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Trigonometry

Appendix A Answers to Selected Exercises and Homework Problems

1 Triangles and Circles
1.1 Angles and Triangles
Homework 1.1

1.

Answer.
isosceles triangle with vertex angle 30°

3.

Answer.
right triangle with legs 4 and 7

5.

Answer.
An isosceles triangle with one obtuse angle

7.

Answer.
θ=108.8

9.

Answer.
α=29

11.

Answer.
β=77

13.

Answer.
α=12

15.

Answer.
θ=65

17.

Answer.
θ=12

19.

Answer.
ψ=73

21.

Answer.
ϕ=88

23.

Answer.
  1. ϕ=120
  2. ϕ=160
  3. ϕ=α+β
  4. An exterior angle is equal to the sum of the opposite interior angles.

25.

Answer.
θ=72,ϕ=54

27.

Answer.
θ=100,ϕ=30

29.

Answer.
  1. 180
  2. 90
  3. a right triangle

31.

Answer.
  1. They are base angles of an isosceles triangle.
  2. They are base angles of an isosceles triangle.
  3. OAB corresponds to θ of Problem 29, and OBC corresponds to ϕ of Problem 29.

33.

Answer.
α=30,β=60

35.

Answer.
x=47,y=133

37.

Answer.
x=60,y=15

39.

Answer.
x=100,y=16

41.

Answer.
x=90,y=55

43.

Answer.
x=50,y=80

45.

Answer.
  1. 1=4,3=5
  2. 180
  3. In the equation 4+2+5=180, substitute 1 for 4, and substitute 3 for 5 to conclude that the sum of the angles in the triangle is 180.

47.

Answer.
1=130 because vertical angles are equal. 2=50 because it makes a straight angle with a 130 angle. 3=65 because it is a base angle of an isosceles triangle whose vertex angle is 50. 4=65 for the same reason. 5=25 because it is complementary to 4.

1.2 Similar Triangles
Homework 1.2

1.

Answer.
PQTSRT, x=7, y=3,α=18

3.

Answer.
PREURN,z=12, θ=10, ϕ=70

5.

Answer.
river
ABTABC, so AT=AC

7.

Answer.
Similar. Corresponding sides are proportional.

9.

Answer.
Similar. Corresponding angles are equal.

11.

Answer.
A=37,B=37

13.

Answer.
h=12

15.

Answer.
p=35

17.

Answer.
g=84

19.

Answer.
h=30

21.

Answer.
154 feet

23.

Answer.
1 mile

25.

Answer.
17.1 square feet

27.

Answer.
y=1217x

29.

Answer.
h=7.5

31.

Answer.
c=15

33.

Answer.
s=6

35.

Answer.
y=35x

37.

Answer.
y=5+34x

39.

Answer.
  1. B=70, CAD=70, DAB=20
  2. DBA and DAC. The hypotenuse is BC in ABC, BA in DBA, and AC in DAC. The short leg is AB in ABC, DB in DBA, and DA in DAC. The longer leg is AC in ABC, DA in DBA, and DC in DAC.

1.3 Circles
Homework 1.3

1.

Answer.
13 miles

3.

Answer.
10, 10.00

5.

Answer.
458.94

7.

Answer.
5

9.

Answer.
25

11.

Answer.
5

13.

Answer.
triangle on grid
  24.7

15.

Answer.
  1. (x+3)2+(y4)2
  2. (x+3)2+(y4)2=5

17.

Answer.
The distance between the points (x,y) and (4,1) is 3 units.

19.

Answer.
  1. 62 cm
  2. 8.49 cm

21.

Answer.
  1. 25π sq in
  2. 78.54 sq in

23.

Answer.
  1. approximation
  2. approximation
  3. approximation
  4. exact

25.

Answer.
  1. x 5 4 3 2 1 0 1 2 3 4 5
    y 0 ±3 ±4 ±21 ±26 ±5 ±26 ±21 ±4 ±3 0
  2. circle on grid

27.

Answer.
  1. circle on grid
  2. x2+y2=36

29.

Answer.
  1. circle on grid
  2. x2+y2<9

31.

