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8.E: Further Applications of Trigonometry (Exercises)

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8.1: Non-right Triangles: Law of Sines

Verbal

1) Describe the altitude of a triangle.

Answer

The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a 90 angle.

2) Compare right triangles and oblique triangles.

3) When can you use the Law of Sines to find a missing angle?

Answer

When the known values are the side opposite the missing angle and another side and its opposite angle.

4) In the Law of Sines, what is the relationship between the angle in the numerator and the side in the denominator?

5) What type of triangle results in an ambiguous case?

Answer

A triangle with two given sides and a non-included angle.

Algebraic

For the exercises 6-10, assume α is opposite side a, β is opposite side b, and γ is opposite side c. Solve each triangle, if possible. Round each answer to the nearest tenth.

6) α=43,γ=69,a=20

7) α=35,γ=73,c=20

Answer

β=72,a12.0,b19.9

8) α=60,β=60,γ=60

9) a=4,α=60,β=100

Answer

γ=20,b4.5,c1.6

10) b=10,β=95,γ=30

11) Find side b when A=37,B=49,c=5

Answer

b3.78

12) Find side a when A=132,C=23,b=10

13) Find side c when B=37,C=21,b=23

Answer

c13.70

For the exercises 14-23, assume α is opposite side a, β is opposite side b, and γ is opposite side c. Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.

14) α=119,a=14,b=26

15) γ=113,b=10,c=32

Answer

one triangle, α50.3,β16.7,a26.7

16) b=3.5,c=5.3,γ=80

17) a=12,c=17,α=35

Answer

two triangles, γ54.3,β90.7,b20.9 or γ125.7,β19.3,b6.9

18) a=20.5,b=35.0,β=25

19) a=7,c=9,α=43

Answer

two triangles, β75.7,γ61.3,b9.9 or β18.3,γ118.7,b3.2

20) a=7,b=3,β=24

21) b=13,c=5,γ=10

Answer

two triangles, α143.2,β26.8,a17.3 or α16.8,β153.2,a8.3

22) a=2.3,c=1.8,γ=28

23) β=119,b=8.2,a=11.3

Answer

no triangle possible

For the exercises 24-26, use the Law of Sines to solve, if possible, the missing side or angle for each triangle or triangles in the ambiguous case. Round each answer to the nearest tenth.

24) Find angle A when a=24,b=5,B=22

25) Find angle A when a=13,b=6,B=20

Answer

A47.8 or A132.2

26) Find angle B when A=12,a=2,b=9

For the exercises 27-30, find the area of the triangle with the given measurements. Round each answer to the nearest tenth.

27) a=5,c=6,β=35

Answer

8.6

28) b=11,c=8,α=28

29) a=32,b=24,γ=75

Answer

370.9

30) a=7.2,b=4.5,γ=43

Graphical

For the exercises 31-36, find the length of side x. Round to the nearest tenth.

31)

CNX_Precalc_Figure_08_01_201.jpg

Answer

12.3

32)

CNX_Precalc_Figure_08_01_202.jpg

33)

CNX_Precalc_Figure_08_01_203.jpg

Answer

12.2

34)

CNX_Precalc_Figure_08_01_204.jpg

35)

CNX_Precalc_Figure_08_01_205.jpg

Answer

16.0

36)

CNX_Precalc_Figure_08_01_206.jpg

For the exercises 37-,42 find the measure of angle x, if possible. Round to the nearest tenth.

37)

CNX_Precalc_Figure_08_01_207.jpg

Answer

29.7

38)

CNX_Precalc_Figure_08_01_208.jpg

39)

CNX_Precalc_Figure_08_01_209.jpg

Answer

x=76.9 or x=103.1

40)

CNX_Precalc_Figure_08_01_210.jpg

41) Notice that x is an obtuse angle.

CNX_Precalc_Figure_08_01_211.jpg

Answer

110.6

42)

CNX_Precalc_Figure_08_01_212.jpg

For the exercises 43-49, find the area of each triangle. Round each answer to the nearest tenth.

43)

CNX_Precalc_Figure_08_01_214.jpg

Answer

A39.4,C47.6,BC20.7

44)

CNX_Precalc_Figure_08_01_215.jpg

45)

CNX_Precalc_Figure_08_01_216.jpg

Answer

57.1

46)

CNX_Precalc_Figure_08_01_217.jpg

47)

CNX_Precalc_Figure_08_01_218.jpg

Answer

42.0

48)

CNX_Precalc_Figure_08_01_219.jpg

49)

CNX_Precalc_Figure_08_01_220.jpg

Answer

430.2

Extensions

50) Find the radius of the circle in the Figure below. Round to the nearest tenth.

CNX_Precalc_Figure_08_01_221.jpg
Figure 50.

51)  Find the diameter of the circle in the Figure below. Round to the nearest tenth.

CNX_Precalc_Figure_08_01_222.jpg

Answer

10.1

52) Find mADC in the Figure below. Round to the nearest tenth.

CNX_Precalc_Figure_08_01_223.jpg
Figure 52.

53)  Find side AD in the Figure below. Round to the nearest tenth.

CNX_Precalc_Figure_08_01_224.jpg

Answer

AD13.8

54) Solve both triangles in the Figure below. Round each answer to the nearest tenth.

CNX_Precalc_Figure_08_01_225.jpg
Figure 54.

55)  Find side AB in the parallelogram shown below. Round to the nearest tenth.

CNX_Precalc_Figure_08_01_226.jpg

Answer

AB2.8

56) Solve the triangle in the Figure below. (Hint: Draw a perpendicular from H to JK. Round each answer to the nearest tenth.

CNX_Precalc_Figure_08_01_227.jpg
Figure 56.

57)  Solve the triangle in the Figure below. (Hint: Draw a perpendicular from N to LM. Round each answer to the nearest tenth.

CNX_Precalc_Figure_08_01_228.jpg

Answer

L49.7,N56.3,LN5.8

58) In the Figure below, ABCD is not a parallelogram. m is obtuse. Solve both triangles. Round each answer to the nearest tenth.

CNX_Precalc_Figure_08_01_229.jpg

Real-World Applications

59) A pole leans away from the sun at an angle of 7 to the vertical, as shown in the Figure below. When the elevation of the sun is 55, the pole casts a shadow 42 feet long on the level ground. How long is the pole? Round the answer to the nearest tenth.

CNX_Precalc_Figure_08_01_231.jpg

Answer

51.4 ft

60) To determine how far a boat is from shore, two radar stations 500 feet apart find the angles out to the boat, as shown in the Figure below. Determine the distance of the boat from station A and the distance of the boat from shore. Round your answers to the nearest whole foot.

CNX_Precalc_Figure_08_01_232.jpg
Figure 60.

61) The Figure below shows a satellite orbiting Earth. The satellite passes directly over two tracking stations A and B, which are 69 miles apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 86.2 and 83.9 respectively. How far is the satellite from station A and how high is the satellite above the ground? Round answers to the nearest whole mile.

CNX_Precalc_Figure_08_01_233.jpg

Answer

The distance from the satellite to station A is approximately 1716 miles. The satellite is approximately 1706 miles above the ground.

62) A communications tower is located at the top of a steep hill, as shown in the Figure below. The angle of inclination of the hill is 67. A guy wire is to be attached to the top of the tower and to the ground, 165 meters downhill from the base of the tower. The angle formed by the guy wire and the hill is 16. Find the length of the cable required for the guy wire to the nearest whole meter.

CNX_Precalc_Figure_08_01_234.jpg
Figure 62.

63) The roof of a house is at a 20 angle. An 8-foot solar panel is to be mounted on the roof and should be angled 38 relative to the horizontal for optimal results. (See the Figure below). How long does the vertical support holding up the back of the panel need to be? Round to the nearest tenth.

