of 1 mph. The boat goes 3 mph in still water. a) Write the distance the boats travels as a function of the angle. b) Write the actual speed of the boat as a function of and. c) Write the time for the trip as a function of. Find the angle for which the student will cross the river in the shortest amount of time. 1 mph V 3 mph y V a) The vector in still water is given by: V Vscos 90 3sin Vy 3cos V 3sin i 3cos j The vector of the current: V c 1i Therefore, the actual direction and speed is determined by the vector: v 3sin 1 i 3cos j 34

35 The number t of hours it takes the boat to cross the river: t d 0. v 3cos y The distance of the boat travels: d tv 0. 3sin 1 3cos 3cos 0. 3sin 1 3cos 3cos 3cos 3sin cos 0. tan 1 v tan v y 3sin 1 3cos 0. tan 1 sec tan 1 0. sec 0. sec b) The actual speed d t 0. sec 0. 3cos 3cos sec c) The shortest time for the boat to cross river is straight ahead which = 0 t sec 3cos

A male uses three elephants to pull a very large log out of the jungle. The papa elephant pulls with 800 lb. of force, the mama elephant pulls with 500 lb.

36 A male uses three elephants to pull a very large log out of the jungle. The papa elephant pulls with 800 lb. of force, the mama elephant pulls with 500 lb. of force, and the baby elephant pulls with 00 lb. force. The angles between the forces are shown in the figure. What is the magnitude of the resultant of all three forces? If mama is pulling due east, then in what direction will the log move? vp 800 cos 30i 800sin 30 j vm 500i v 00cos 0i 00sin 0 j b The total force is determined by: F v v v p m b 800 cos sin cos 0 00sin 0 i j i i j 800 cos cos 0 i 800sin 30 00sin i j 30 v p 0 j v b v m 800 lb 500 lb 00 lb The magnitude of the resultant: F lbs The direction: tan E1 3.5 N 36

37 Section 4.4 Trigonometric Form of Comple Numbers Write 3 i in trigonometric form. (Use radian measure) 3 3 i y 1 r 3 1 y tan The reference angle for is 6 and the angle is in quadrant II. Therefore, i cis 5 6 Write 3 4i in trigonometric form. r 3 ( 4) 5 34i tan The angle is in quadrant II; therefore, i 5 cis17 37

38 Write 1 0i in trigonometric form. r i tan The angle is in quadrant III; therefore, i 9 cis3.6 Write 11 i in trigonometric form. r i tan The angle is in quadrant I; therefore, i 5 5 cis10.3 Write 4cos30 i sin 30 in standard form. 4 3 cos30 sin i i 3 i 38

39 Write cis 7 in standard form. 4 cis 7 cos 7 isin i 1 i Find the quotient 0cis 75 4cis 40 0cis 75 0 cis cis cis 35 5 cos35 i sin i. Write the result in rectangular form. 39

40 Divide z 1 i 1 3 by z 3 i. Write the result in rectangular form. z 1 z 1 i 3 or 3 i r i 3 : 1 3 tan 1 3 r i : tan i 3 3i z cis i 3i z cis i i i cis i cis i cis i cos isin

41 Section 4.5 Polar Coordinates Convert to rectangular coordinates, 60, 60 cos60 isin i 1 i 3 Convert to rectangular coordinates, 5, 5 cos( 5 ) i sin( 5 ) 1 1 i 1 i Convert to rectangular coordinates 4 3, 4 3, 4 3 cos isin i

42 Convert to polar coordinates r , 3 tan tan , 3 r The angle is in quadrant III; therefore, , 3 3, 5 Convert to polar coordinates, 3, 3 r r tan tan The angle is in quadrant IV; therefore, , 3 4, 300 Convert to polar coordinates r 0, 0 tan 1 0 0, 0,, 0 r 0 0 4

43 Convert to polar coordinates 1, 3 r 0 0 r 1 3 1, tan 1 3 The angle is in quadrant III; therefore, 1, 3 4, Write the equation in rectangular coordinates r 4 y 4 r 4 Write the equation in rectangular coordinates r 6cos r 6 r r 6 r 6cos y 6 43

44 Write the equation in rectangular coordinates r 4cos r 4 cos sin cos sin r 4 y r r y 4 r r 4 y r r 4 4 y 4 y 4 4y r y y r Write the equation in rectangular coordinates r cos sin r cos sin cos sin r y r r r y r r y y r 44

45 Write the equation in polar coordinates y 5 rcos rsin 5 r cos y r sin r cos sin 5 r 5 cos sin Write the equation in polar coordinates y 9 r 9 y 9 r y Write the equation in polar coordinates r 4rcos r r 4r cos r r 4cos y 4 Write the equation in polar coordinates y y r cos y r sin rsin rcos sin cos 45

46 Section De Moivre s Theorem Find 1 i 8 and epress the result in rectangular form. 1 1 r 1i tan i cis 4 8 1i cis cis cis 16 cos isin 16 1 i0 16 Find 1 i 10 and epress the result in rectangular form. 10 1i cis cis cis 5 3cis 3 cos isin 3 0 i 3i 46

47 Find fifth roots of z 1 i 3 and epress the result in rectangular form. r 1 3 1i tan i 3 cis60 1/5 1/5 5 cis 60360k cis 1 7 k If k 0 5 cis cis1 If k 1 5 cis cis84 If k 5 cis cis156 If k 3 5 cis cis8 If k 4 5 cis cis300 Find the fourth roots of z 16cis60 4 z 416 cis 60360k 4 4 cis 15 90k If k 0 cis cis1 5 If k 1 cis cis10 5 If k cis cis19 5 If k 3 cis cis85 47

48 Find the cube roots of 7. cis 1/ cis 0 360k cis 010 k If k 0 z 3 cis cis0 1 If k 1z 3 cis 0 10 cis10 If k z 3 cis 010 cis40 Find all comple number solutions of r ( 1) tan cis 1 cis 1/3 1 1/3 cis k 3 3 cis k Ifk0cis cis c 3 If k 1 cis is cis If k cis cis cos isin i 3 3 cos isin cos isin i

49 Find 5 30 cis 5 5 cis30 cis 5(30 ) 3 cos150 isin i i 49

Ch6prac 1.Find the degree measure of the angle with the given radian measure. (Round your answer to the nearest whole number.) -2

Ch6prac 1.Find the degree measure of the angle with the given radian measure. (Round your answer to the nearest whole number.) -2 2. Find the degree measure of the angle with the given radian measure.