Answer.
  1. No real value of y can satisfy x2+y2=16 unless 4x4
  2. The graph has no points where x>4 and no points where x<4

33.

Answer.
10

35.

Answer.
  1. circle
  2. 12π

37.

Answer.
  1. circle
  2. 4π

39.

Answer.
(25,4),(25,4)
circle

41.

Answer.
P(12,32),Q(12,32),R(34,74),S(34,74)

43.

Answer.
  1. 45
  2. 5π ft
  3. 50π sq ft

45.

Answer.
  1. 25
  2. 40π sq ft
  3. 8π ft

47.

Answer.
  1. 110
  2. π10 sq km
  3. π5 km

49.

Answer.
  1. 56
  2. 15π2 sq m
  3. 5π m

51.

Answer.
2070 miles

53.

Answer.
  1. 54,000 miles
  2. 2240 mph

55.

Answer.
  1. (x3)2+(y+2)2=36
  2. (xh)2+(yk)2=r2

1.4 Chapter 1 Summary and Review
Chapter 1 Review Problems

1.

Answer.
triangle

3.

Answer.
triangle

5.

Answer.
α=β=γ=60

7.

Answer.
ϕ=ω=79

9.

Answer.
θ=65, ϕ=25

11.

Answer.
δ=30, γ=60

13.

Answer.
σ=39, ω=79

15.

Answer.
α=5137, β=6427

17.

Answer.
ABCEDC, α=40, β=130, x=32

19.

Answer.
Yes, three pairs of equal angles

21.

Answer.
Yes, three pairs of equal angles

23.

Answer.
13

25.

Answer.
18

27.

Answer.
y=5x2

29.

Answer.
y=7x3

31.

Answer.
y=x3

33.

Answer.
y=12x5

35.

Answer.
α=70

37.

Answer.
14 ft

39.

Answer.
334 in

41.

Answer.
All side have length 61, opposite sides have slopes 56 and 65

43.

Answer.
AC=BC=18

45.

Answer.
  1. (x2)2+(y5)2=3
  2. (x2)2+(y5)2=9

47.

Answer.
458.944 cm

49.

Answer.
(13,223),(13,223)

51.

Answer.
  1. 4π ft
  2. 20π ft2

53.

Answer.
  1. 45,60
  2. 49π8 in2,6π in2 Delbert
  3. 79π4 in, 2π in, Francine

2 The Trigonometric Ratios
2.1 Side and Angle Relationships
Homework 2.1

1.

Answer.
The sum of the angles is not 180.

3.

Answer.
The exterior angle is not equal to the sum of the opposite interior angles.

5.

Answer.
The sum of the acute angles is not 90.

7.

Answer.
The largest side is not opposite the largest angle.

9.

Answer.
The Pythagorean theorem is not satisfied.

11.

Answer.
52+122=132, but the angle opposite the side of length 13 is 85.

13.

Answer.
4<x<16

15.

Answer.
0<x<16

17.

Answer.
21 in

19.

Answer.
62 in

21.

Answer.
The rectangle is 610 inches by 1810 inches.

23.

Answer.
29

25.

Answer.
3

27.

Answer.
No

29.

Answer.
Yes

31.

Answer.
No

33.

Answer.
The distance from (0,0) to (3,3) is 32, and the distance from (3,3) to (6,0) is also 32, so the triangle is isosceles. The distance from (0,0) to (6,0) is 6, and (32)2+(32)2=62 so the triangle is a right triangle.

35.

Answer.
25 ft

37.

Answer.
α=30,β=60,h=3

39.

Answer.
83 in

41.

Answer.
  1. No
  2. Yes

43.

Answer.
  1. (1,0) and (1,0); 2
  2. (p+1)2+q2 and (p1)2+q2
  3. ((p+1)2+q2)2+((p1)2+q2)2=p2+2p+1+q2+p22p+1+q2=2p2+2+2q2=2+2(p2+q2)=2+2(1)=4

2.2 Right Triangle Trigonometry
Homework 2.2

1.

Answer.
  1. 0.91
  2. 0.91
  3. 0.9063

3.

Answer.
  1. 0.77
  2. 0.77
  3. 0.7660

5.

Answer.
  1. 41314.42
  2. sinθ=0.5547, cosθ=0.8321, tanθ=0.6667

7.