CNX_Precalc_Figure_08_01_235.jpg

Answer

2.6 ft

64) Similar to an angle of elevation, an angle of depression is the acute angle formed by a horizontal line and an observer’s line of sight to an object below the horizontal. A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 6.6 km apart, to be 37 and 44 as shown in the Figure below. Find the distance of the plane from point A to the nearest tenth of a kilometer.

CNX_Precalc_Figure_08_01_236.jpg
Figure 64.

65) A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 4.3 km apart, to be 32 and 56, as shown in the Figure below. Find the distance of the plane from point A to the nearest tenth of a kilometer.

CNX_Precalc_Figure_08_01_237.jpg

Answer

5.6 km

66) In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they find the angle of elevation from the street to the top of the building to be 39. They then move 300 feet closer to the building and find the angle of elevation to be 50. Assuming that the street is level, estimate the height of the building to the nearest foot.

67) In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they find the angle of elevation from the street to the top of the building to be 35. They then move 250 feet closer to the building and find the angle of elevation to be 53. Assuming that the street is level, estimate the height of the building to the nearest foot.

Answer

371 ft

68) Points A and B are on opposite sides of a lake. Point C is 97 meters from A. The measure of BAC is determined to be 101, and the measure of ACB is determined to be 53. What is the distance from A to B,rounded to the nearest whole meter?

69) A man and a woman standing 312 miles apart spot a hot air balloon at the same time. If the angle of elevation from the man to the balloon is 27, and the angle of elevation from the woman to the balloon is 41, find the altitude of the balloon to the nearest foot.

Answer

5936 ft

70) Two search teams spot a stranded climber on a mountain. The first search team is 0.5 miles from the second search team, and both teams are at an altitude of 1 mile. The angle of elevation from the first search team to the stranded climber is 15. The angle of elevation from the second search team to the climber is 22. What is the altitude of the climber? Round to the nearest tenth of a mile.

71) A street light is mounted on a pole. A 6-foot-tall man is standing on the street a short distance from the pole, casting a shadow. The angle of elevation from the tip of the man’s shadow to the top of his head of 28. A 6-foot-tall woman is standing on the same street on the opposite side of the pole from the man. The angle of elevation from the tip of her shadow to the top of her head is 28. If the man and woman are 20 feet apart, how far is the street light from the tip of the shadow of each person? Round the distance to the nearest tenth of a foot.

Answer

24.1 ft

72) Three cities, A, B, and C, are located so that city A is due east of city B. If city C is located 35 west of north from city B and is 100 miles from city A and 70 miles from city B, how far is city A from city B? Round the distance to the nearest tenth of a mile.

73) Two streets meet at an 80 angle. At the corner, a park is being built in the shape of a triangle. Find the area of the park if, along one road, the park measures 180 feet, and along the other road, the park measures 215 feet.

Answer

19,056 ft2

74) Brian’s house is on a corner lot. Find the area of the front yard if the edges measure 40 and 56 feet, as shown in the Figure below.

Fig 8.1.74.png

75) The Bermuda triangle is a region of the Atlantic Ocean that connects Bermuda, Florida, and Puerto Rico. Find the area of the Bermuda triangle if the distance from Florida to Bermuda is 1030 miles, the distance from Puerto Rico to Bermuda is 980 miles, and the angle created by the two distances is 62.

Answer

445,624 square miles

76) A yield sign measures 30 inches on all three sides. What is the area of the sign?

77) Naomi bought a modern dining table whose top is in the shape of a triangle. Find the area of the table top if two of the sides measure 4 feet and 4.5 feet, and the smaller angles measure 32 and 42, as shown in the Figure below.

fig 8.1.77.png

Answer

8.65 ft2

8.2: Non-right Triangles - Law of Cosines

Verbal

1) If you are looking for a missing side of a triangle, what do you need to know when using the Law of Cosines?

Answer

two sides and the angle opposite the missing side.

2) If you are looking for a missing angle of a triangle, what do you need to know when using the Law of Cosines?

3) Explain what s represents in Heron’s formula.

Answer

s is the semi-perimeter, which is half the perimeter of the triangle.

 

4) Explain the relationship between the Pythagorean Theorem and the Law of Cosines.

5) When must you use the Law of Cosines instead of the Pythagorean Theorem?

Answer

The Law of Cosines must be used for any oblique (non-right) triangle.

Algebraic

For the exercises 6-15, assume α is opposite side a, β is opposite side b, and γ is opposite side c. If possible, solve each triangle for the unknown side. Round to the nearest tenth.

6) γ=41.2,a=2.49,b=3.13

7) α=120,b=6,c=7

Answer

11.3

8) β=58.7,a=10.6,c=15.7

9) α=115,a=18,b=23

Answer

34.7

10) α=119,a=26,b=14

11) γ=113,b=10,c=32

Answer

26.7

12) β=67,a=49,b=38

13) α=43.1,a=184.2,b=242.8

Answer

257.4

14) α=36.6,a=186.2,b=242.2

15) β=50,a=105,b=45

Answer

not possible

For the exercises 16-20, use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.

16) a=42,b=19,c=30; find angle A.

17) a=14,b=13,c=20; find angle C.

Answer

95.5

18) a=16,b=31,c=20; find angle B.

19) a=13,b=22,c=28; find angle A.

Answer

26.9

20) a=108,b=132,c=160; find angle C.

For the exercises 21-26, solve the triangle. Round to the nearest tenth.

21) A=35,b=8,c=11

Answer

B45.9,C99.1,a6.4

22) B=88,a=4.4,c=5.2

23) C=121,a=21,b=37

Answer

A20.6,B38.4,c51.1

24) a=13,b=11,c=15

25) a=3.1,b=3.5,c=5

Answer

A37.8,B43.8,C98.4

26) a=51,b=25,c=29

For the exercises 27-,31 use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.

27) Find the area of a triangle with sides of length 18 in, 21 in, and 32 in. Round to the nearest tenth.

Answer

177.56 in2

28) Find the area of a triangle with sides of length 20 cm, 26 cm, and 37 cm. Round to the nearest tenth.

29) a=12 m, b=13 m, c=14 m

Answer

0.04 m2

30) a=12.4 ft, b=13.7 ft, c=20.2 ft

31) a=1.6 yd, b=2.6 yd, c=4.1 yd

Answer

0.91 yd2

Graphical

For the exercises 32-37, find the length of side x. Round to the nearest tenth.

32)

fig 8E 8.2.32.png

33)

fig 8E 8.2.33.png

Answer

3.0

34)

fig 8E 8.2.34.png

35)

fig 8E 8.2.35.png

Answer

29.1

36)

fig 8E 8.2.36.png

37)

fig 8E 8.2.37.png

Answer

0.5

For the exercises 38-41, find the measurement of angle A

38)

fig 8E 8.2.38.png

39)

fig 8E 8.2.39.png

Answer

70.7

40)

fig 8E 8.2.40.png

41)

fig 8E 8.2.41.png

Answer

77.4

42) Find the measure of each angle in the triangle shown in the Figure below. Round to the nearest tenth.

fig 8E 8.2.42.png

For the exercises 43-46, solve for the unknown side. Round to the nearest tenth.

43)

fig 8E 8.2.43.png

Answer

25.0

44)

fig 8E 8.2.44.png

45)

fig 8E 8.2.45.png

Answer

9.3

46)

fig 8E 8.2.46.png

For the exercises 47-51, find the area of the triangle. Round to the nearest hundredth.