Answer.
  1. 41515.49
  2. sin(θ)=0.9682, cos(θ)=0.2500, tan(θ)=3.8730

9.

Answer.
  1. 26716.37
  2. sin(θ)=0.2116, cos(θ)=0.9774, tan(θ)=0.2165

11.

Answer.
triangles
(Answers may vary)

13.

Answer.
triangles
(Answers may vary)

15.

Answer.
triangles
(Answers may vary)

17.

Answer.
14.41

19.

Answer.
37.86

21.

Answer.
86.08

23.

Answer.
triangle

25.

Answer.
triangle

27.

Answer.
  1. triangle
  2. tan(54.8)=h20, 170.1 yd

29.

Answer.
  1. triangle
  2. tan(36.2)=260d, 355.2 ft

31.

Answer.
  1. triangle
  2. sin(48)=a1500, 1114.7 m

33.

Answer.
  1. triangle
  2. cos(38)=1800x, 2284.2 m

35.

Answer.
x=82tan(θ)

37.

Answer.
x=11 sin(θ)

39.

Answer.
x=9cos(θ)

41.

Answer.
36 sin(25)15.21

43.

Answer.
46 sin(20)15.73

45.

Answer.
12 sin(40)7.71

47.

Answer.
     sin(θ) cos(θ) tan(θ)
θ 35 45 34
ϕ 45 35 43

49.

Answer.
     sin(θ) cos(θ) tan(θ)
θ 15 25 12
ϕ 25 15 2

51.

Answer.
  1. θ and ϕ are complements.
  2. sin(θ)=cos(ϕ) and cos(θ)=sin(ϕ). The side opposite θ is the side adjacent to ϕ, and vice versa.

53.

Answer.
  1. As θ increases, tan(θ) increases also. The side opposite θ increases in length while the side adjacent to θ remains fixed.
  2. As θ increases, cos(θ) decreases. The side adjacent to θ remains fixed while the hypotenuse increases in length.

55.

Answer.
As θ decreases toward 0, the side opposite θ approaches a length of 0, so sin (θ) approaches 0. But as θ increases toward 90, the length of the side opposite θ approaches the length of the hypotenuse, so sin(θ) approaches 1.

57.

Answer.
The triangle is not a right tringle.

59.

Answer.
2120 is the ratio of hypotenuse to the adjacent side, which is the reciprocal of cos(θ).

61.

Answer.
  1. 0.2358
  2. sine
  3. 48
  4. 77

63.

Answer.
  1. 512
  2. 3
  3. 23
  4. 27

65.

Answer.
Although the triangles may differ in size, the ratio of the side adjacent to the angle to the hypotenuse of the triangle remains the same because the triangles would all be similar, and hence corresponding sides are proportional.

67.

Answer.
  1. 23
  2. 23
  3. triangle

2.3 Solving Right Triangles
Homework 2.3

1.

Answer.
A=61, a=25.26, c=28.88

3.

Answer.
A=68, a=0.93, b=0.37

5.

Answer.
  1. triangle
  2. B=48,  a=17.4,  b=19.3

7.

Answer.
  1. triangle
  2. A=57,  b=194.4,  c=357.7

9.

Answer.
  1. triangle
  2. B=78,  b=18.8,  c=19.2

11.

Answer.
  1. triangle
    • Solve sin(53.7)=8.2c for c.
    • Solve tan(53.7)=8.2a for a.
    • Subtract 53.7 from 90 to find A.

13.

Answer.
  1. triangle
    • Solve cos(25)=40c for c.
    • Solve tan(25)=a40 for a.
    • Subtract 25 from 90 to find B.

15.

Answer.
  1. triangle
    • Solve sin(64.5)=a24 for a.
    • Solve cos(64.5)=b24 for b.
    • Subtract 64.5 from 90 to find B.

17.

Answer.
74.2

19.

Answer.
56.4

21.

Answer.
66.0

23.

Answer.
11.5
triangle

25.

Answer.
56.3
triangle

27.

Answer.
73.5
triangle

29.

Answer.
cos(15)=0.9659  and  cos1(0.9659)=15

31.