47)

fig 8E 8.2.47.png

Answer

43.52

48)

fig 8E 8.2.48.png

49)

fig 8E 8.2.49.png

Answer

1.41

50)

fig 8E 8.2.50.png

51)

fig 8E 8.2.51.png

Answer

0.14

Extensions

52) A parallelogram has sides of length 16 units and 10 units. The shorter diagonal is 12 units. Find the measure of the longer diagonal.

53) The sides of a parallelogram are 11 feet and 17 feet. The longer diagonal is 22 feet. Find the length of the shorter diagonal.

Answer

18.3

54) The sides of a parallelogram are 28 centimeters and 40 centimeters. The measure of the larger angle is100. Find the length of the shorter diagonal.

55) A regular octagon is inscribed in a circle with a radius of 8 inches. (See Figure below.) Find the perimeter of the octagon.

fig 8E 8.2.55.png

Answer

48.98

56) A regular pentagon is inscribed in a circle of radius 12 cm. (See Figure below.) Find the perimeter of the pentagon. Round to the nearest tenth of a centimeter.

fig 8E 8.2.56.png

For the exercises 57-58, suppose that x2=25+3660cos(52) represents the relationship of three sides of a triangle and the cosine of an angle.

57) Draw the triangle.

Answer

fig 8E 8.2.57.png

58) Find the length of the third side.

For the exercises 59-61, find the area of the triangle.

59)

fig 8E 8.2.59.png

Answer

7.62

60)

fig 8E 8.2.60.png

61)

fig 8E 8.2.61.png

Answer

85.1

Real-World Applications

62) A surveyor has taken the measurements shown in the Figure below. Find the distance across the lake. Round answers to the nearest tenth.

fig 8E 8.2.62.png
Figure below (not to scale). Find the distance between the two cities. Round answers to the nearest tenth.

fig 8E 8.2.63.png

Answer

24.0 km

64) An airplane flies 220 miles with a heading of 40, and then flies 180 miles with a heading of 170. How far is the plane from its starting point, and at what heading? Round answers to the nearest tenth.

65) A 113-foot tower is located on a hill that is inclined 34 to the horizontal, as shown in the Figure below. A guy-wire is to be attached to the top of the tower and anchored at a point 98 feet uphill from the base of the tower. Find the length of wire needed.

fig 8E 8.2.65.png

Answer

99.9 ft

66) Two ships left a port at the same time. One ship traveled at a speed of 18 miles per hour at a heading of 320. The other ship traveled at a speed of 22 miles per hour at a heading of 194. Find the distance between the two ships after 10 hours of travel.

67) The graph in the Figure below represents two boats departing at the same time from the same dock. The first boat is traveling at 18 miles per hour at a heading of 327 and the second boat is traveling at 4 miles per hour at a heading of 60. Find the distance between the two boats after 2 hours.

fig 8E 8.2.67.png

Answer

37.3 miles

68) A triangular swimming pool measures 40 feet on one side and 65 feet on another side. These sides form an angle that measures 50. How long is the third side (to the nearest tenth)?

69) A pilot flies in a straight path for 1 hour 30 min. She then makes a course correction, heading 10 to the right of her original course, and flies 2 hours in the new direction. If she maintains a constant speed of 680 miles per hour, how far is she from her starting position?

Answer

2371 miles

70) Los Angeles is 1,744 miles from Chicago, Chicago is 714 miles from New York, and New York is 2,451 miles from Los Angeles. Draw a triangle connecting these three cities, and find the angles in the triangle.

71) Philadelphia is 140 miles from Washington, D.C., Washington, D.C. is 442 miles from Boston, and Boston is 315 miles from Philadelphia. Draw a triangle connecting these three cities and find the angles in the triangle.

Answer

fig 8E 8.2.71.png

72) Two planes leave the same airport at the same time. One flies at 20 east of north at 500 miles per hour. The second flies at 30 east of south at 600 miles per hour. How far apart are the planes after 2 hours?

73) Two airplanes take off in different directions. One travels 300 mph due west and the other travels 25 north of west at 420 mph. After 90 minutes, how far apart are they, assuming they are flying at the same altitude?

Answer

599.8 miles

74) A parallelogram has sides of length 15.4 units and 9.8 units. Its area is 72.9 square units. Find the measure of the longer diagonal.

75) The four sequential sides of a quadrilateral have lengths 4.5 cm, 7.9 cm, 9.4 cm, and 12.9 cm. The angle between the two smallest sides is 117. What is the area of this quadrilateral?

Answer

65.4 cm2

76) The four sequential sides of a quadrilateral have lengths 5.7 cm, 7.2 cm, 9.4 cm, and 12.8 cm. The angle between the two smallest sides is 106. What is the area of this quadrilateral?

77) Find the area of a triangular piece of land that measures 30 feet on one side and 42 feet on another; the included angle measures 132. Round to the nearest whole square foot.

Answer

468 ft2

78) Find the area of a triangular piece of land that measures 110 feet on one side and 250 feet on another; the included angle measures 85. Round to the nearest whole square foot.

8.3: Polar Coordinates

Verbal

1) How are polar coordinates different from rectangular coordinates?

Answer

For polar coordinates, the point in the plane depends on the angle from the positive x-axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations

2) How are the polar axes different from the x- and y-axes of the Cartesian plane?

3) Explain how polar coordinates are graphed.

Answer

Determine θ for the point, then move r units from the pole to plot the point. If r is negative, move r units from the pole in the opposite direction but along the same angle. The point is a distance of r away from the origin at an angle of θ from the polar axis.

4) How are the points (3,π2) and (3,π2) related?

5) Explain why the points (3,π2) and (3,π2) are the same.

Answer

The point (3,π2) has a positive angle but a negative radius and is plotted by moving to an angle of π2 and then moving 3 units in the negative direction. This places the point 3 units down the negative y-axis. The point (3,π2) has a negative angle and a positive radius and is plotted by first moving to an angle of π2 and then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the negative y-axis.

Algebraic

6) (7,7π6)

7) (5,π)

Answer

(5,0)

8) (6,π4)

9) (3,π6)

Answer

(332,32)

10) (4,7π4)

For the exercises 11-15, convert the given Cartesian coordinates to polar coordinates with r>0 and 0θ2π. Remember to consider the quadrant in which the given point is located.

11) (4,2)

Answer

(25,0.464)

12) (4,6)

13) (3,5)

Answer

(34,5.253)

14) (10,13)

15) (8,8)

Answer

(82,π4)

For the exercises 16-27, convert the given Cartesian equation to a polar equation.

16) x=3

17) y=4

Answer

r=4cscθ

18) y=4x2

19) y=2x4

Answer

r=3sinθ2cos4θ

20) x2+y2=4y

21) x2+y2=3x

Answer

r=3cosθ

22) x2y2=x

23) x2y2=3y

Answer

r=3sinθcos(2θ)

24) x2+y2=9

25) x2=9y

Answer

r=9sinθcos2θ

26) y2=9x

27) 9xy=1

Answer

r=19cosθsinθ

For the exercises 28-39, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

28) r=3sinθ

29) r=4cosθ

Answer

x2+y2=4x or (x2)24+y24=1; circle

30) r=4sinθ+7cosθ

31) r=6cosθ+3sinθ

Answer

3y+x=6; line

32) r=2secθ

33) r=3cscθ

Answer

y=3; line

34) r=rcosθ+2

35) r2=4secθcscθ

Answer

xy=4; hyperbola

36) r=4

37) r2=4

Answer

x2+y2=4; circle

38) r=14cosθ3sinθ

39) r=3cosθ5sinθ

Answer

x5y=3; line

Graphical

For the exercises 40-44, find the polar coordinates of the point.