Answer.
tan(65)=2.1445  and  tan1(2.1445)=65

33.

Answer.
sin1(0.6)36.87 is the angle whose sine is 0.6. (sin6)19.5668 is the reciprocal of sin(6).

35.

Answer.
  1. triangle
  2. sin(θ)=18063(2458), 14.6

37.

Answer.
  1. triangle
  2. tanθ=3210, 72.6

39.

Answer.
  1. triangle
  2. c=101031.6,  A34.7,  B55.3

41.

Answer.
  1. triangle
  2. a=256.2816.0,  A56.5,  B33.5

43.

Answer.
  1. triangle
  2. tan1(2630)40.9,  9116763612.6 cm

45.

Answer.
  1. triangle
  2. 6415 km

47.

Answer.
  1. triangle
  2. 462.9 ft

49.

Answer.
(a) and (b)

51.

Answer.
(a) and (d)

53.

Answer.
320.8660

55.

Answer.
13=330.5774

57.

Answer.
1.0000

59.

Answer.
θ    0       30       45       60       90   
sin(θ) 0 12 22 32 1
cos(θ) 1 32 22 12 0
tan(θ) 0 13 1 3 undefined

61.

Answer.
  1. smaller
  2. larger
  3. larger

63.

Answer.
a=33, b=3, B=30

65.

Answer.
a=b=42, B=45

67.

Answer.
e=4, f=43, F=120

69.

Answer.
d=23, e=22,f=2+6, F=75

71.

Answer.
a=20, b=20, c=202

73.

Answer.
  1. 323 cm
  2. 1283 sq cm

75.

Answer.
  1. 10 sq cm
  2. 102 sq cm
  3. 103 sq cm

77.

Answer.
  1. 64 sq in
  2. 42 by 42, area 32 sq in
  3. 2:1

2.4 Chapter 2 Summary and Review
Chapter 2 Review Problems

1.

Answer.
If C>93, then A+B+C>180

3.

Answer.
If A<B<58, then A+B+C<180

5.

Answer.
If C>50, then A+B+C>180

7.

Answer.
triangle

9.

Answer.
a=97

11.

Answer.
c=52

13.

Answer.
Yes

15.

Answer.
θ=35.26

17.

Answer.
No. a=6, c=10 or a=9, c=15

19.

Answer.
  1. w=86.05
  2. sin(θ)=0.7786, cos(θ)=0.6275, tan(θ)=1.2407

21.

Answer.
  1. y=16.52
  2. sin(θ)=0.6957, cos(θ)=0.7184, tan(θ)=0.9684

23.

Answer.
a=7.89

25.

Answer.
x=3.57

27.

Answer.
b=156.95

29.

Answer.
A=30, a=2333, c=4633

31.

Answer.
F=105, d=102, e=20, f=10+103

33.

Answer.
3 cm

35.

Answer.
43.30 cm

37.

Answer.
15.92 m

39.

Answer.
114.02 ft, 37.87

41.

Answer.
  1. 60.26
  2. 60.26
  3. m=74=tan(θ)

43.

Answer.
  1. c2
  2. ba, (ba)2
  3. 12ab
  4. 4(12ab)+(ab)2=2ab+b22ab+a2=a2+b2

3 Laws of Sines and Cosines
3.1 Obtuse Angles
Homework 3.1

1.

Answer.
  1. 150
  2. 135
  3. 60
  4. 155
  5. 15
  6. 70

3.

Answer.
  1. (5,2)
  2. 29
  3. cos(θ)=529,  sin(θ)=229,  tan(θ)=25

5.

Answer.
  1. (4,7)
  2. 65
  3. cos(θ)=465,  sin(θ)=765,  tan(θ)=74

7.

Answer.
  1. sin(θ)=997,  cos(θ)=497
  2. angle
  3. sin(180θ)=997,  cos(180θ)=497
  4. θ=66,  180θ=114

9.

Answer.
  1. sin(θ)=889,  cos(θ)=589
  2. angle
  3. sin(180θ)=889,  cos(180θ)=589
  4. θ=122,  180θ=58

11.

Answer.
  1. angle
  2. cos(θ)=513,  sin(θ)=1213,  tan(θ)=125
  3. 112.6

13.