40)

CNX_Precalc_Figure_08_03_201n.jpg

41)

CNX_Precalc_Figure_08_03_202n.jpg

Answer

(3,3π4)

42)

CNX_Precalc_Figure_08_03_203n.jpg

43)

CNX_Precalc_Figure_08_03_204n.jpg

Answer

(5,π)

44)

CNX_Precalc_Figure_08_03_205n.jpg

For the exercises 45-54, plot the points.

45) (2,π3)

Answer

CNX_Precalc_Figure_08_03_206.jpg

46) (1,π2)

47) (3.5,7π4)

Answer

CNX_Precalc_Figure_08_03_208.jpg

48) (4,π3)

49) (5,π2)

Answer

CNX_Precalc_Figure_08_03_210.jpg

50) (4,5π4)

51) (3,5π6)

Answer

CNX_Precalc_Figure_08_03_212.jpg

52) (1.5,7π6)

53) (2,π4)

Answer

CNX_Precalc_Figure_08_03_214.jpg

54) (1,3π2)

For the exercises 55-61, convert the equation from rectangular to polar form and graph on the polar axis.

55) 5xy=6

Answer

r=65cosθsinθ

CNX_Precalc_Figure_08_03_222.jpg

56) 2x+7y=3

57) x2+(y1)2=1

Answer

r=2sinθ

CNX_Precalc_Figure_08_03_224.jpg

58) (x+2)2+(y+3)2=13

59) x=2

Answer

r=2cosθ

CNX_Precalc_Figure_08_03_226.jpg

60) x2+y2=5y

61) x2+y2=3x

Answer

r=3cosθ

CNX_Precalc_Figure_08_03_228.jpg

For the exercises 62-68, convert the equation from polar to rectangular form and graph on the rectangular plane.

62) r=6

63) r=4

Answer

x2+y2=16

CNX_Precalc_Figure_08_03_230.jpg

64) θ=2π3

65) θ=π4

Answer

y=x

CNX_Precalc_Figure_08_03_232.jpg

66) r=secθ

67) r=10sinθ

Answer

x2+(y+5)2=25

CNX_Precalc_Figure_08_03_234.jpg

68) r=3cosθ

Technology

69) Use a graphing calculator to find the rectangular coordinates of (2,π5).Round to the nearest thousandth.

Answer

(1.618,1.176)

70) Use a graphing calculator to find the rectangular coordinates of (3,3π7). Round to the nearest thousandth.

71) Use a graphing calculator to find the polar coordinates of (7,8) in degrees. Round to the nearest thousandth.

Answer

(10.630,131.186)

72) Use a graphing calculator to find the polar coordinates of (3,4) in degrees. Round to the nearest hundredth.

73) Use a graphing calculator to find the polar coordinates of (2,0) in degrees. Round to the nearest hundredth.

Answer

(2,3.14) or (2,π)

Extensions

74) Describe the graph of r=asecθ; a>0.

75) Describe the graph of r=asecθ; a<0.

Answer

A vertical line with a units left of the y-axis.

76) Describe the graph of r=acscθ; a>0.

77) Describe the graph of r=acscθ; a<0.

Answer

A horizontal line with a units below the x-axis.

78) What polar equations will give an oblique line?

For the exercises 79-84, graph the polar inequality.

79) r<4

Answer

CNX_Precalc_Figure_08_03_216.jpg

80) 0θπ4

81) θ=π4,r2

Answer

CNX_Precalc_Figure_08_03_218.jpg

82) θ=π4,r3

83) 0θπ3,r<2

Answer

CNX_Precalc_Figure_08_03_220.jpg

84) π6<θπ3,3<r<2

8.4: Polar Coordinates - Graphs

Verbal

1) Describe the three types of symmetry in polar graphs, and compare them to the symmetry of the Cartesian plane.

Answer

Symmetry with respect to the polar axis is similar to symmetry about the x-axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line θ=π2 is similar to symmetry about the y-axis.

2) Which of the three types of symmetries for polar graphs correspond to the symmetries with respect to the x-axis, y-axis, and origin?

3) What are the steps to follow when graphing polar equations?

Answer

Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, limaçon, lemniscate, etc., then plot points at θ=0, π2, π, and 3π2 and sketch the graph.

4) Describe the shapes of the graphs of cardioids, limaçons, and lemniscates.

5) What part of the equation determines the shape of the graph of a polar equation?

Answer

The shape of the polar graph is determined by whether or not it includes a sine, a cosine, and constants in the equation.

Graphical

For the exercises 6-15, test the equation for symmetry.

6) r=5cos3θ

7) r=33cosθ

Answer

symmetric with respect to the polar axis

8) r=3+2sinθ

9) r=3sin2θ

Answer

symmetric with respect to the polar axis, symmetric with respect to the line θ=π2, symmetric with respect to the pole

10) r=4

11) r=2θ

Answer

no symmetry

12) r=4cosθ2

13) r=2θ

Answer

no symmetry

14) r=31cos2θ

15) r=5sin2θ

Answer

symmetric with respect to the pole

For the exercises 16-43, graph the polar equation. Identify the name of the shape.

16) r=3cosθ

17) r=4sinθ

Answer

circle

CNX_Precalc_Figure_08_04_202.jpg

18) r=2+2cosθ

19) r=22cosθ

Answer

cardioid

CNX_Precalc_Figure_08_04_204.jpg

20) r=55sinθ

21) r=3+3sinθ

Answer

cardioid

CNX_Precalc_Figure_08_04_206.jpg

22) r=3+2sinθ

23) r=7+4sinθ

Answer

one-loop/dimpled limaçon

CNX_Precalc_Figure_08_04_208.jpg

24) r=4+3cosθ

25) r=5+4cosθ

Answer

one-loop/dimpled limaçon

CNX_Precalc_Figure_08_04_210.jpg

26) r=10+9cosθ

27) r=1+3sinθ

Answer

inner loop/ two-loop limaçon

CNX_Precalc_Figure_08_04_212.jpg

28) r=2+5sinθ

29) r=5+7sinθ

Answer

inner loop/ two-loop limaçon

CNX_Precalc_Figure_08_04_214.jpg

30) r=2+4cosθ

31) r=5+6cosθ

Answer

inner loop/ two-loop limaçon

CNX_Precalc_Figure_08_04_216.jpg

32) r2=36cos(2θ)

33) r2=10cos(2θ)

Answer

lemniscate

CNX_Precalc_Figure_08_04_218.jpg

34) r2=4sin(2θ)

35) r2=10sin(2θ)

Answer

lemniscate

CNX_Precalc_Figure_08_04_220.jpg

36) r=3sin(2θ)

37) r=3cos(2θ)

Answer

rose curve

CNX_Precalc_Figure_08_04_222.jpg

38) r=5sin(3θ)

39) r=4sin(4θ)

Answer

rose curve

CNX_Precalc_Figure_08_04_224.jpg

40) r=4sin(5θ)

41) r=θ

Answer

Archimedes’ spiral

CNX_Precalc_Figure_08_04_226.jpg

42) r=2θ

43) r=3θ

Answer

Archimedes’ spiral

CNX_Precalc_Figure_08_04_228.jpg

Technology

For the exercises 44-53, use a graphing calculator to sketch the graph of the polar equation.

44) r=1θ

45) r=1θ

Answer

CNX_Precalc_Figure_08_04_231.jpg

46) r=2sinθtanθ, a cissoid

47) r=21sin2θ, a hippopede

Answer

CNX_Precalc_Figure_08_04_233.jpg

48) r=5+cos(4θ)

49) r=2sin(2θ)

Answer

CNX_Precalc_Figure_08_04_235.jpg

50) r=θ2

51) r=θ+1

Answer

CNX_Precalc_Figure_08_04_237.jpg

52) r=θsinθ

53) r=θcosθ

Answer

CNX_Precalc_Figure_08_04_239.jpg

For the exercises 54-63, use a graphing utility to graph each pair of polar equations on a domain of [0,4π]and then explain the differences shown in the graphs.