Answer.
  1. angle
  2. cos(θ)=35,  tan(θ)=34
  3. 143.1

15.

Answer.
  1. angle
  2. sin(θ)=11211,  tan(θ)=1123
  3. 74.2

17.

Answer.
  1. angle
  2. sin(θ)=137,  cos(θ)=637
  3. 170.5

19.

Answer.
  1. angle
  2. sin(θ)=417,  cos(θ)=117
  3. 76.0

21.

Answer.
θ    0       30       45       60       90       120       135       150       180   
cos(θ) 1 32 12 12 0 12 12  32 1
sin(θ) 0 12 12 32 1 32 12 12 0
tan(θ) 0 13 1 3 undefined 3 1 13 0

23.

Answer.
  1. sin(θ)=sin(180θ)
  2. cos(θ)=cos(180θ)
  3. tan(θ)=tan(180θ)

25.

Answer.
  1. θ41.4,  ϕ138.6
  2. angles
  3. sin(θ)=sin(ϕ)=74

27.

Answer.
  1. θ81.2,  ϕ98.8
  2. angles
  3. sin(θ)=sin(ϕ)=1562794000.9883

29.

Answer.
44.4 and 135.6

31.

Answer.
57.1 and 122.9

33.

Answer.
41.8 and 138.2

35.

Answer.
sin(123)=q,  cos(33)=q,  cos(147)=q

37.

Answer.
cos(106)=m,  sin(16)=m,  sin(164)=m

39.

Answer.
  1. angles
  2. (4,3), (8,6)
  3. y=tan1(34)36.87
  4. angles
    (4,3), (8,6); 143.13

41.

Answer.
  1. b=8 in, h=33 in
  2. 123 sq in

43.

Answer.
  1. b=6322 mi, h=322 mi
  2. 18294 sq mi

45.

Answer.
angle
  1. (1,3)
  2. (3,3)

47.

Answer.
triangle
  1. (3,3)
  2. (5,5)

49.

Answer.
20.71 sq m

51.

Answer.
55.51 sq cm

55.

Answer.
38.04 sq units

57.

Answer.
13,851.3 sq ft

59.

Answer.
  1. (74.97,59.00)
  2. BC=141.97,  PC=59.00
  3. 153.74

61.

Answer.
514

63.

Answer.
Bob found an acute angle. The obtuse angle is the supplement of 17.46, or 162.54.

65.

Answer.
  1. triangle
  2. cos(θ)=x3,  sin(θ)=9x23,  tan(θ)=9x2x

67.

Answer.
  1. triangle
  2. cos(θ)=4y22,  sin(θ)=y2,  tan(θ)=y4y2

69.

Answer.
  1. triangle
  2. cos(θ)=11+m2,  sin(θ)=m1+m2,  tan(θ)=m

3.2 The Law of Sines
Homework 3.2

1.

Answer.
x=7.85

3.

Answer.
q=33.81

5.

Answer.
d=28.37

7.

Answer.
θ=30.80

9.

Answer.
θ=126.59

11.

Answer.
β=37.14

13.

Answer.
triangle
a=4.09,  c=9.48,  C=115

15.

Answer.
triangle
b=2.98,  A=36.54,  B=99.46

17.

Answer.
triangle
a=43.55,  b=54.62,  C=99

19.

Answer.
a.
triangle
b. 808.1 ft

21.

Answer.
a.
triangle
b. 68.2 km

23.

Answer.
a.
triangle
b.1.23 mi + 0.99 mi; 0.22 mi

25.

Answer.
a.
triangle
b. 322.6 m

27.

Answer.
  1. 1
  2. 66
  3. 2617.2 ft
  4. 1022.6 ft

29.

Answer.
540,000 AU 8.1×1013 km

31.

Answer.
750,000 AU 1.1×1014 km

33.

Answer.
  1. 32
  2. No, a is too short.
  3. 2
  4. 1

35.

Answer.
  1. 1,
    triangle
  2. 0,
    triangle
  3. 2,
    triangle
  4. 1,
    triangle

37.