54) r=θ,r=θ

55) r=θ,r=θ+sinθ

Answer

They are both spirals, but not quite the same.

56) r=sinθ+θ,r=sinθθ

57) r=2sin(θ2),r=θsin(θ2)

Answer

Both graphs are curves with 2 loops. The equation with a coefficient of θ has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to 4π to get a better picture.

58) r=sin(cos(3θ)),r=sin(3θ)

59) On a graphing utility, graph r=sin(165θ) on [0,4π], [0,8π], [0,12π] and [0,16π].Describe the effect of increasing the width of the domain.

Answer

When the width of the domain is increased, more petals of the flower are visible.

60) On a graphing utility, graph and sketch r=sinθ+(sin(52θ))3 on [0,4π].

61) On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs. r1=3sin(3θ)r2=2sin(3θ)r3=sin(3θ)

Answer

The graphs are three-petal, rose curves. The larger the coefficient, the greater the curve’s distance from the pole.

62) On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs. r1=3+3cos(θ)r2=2+2cos(θ)r3=1+cos(θ)

63) On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs. r1=3θr2=2θr3=θ

Answer

The graphs are spirals. The smaller the coefficient, the tighter the spiral.

Extensions

For the exercises 64-72, draw each polar equation on the same set of polar axes, and find the points of intersection.

64) r1=3+2sinθ,r2=2

65) r1=64cosθ,r2=4

Answer

(4,π3),(4,5π3)

66) r1=1+sinθ,r2=3sinθ

67) r1=1+cosθ,r2=3cosθ

Answer

(32,π3),(32,5π3)

68) r1=cos(2θ),r2=sin(2θ)

69) r1=sin2(2θ),r2=1cos(4θ)

Answer

(0,π2),(0,π),(0,3π2),(0,2π)

70) r1=3,r2=2sin(θ)

71) r21=sinθ,r22=cosθ

Answer

(482,π4), (482,5π4), and at θ=3π4, 7π4 since r is squared

72) r1=1+cosθ,r2=1sinθ

8.5: Polar Form of Complex Numbers

Verbal

1) A complex number is a+bi. Explain each part.

Answer

a is the real part, b is the imaginary part, and i=1

2) What does the absolute value of a complex number represent?

3) How is a complex number converted to polar form?

Answer

Polar form converts the real and imaginary part of the complex number in polar form using x=rcosθ and y=rsinθ.

4) How do we find the product of two complex numbers?

5) What is De Moivre’s Theorem and what is it used for?

Answer

zn=rn(cos(nθ)+isin(nθ))

It is used to simplify polar form when a number has been raised to a power.

Algebraic

For the exercises 6-11, find the absolute value of the given complex number.

6) 5+3i

7) 7+i

Answer

52

8) 33i

9) 26i

Answer

38

10) 2i

11) 2.23.1i

Answer

14.45

For the exercises 12-16, write the complex number in polar form.

12) 2+2i

13) 84i

Answer

45cis(333.4)

14) 1212i

15) 3+i

Answer

2cis(π6)

16) 3i

For the exercises 17-22, convert the complex number from polar to rectangular form.

17) z=7cis(π6)

Answer

732+i72

18) z=2cis(π3)

19) z=4cis(7π6)

Answer

232i

20) z=7cis(25)

21) z=3cis(240)

Answer

1.5i332

22) z=2cis(100)

For the exercises 23-28, find z1z2 in polar form.

23) z1=23cis(116);z2=2cis(82)

Answer

43cis(198)

24) z1=2cis(205);z2=22cis(118)

25) z1=3cis(120);z2=14cis(60)

Answer

34cis(180)

26) z1=3cis(π4);z2=5cis(π6)

27) z1=5cis(5π8);z2=15cis(π12)

Answer

53cis(17π24)

28) z1=4cis(π2);z2=2cis(π4)

For the exercises 29-,34 find z1z2 in polar form.

29) z1=21cis(135);z2=3cis(65)

Answer

7cis(70)

30) z1=2cis(90);z2=2cis(60)

31) z1=15cis(120);z2=3cis(40)

Answer

5cis(80)

32) z1=6cis(π3);z2=2cis(π4)

33) z1=52cis(π);z2=2cis(2π3)

Answer

5cis(π3)

34) z1=2cis(3π5);z2=3cis(π4)

For the exercises 35-40, find the powers of each complex number in polar form.

35) Find z3 when z=5cis(45).

Answer

125cis(135)

36) Find z4 when z=2cis(70).

37) Find z2 when z=3cis(120).

Answer

9cis(240)

38) Find z2 when z=4cis(π4).

39) Find z4 when z=cis(3π16).

Answer

cis(3π4)

40) Find z3 when z=3cis(5π3).

For the exercises 41-45, evaluate each root.

41) Evaluate the cube root of z when z=27cis(240).

 

Answer

3cis(80),3cis(200),3cis(320)

42) Evaluate the square root of z when z=16cis(100).

43) Evaluate the cube root of z when z=32cis(2π3).

Answer

234cis(2π9),234cis(8π9),234cis(14π9)

44) Evaluate the square root of z when z=32cis(π).

45) Evaluate the cube root of z when z=8cis(7π4).

Answer

22cis(7π8),22cis(15π8)

Graphical

For the exercises 46-55, plot the complex number in the complex plane.

46) 2+4i

47) 33i

Answer

CNX_Precalc_Figure_08_05_202.jpg

48) 54i

49) 15i

Answer

CNX_Precalc_Figure_08_05_204.jpg

50) 3+2i

51) 2i

Answer

CNX_Precalc_Figure_08_05_206.jpg

52) 4

53) 62i

Answer

CNX_Precalc_Figure_08_05_208.jpg

54) 2+i

55) 14i

Answer

CNX_Precalc_Figure_08_05_210.jpg

Technology

For the exercises 56-, find all answers rounded to the nearest hundredth.

56) Use the rectangular to polar feature on the graphing calculator to change 5+5i to polar form.

57) Use the rectangular to polar feature on the graphing calculator to change 32i to polar form.

Answer

3.61e0.59i

58) Use the rectangular to polar feature on the graphing calculator to change 38i to polar form.

59) Use the polar to rectangular feature on the graphing calculator to change 4cis(120) to rectangular form.

Answer

2+3.46i

60) Use the polar to rectangular feature on the graphing calculator to change 2cis(45) to rectangular form.

61) Use the polar to rectangular feature on the graphing calculator to change 5cis(210) to rectangular form.

Answer

4.332.50i

8.6: Parametric Equations

Verbal

1) What is a system of parametric equations?

Answer

A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example, x=f(t) and y=f(t).

2) Some examples of a third parameter are time, length, speed, and scale. Explain when time is used as a parameter.

3) Explain how to eliminate a parameter given a set of parametric equations.

Answer

Choose one equation to solve for t, substitute into the other equation and simplify.

4) What is a benefit of writing a system of parametric equations as a Cartesian equation?

5) What is a benefit of using parametric equations?

Answer

Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately the equations are functions.

6) Why are there many sets of parametric equations to represent on Cartesian function?

Algebraic

For the exercises 7-25, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation.

7) {x(t)=5ty(t)=82t

Answer

y=2+2x

8) {x(t)=63ty(t)=10t

9) {x(t)=2t+1y(t)=3t

Answer

y=3x12

10) {x(t)=3t1y(t)=2t2

11) {x(t)=2ety(t)=15t

Answer

x=2e1y5 or y=15ln(x2)

12) {x(t)=e2ty(t)=2et

13) {x(t)=4log(t)y(t)=3+2t

Answer

x=4log(y32)

14) {x(t)=log(2t)y(t)=t1

15) {x(t)=t3ty(t)=2t

Answer

x=(y2)3y2

16) {x(t)=tt4y(t)=t+2

17) {x(t)=e2ty(t)=e6t

Answer

y=x3

18) {x(t)=t5y(t)=t10

19) {x(t)=4costy(t)=5sint

Answer

(x4)2+(y5)2=1

20) {x(t)=3sinty(t)=6cost

21) {x(t)=2cos2ty(t)=sint

Answer

y2=112x

22) {x(t)=cost+4y(t)=2sin2t

23) {x(t)=t1y(t)=t2

Answer

y=x2+2x+1

24) {x(t)=ty(t)=t3+1

25) {x(t)=2t1y(t)=t32

Answer

y=(x+12)32

For the exercises 26-29, rewrite the parametric equation as a Cartesian equation by building an xy table.

26) {x(t)=2t1y(t)=t+4

27) {x(t)=4ty(t)=3t+2

Answer

y=3x+14

28) {x(t)=2t1y(t)=5t

29) {x(t)=4t1y(t)=4t+2

Answer

y=x+3

For the exercises 30-33, parameterize (write parametric equations for) each Cartesian equation by setting x(t)=t or by setting y(t)=t.

30) y(x)=3x2+3

31) y(x)=2sinx+1

Answer

{x(t)=ty(t)=2sint+1

32) x(y)=3log(y)+y

33) x(y)=y+2y

Answer

{x(t)=t+2ty(t)=t

For the exercises 34-41, parameterize (write parametric equations for) each Cartesian equation by using x(t)=acost and y(t)=bsint. Identify the curve.

34) x24+x29=1

35) x216+x236=1

Answer

{x(t)=4costy(t)=6sint;Ellipse

36) x2+y2=16

37) x2+y2=10

Answer

{x(t)=10costy(t)=10sint;Circle

38) Parameterize the line from (3,0) to (2,5) so that the line is at (3,0) at t=0, and at (2,5) at t=1.

39) Parameterize the line from (1,0) to (3,2) so that the line is at (1,0) at t=0, and at (3,2) at t=1.

Answer

{x(t)=1+4ty(t)=2t

40) Parameterize the line from (1,5) to (2,3) so that the line is at (1,5) at t=0, and at (2,3) at t=1.

41) Parameterize the line from (4,1) to (6,2) so that the line is at (4,1) at t=0, and at (6,2) at t=1.

Answer

{x(t)=4+2ty(t)=13t

Technology

For the exercises 42-43, use the table feature in the graphing calculator to determine whether the graphs intersect.

42) {x1(t)=3ty1(t)=2t1and{x2(t)=t+3y2(t)=4t4

43) {x1(t)=t2y1(t)=2t1and{x2(t)=t+6y2(t)=t+1

Answer

yes, at t=2

For the exercises 44-46, use a graphing calculator to complete the table of values for each set of parametric equations.

44) {x1(t)=3t23t+7y1(t)=2t+3

t x y
-1    
0    
1    

45) {x1(t)=t24y1(t)=2t21

t x y
1    
2    
3    
Answer
t x y
1 -3 1
2 0 7
3 5 17

46) {x1(t)=t4y1(t)=t3+4

t x y
-1    
0    
1    
2    

Extensions

47) Find two different sets of parametric equations for y=(x+1)2.

Answer

answers may vary: {x(t)=t1y(t)=t2and{x(t)=t+1y(t)=(t+2)2

48) Find two different sets of parametric equations for y=3x2.

49) Find two different sets of parametric equations for y=x24x+4.

Answer

answers may vary: {x(t)=ty(t)=t24t+4and{x(t)=t+2y(t)=t2

8.7: Parametric Equations - Graphs

Verbal

1) What are two methods used to graph parametric equations?

Answer

plotting points with the orientation arrow and a graphing calculator

2) What is one difference in point-plotting parametric equations compared to Cartesian equations?

3) Why are some graphs drawn with arrows?

Answer

The arrows show the orientation, the direction of motion according to increasing values of t.

4) Name a few common types of graphs of parametric equations.

5) Why are parametric graphs important in understanding projectile motion?

Answer

The parametric equations show the different vertical and horizontal motions over time.

Graphical

For the exercises 6-11, graph each set of parametric equations by making a table of values. Include the orientation on the graph.

6) {x(t)=ty(t)=t21

t x y
-3    
-2    
-1    
0    
1    
2    
3

7) {x(t)=t1y(t)=t2

t -3 -2 -1 0 1 2
x            
y            
Answer

CNX_Precalc_Figure_08_07_202.jpg

8) {x(t)=2+ty(t)=32t

t -2 -1 0 1 2 3
x            
y            

9) {x(t)=22ty(t)=3+t

t -3 -2 -1 0 1
x          
y          
Answer

CNX_Precalc_Figure_08_07_204.jpg

10) {x(t)=t3y(t)=t+2

t -2 -1 0 1 2
x          
y          

11) {x(t)=t2y(t)=t+3

t -2 -1 0 1 2
x          
y          
Answer

CNX_Precalc_Figure_08_07_206.jpg

For the exercises 12-22, sketch the curve and include the orientation.

12) {x(t)=ty(t)=t

13) {x(t)=ty(t)=t

Answer

CNX_Precalc_Figure_08_07_208.jpg

14) {x(t)=5|t|y(t)=t+2

15) {x(t)=t+2y(t)=5|t|

Answer

CNX_Precalc_Figure_08_07_210.jpg

16) {x(t)=4sinty(t)=2cost

17) {x(t)=2sinty(t)=4cost

Answer

CNX_Precalc_Figure_08_07_212.jpg

18) {x(t)=3cos2ty(t)=3sint

19) {x(t)=3cos2ty(t)=3sin2t

Answer

CNX_Precalc_Figure_08_07_214.jpg

20) {x(t)=secty(t)=tant

21) {x(t)=secty(t)=tan2t

Answer

CNX_Precalc_Figure_08_07_216.jpg

22) {x(t)=1e2ty(t)=et

For the exercises 23-27, graph the equation and include the orientation. Then, write the Cartesian equation.

23) {x(t)=t1y(t)=t2

Answer

CNX_Precalc_Figure_08_07_218.jpg

24) {x(t)=t3y(t)=t+3

25) {x(t)=2costy(t)=sint

Answer

CNX_Precalc_Figure_08_07_220.jpg

26) {x(t)=7costy(t)=7sint

27) {x(t)=e2ty(t)=et

Answer

CNX_Precalc_Figure_08_07_222.jpg

For the exercises 28-33, graph the equation and include the orientation.

28) x=t2,y=3t,0t5

29) x=2t,y=t2,5t5

Answer

CNX_Precalc_Figure_08_07_224.jpg

30) x=t,y=25t2,0<t5

31) x(t)=t,y(t)=t,t0

Answer

CNX_Precalc_Figure_08_07_226.jpg

32) x=2cost,y=6sint,0tπ

33) x=sect,y=tant,π2<t<π2

Answer

CNX_Precalc_Figure_08_07_228.jpg

For the exercises 34-41, use the parametric equations for integers a and b: x(t)=acos((a+b)t)y(t)=acos((ab)t)

34) Graph on the domain [π,0], where a=2 and b=1, and include the orientation.

35) Graph on the domain [π,0], where a=3 and b=2, and include the orientation.

Answer

CNX_Precalc_Figure_08_07_230.jpg

36) Graph on the domain [π,0], where a=4 and b=3, and include the orientation.

37) Graph on the domain [π,0], where a=5 and b=4, and include the orientation.

Answer

CNX_Precalc_Figure_08_07_232.jpg

38) If a is 1 more than b, describe the effect the values of a and b have on the graph of the parametric equations.

 

39) Describe the graph if a=100 and b=99.

Answer

There will be 100 back-and-forth motions.

40) What happens if b is 1 more than a? Describe the graph.

41) If the parametric equations x(t)=t2 and y(t)=63t have the graph of a horizontal parabola opening to the right, what would change the direction of the curve?

Answer

Take the opposite of the x(t) equation.

For the exercises 42-46, describe the graph of the set of parametric equations.

42) x(t)=t2 and y(t) is linear

43) y(t)=t2 and x(t) is linear

Answer

The parabola opens up.

44) y(t)=t2 and x(t) is linear

45) Write the parametric equations of a circle with center (0,0), radius 5, and a counterclockwise orientation.

Answer

{x(t)=5costy(t)=5sint

46) Write the parametric equations of an ellipse with center (0,0),major axis of length 10, minor axis of length 6, and a counterclockwise orientation.

 

For the exercises 47-52, use a graphing utility to graph on the window [3,3] by [3,3] on the domain [0,2π) for the following values ofa and b, and include the orientation. {x(t)=sin(at)y(t)=sin(bt)

47) a=1,b=2

Answer

CNX_Precalc_Figure_08_07_233.jpg

48) a=2,b=1

49) a=3,b=3

Answer

CNX_Precalc_Figure_08_07_235.jpg

50) a=5,b=5

51) a=2,b=5

Answer

CNX_Precalc_Figure_08_07_237.jpg

52) a=5,b=2

Technology

For the exercises 53-56, look at the graphs that were created by parametric equations of the form {x(t)=acos(bt)y(t)=csin(dt)Use the parametric mode on the graphing calculator to find the values of a,b,c, and d to achieve each graph.

53)

CNX_Precalc_Figure_08_07_239.jpg

Answer

a=4,b=3,c=6,d=1

54)

CNX_Precalc_Figure_08_07_240.jpg

55)

CNX_Precalc_Figure_08_07_241.jpg

Answer

a=4,b=2,c=3,d=3

56)

CNX_Precalc_Figure_08_07_242.jpg

For the exercises 57-62, use a graphing utility to graph the given parametric equations.

{x(t)=cost1y(t)=sint+t

{x(t)=cost+ty(t)=sint1

{x(t)=tsinty(t)=cost1

57) Graph all three sets of parametric equations on the domain [0,2π].

Answer

Ex 8.7.57a.png

Ex 8.7.57b.png

Ex 8.7.57c.png

58) Graph all three sets of parametric equations on the domain [0,4π].

59) Graph all three sets of parametric equations on the domain [4π,6π].

Answer

Ex 8.7.59a.png

Ex 8.7.59b.png

Ex 8.7.59c.png

60) The graph of each set of parametric equations appears to “creep” along one of the axes. What controls which axis the graph creeps along?

61) Explain the effect on the graph of the parametric equation when we switched sint and cost.

Answer

The y-intercept changes.

62) Explain the effect on the graph of the parametric equation when we changed the domain.

Extensions

63) An object is thrown in the air with vertical velocity of 20 ft/s and horizontal velocity of 15 ft/s. The object’s height can be described by the equation y(t)=16t2+20t, while the object moves horizontally with constant velocity 15 ft/s. Write parametric equations for the object’s position, and then eliminate time to write height as a function of horizontal position.

Answer

y(x)=16(x15)2+20(x15)

64) A skateboarder riding on a level surface at a constant speed of 9 ft/s throws a ball in the air, the height of which can be described by the equation y(t)=16t2+10t+5. Write parametric equations for the ball’s position, and then eliminate time to write height as a function of horizontal position.

For the exercises 65-69, use this scenario: A dart is thrown upward with an initial velocity of 65 ft/s at an angle of elevation of 52. Consider the position of the dart at any time t. Neglect air resistance.

65) Find parametric equations that model the problem situation.

Answer

{x(t)=64tcos(52)y(t)=16t2+64tsin(52)

66) Find all possible values of x that represent the situation.

67) When will the dart hit the ground?

Answer

approximately 3.2 seconds

68) Find the maximum height of the dart.

69) At what time will the dart reach maximum height?

Answer

1.6 seconds

For the exercises 70-73, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain.

70) An epicycloid{x(t)=14costcos(14t)y(t)=14sint+sin(14t)on the domain [0,2π]

71) An hypocycloid{x(t)=6sint+2sin(6t)y(t)=6cost2cos(6t)on the domain [0,2π]

Answer

Ex 8.7.71.png

72) An hypotrochoid{x(t)=2sint+5cos(6t)y(t)=5cost2sin(6t)on the domain [0,2π]

73) A rose{x(t)=5sin(2t)sinty(t)=5sin(2t)coston the domain [0,2π]

Answer

Ex 8.7.73.png

8.8: Vectors

Verbal

1) What are the characteristics of the letters that are commonly used to represent vectors?

Answer

lowercase, bold letter, usually u,v,w

2) How is a vector more specific than a line segment?

3) What are i and j, and what do they represent?

Answer

They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.

4) What is component form?

5) When a unit vector is expressed as a,b which letter is the coefficient of the i and which the j?

Answer

The first number always represents the coefficient of the i and the second represents the j.

Algebraic

6) Given a vector with initial point (5,2) and terminal point (1,3), find an equivalent vector whose initial point is (0,0). Write the vector in component form a,b.

7) Given a vector with initial point (4,2) and terminal point (3,3), find an equivalent vector whose initial point is (0,0). Write the vector in component form a,b.

Answer

7,5

8) Given a vector with initial point (7,1) and terminal point (1,7), find an equivalent vector whose initial point is (0,0). Write the vector in component form a,b.

For the exercises 9-15, determine whether the two vectors u and v are equal, where u has an initial point P1 and a terminal point P2 and v has an initial point P3 and a terminal point P4.

9) P1=(5,1),P2=(3,2),P3=(1,3),P4=(9,4)

Answer

not equal

10) P1=(2,3),P2=(5,1),P3=(6,1),P4=(9,3)

11) P1=(1,1),P2=(4,5),P3=(10,6),P4=(13,12)

Answer

equal

12) P1=(3,7),P2=(2,1),P3=(1,2),P4=(1,4)

13) P1=(8,3),P2=(6,5),P3=(11,8),P4=(9,10)

Answer

equal

14) Given initial point P1=(3,1) and terminal point P2=(5,2), write the vector v in terms of i and j.

15) Given initial point P1=(6,0) and terminal point P2=(1,3), write the vector v in terms of i and j.

Answer

7i3j

For the exercises 16-17, use the vectors u=i+5j,v=2i3j,w=4ij

16) Find u+(vw)

17) Find 4v+2u

Answer

6i2j

For the exercises 18-21, use the given vectors to compute u+v,uv,2u3v

18) u=2,3,v=1,5

19) u=3,4,v=2,1

Answer

u+v=5,5,uv=1,3,2u3v=0,5

20) Let v=4i+3j. Find a vector that is half the length and points in the same direction as v.

21) Let v=5i+2j. Find a vector that is twice the length and points in the opposite direction as v.

Answer

10i4j

For the exercises 22-27, find a unit vector in the same direction as the given vector.

22) a=3i+4j

23) b=2i+5j

Answer

22929i+52929j

24) c=10ij

25) d=13i+52j

Answer

2229229i+15229229j

26) u=100i+200j

27) u=14i+2j

Answer

7210i+210j

 

For the exercises 28-35, find the magnitude and direction of the vector, 0θ<2π.

28) 0,4

29) 6,5

Answer

|v|=7.810,θ=39.806

30) 2,5

31) 4,6

Answer

|v|=7.211,θ=236.310

32) Given u=3i4j and v=2i+3j, calculate uv.

33) Given u=ij and v=i+5j, calculate uv.

Answer

6

34) Given u=2,4 and v=3,1,calculate uv.

35) Given u=1,6 and v=6,1 ,calculate uv.

Answer

12

Graphical

For the exercises 36-,38 given v, draw v, 3v, and 12v.

36) 2,1

37) 1,4

Answer

CNX_Precalc_Figure_08_08_253.jpg

38) 3,2

For the exercises 39-41, use the vectors shown to sketch u+v, uv, and 2u.

39)

CNX_Precalc_Figure_08_08_204.jpg

Answer

CNX_Precalc_Figure_08_08_205.jpg

40)

CNX_Precalc_Figure_08_08_206.jpg

41)

CNX_Precalc_Figure_08_08_208.jpg

Answer

CNX_Precalc_Figure_08_08_209.jpg

For the exercises 42-43, use the vectors shown to sketch 2u+v.

42)

CNX_Precalc_Figure_08_08_210.jpg

43)

CNX_Precalc_Figure_08_08_212.jpg

Answer

CNX_Precalc_Figure_08_08_213.jpg

For the exercises 44-45, use the vectors shown to sketch u3v.

44)

CNX_Precalc_Figure_08_08_214.jpg

45)

CNX_Precalc_Figure_08_08_216.jpg

Answer

CNX_Precalc_Figure_08_08_217.jpg

For the exercises 46-47, write the vector shown in component form.

46)

CNX_Precalc_Figure_08_08_218.jpg

47)

CNX_Precalc_Figure_08_08_219.jpg

Answer

4,1

48) Given initial point P1=(2,1 and terminal point P2=(1,2) ,write the vector v in terms of i and j, then draw the vector on the graph.

49) Given initial point P1=(4,1 and terminal point P2=(3,2),   write the vector v in terms of i and j. Draw the points and the vector on the graph.

Answer

v=7i+3j

CNX_Precalc_Figure_08_08_221.jpg

50) Given initial point P1=(3,3 and terminal point P2=(3,3),   write the vector v in terms of i and j. Draw the points and the vector on the graph.

Extensions

For the exercises 51-54, use the given magnitude and direction in standard position, write the vector in component form.

51) |v|=6,θ=45

Answer

32i+32j

52) |v|=8,θ=220

53) |v|=2,θ=300

Answer

i3j

54) |v|=5,θ=135

55) A 60-pound box is resting on a ramp that is inclined 12. Rounding to the nearest tenth,

  1. Find the magnitude of the normal (perpendicular) component of the force.
  2. Find the magnitude of the component of the force that is parallel to the ramp.
Answer
  1. 58.7
  2. 12.5

56) A 25-pound box is resting on a ramp that is inclined 8. Rounding to the nearest tenth,

  1. Find the magnitude of the normal (perpendicular) component of the force.
  2. Find the magnitude of the component of the force that is parallel to the ramp.

57) Find the magnitude of the horizontal and vertical components of a vector with magnitude 8 pounds pointed in a direction of 27 above the horizontal. Round to the nearest hundredth.

Answer

x=7.13 pounds, y=3.63 pounds

58) Find the magnitude of the horizontal and vertical components of the vector with magnitude 4 pounds pointed in a direction of 127 above the horizontal. Round to the nearest hundredth.

59) Find the magnitude of the horizontal and vertical components of a vector with magnitude 5 pounds pointed in a direction of 55 above the horizontal. Round to the nearest hundredth.

Answer

x=2.87 pounds, y=4.10 pounds

60) Find the magnitude of the horizontal and vertical components of the vector with magnitude 1 pound pointed in a direction of 8 above the horizontal. Round to the nearest hundredth.

Real-World Applications

61) A woman leaves home and walks 3 miles west, then 2 miles southwest. How far from home is she, and in what direction must she walk to head directly home?

Answer

4.635 miles, 17.764 N of E

62) A boat leaves the marina and sails 6 miles north, then 2 miles northeast. How far from the marina is the boat, and in what direction must it sail to head directly back to the marina?

63) A man starts walking from home and walks 4 miles east, 2 miles southeast, 5 miles south, 4 miles southwest, and 2 miles east. How far has he walked? If he walked straight home, how far would he have to walk?

Answer

17 miles, 10.318 miles

64) A woman starts walking from home and walks 4 miles east, 7 miles southeast, 6 miles south, 5 miles southwest, and 3 miles east. How far has she walked? If she walked straight home, how far would she have to walk?

65) A man starts walking from home and walks 3 miles at 20 north of west, then 5 miles at 10 west of south, then 4 miles at 15 north of east. If he walked straight home, how far would he have to the walk, and in what direction?

Answer

Distance: 2.868, Direction: 86.474 North of West, or 3.526 West of North

66) A woman starts walking from home and walks 6 miles at 40^{\circ} north of east, then 2 miles at 15^{\circ} east of south, then 5 miles at 30^{\circ} south of west. If she walked straight home, how far would she have to walk, and in what direction?

67) An airplane is heading north at an airspeed of 600 km/hr, but there is a wind blowing from the southwest at 80 km/hr. How many degrees off course will the plane end up flying, and what is the plane’s speed relative to the ground?

Answer

4.924^{\circ}, 659 km/hr

68) An airplane is heading north at an airspeed of 500 km/hr, but there is a wind blowing from the northwest at 50 km/hr. How many degrees off course will the plane end up flying, and what is the plane’s speed relative to the ground?

69) An airplane needs to head due north, but there is a wind blowing from the southwest at 60 km/hr. The plane flies with an airspeed of 550 km/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?

Answer

4.424^{\circ}

70) An airplane needs to head due north, but there is a wind blowing from the northwest at 80 km/hr. The plane flies with an airspeed of 500 km/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?

71) As part of a video game, the point (5,7) is rotated counterclockwise about the origin through an angle of \(35^{\circ}\). Find the new coordinates of this point.

Answer

(0.081,8.602)

72) As part of a video game, the point (7,3) is rotated counterclockwise about the origin through an angle of 40^{\circ}. Find the new coordinates of this point.

73) Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 10 mph relative to the car, and the car is traveling 25 mph down the road. If one child doesn't catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?

Answer

21.801^{\circ}, relative to the car’s forward direction

74) Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 8 mph relative to the car, and the car is traveling 45 mph down the road. If one child doesn't catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?

75) A 50-pound object rests on a ramp that is inclined 19^{\circ}. Find the magnitude of the components of the force parallel to and perpendicular to (normal) the ramp to the nearest tenth of a pound.

Answer

parallel: 16.28, perpendicular: 47.28 pounds

76) Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it upward, and 5 pounds acting on it 45^{\circ} from the horizontal. What single force is the resultant force acting on the body?

77) Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it ─135^{\circ} from the horizontal, and 5 pounds acting on it directed 150^{\circ} from the horizontal. What single force is the resultant force acting on the body?

Answer

19.35 pounds, 231.54^{\circ} from the horizontal

78) The condition of equilibrium is when the sum of the forces acting on a body is the zero vector. Suppose a body has a force of 2 pounds acting on it to the right, 5 pounds acting on it upward, and 3 pounds acting on it 45^{\circ} from the horizontal. What single force is needed to produce a state of equilibrium on the body?

79) Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it \(30^{\circ}\) from the horizontal. What single force is needed to produce a state of equilibrium on the body? Draw the vector.

Answer

5.1583 pounds, 75.8^{\circ} from the horizontal

Contributors and Attributions

 


This page titled 8.E: Further Applications of Trigonometry (Exercises) is shared under a not declared license and was authored, remixed, and/or curated by OpenStax.

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