Answer.
  1. C=25.37, B=114.63, b=16.97
  2. C=58.99, B=81.01, b=9.22 or C=121.01, B=18.99, b=3.04
  3. no solution
  4. 5.14

39.

Answer.
A=40.44, B=114.56 or A=139.56, B=15.44

41.

Answer.
C=37.14, A=93.86

43.

Answer.
1299 yd or 277.2 yd

45.

Answer.
  1. 11.79
  2. 24.16
  3. 24.16

47.

Answer.
triangle
  1. 12absin(C)
  2. 12acsin(B)
  3. 12bcsin(A)

49.

Answer.
  1. triangle
  2. b=hsin(A)
  3. h=asin(B)
  4. b=asin(B)sin(A)
  5. ii

3.3 The Law of Cosines
Homework 3.3

1.

Answer.
  1. 7470cos(θ)
  2. 12.78
  3. 135.22

3.

Answer.
  1. a2+c2b22ac
  2. 0.4

5.

Answer.
  1. b28cos(α)b65=0
  2. 11.17, 5.82

7.

Answer.
7.70

9.

Answer.
13.44

11.

Answer.
5.12

13.

Answer.
133.43

15.

Answer.
40.64

17.

Answer.
A=91.02, B=37.49, C=51.49

19.

Answer.
A=34.34, B=103.49, C=42.17

21.

Answer.
6.30 or 2.70

23.

Answer.
29.76 or 5.91

25.

Answer.
16.00

27.

Answer.
Law of Cosines: 612=292+46222946cos(ϕ)

29.

Answer.
Law of Sines: asin(46)=16sin(25)

31.

Answer.
First the Law of Cosines: x2=472+29224729cos(81), then either the Law of Sines: sin(θ)47=sin(81)x or the Law of Cosines: 472=x2+2922x29cos(θ)

33.

Answer.
Law of Cosines: 92=42+z224zcos(28), or use the Law of Sines first to find the (acute) angle opposite the side of length 4, then find the angle opposite the side of length z by subtracting the sum of the known angles from 180, then using the Law of Sines again.

35.

Answer.
  1. triangle
  2. b=16.87,  A=85.53,  C=47.47

37.

Answer.
  1. triangle
  2. A=58.41, B=48.19, C=73.40

39.

Answer.
  1. triangle
  2. a=116.52,  A=85.07,  C=56.93 or a=37.93,  A=18.93,  C=123.07

41.

Answer.
  1. triangle
  2. a=7.76,  b=8.97,  C=39

43.

Answer.
  1. triangle
  2. 1383.3 m

45.

Answer.
  1. triangle
  2. 2123 mi, 168.43 east of north

47.

Answer.
  1. triangle
  2. 7.74 west of south, 917.9 mi

49.

Answer.
  1. triangle
  2. 92.99 ft

51.

Answer.
147.73 cm2

53.

Answer.
10.53

55.

Answer.
4.08

57.

Answer.
  1. First figure: bx is the base of the small right triangle. Second: x is the horizontal distance between P and the x-axis, so b+(x) or bx is the base of the large right triangle. Third: x=0, and b is the base of a right triangle.
  2. First: x and y are the legs of a right triangle, a is the hypotenuse. Second: x and y are the legs of a right triangle with hypotenuse a. Third: x=0 and y=a
  3. x=acos(C)

59.

Answer.
b2+c2=(a2+c22accos(B))+(a2+b22bccos(C))=2a2+b2+c22a(ccos(B)+bcos(C))
so 2a2=2a(ccos(B)+bcos(C)), and dividing both sides by 2a yields a=(ccos(B)+bcos(C)

61.

Answer.
For the first equation, start with the Law of Cosines in the form
a2=b2+c22bccos(A)
Add 2ab+2bccos(A)a2 to both sides of the equation, factor the right side, then divide both sides by 2bc.
For the second equation, start with the Law of Cosines in the form
b2+c22bccos(A)=a2
Add 2bcb2c2 to both sides of the equation, factor the right side, then divide both sides by 2bc.

3.4 Chapter 3 Summary and Review
Chapter 3 Review Problems

1.

Answer.
12, ±32

3.

Answer.
  1. triangle
  2. 49.33
  3. triangle
    114

5.

Answer.
  1. triangle
  2. cos(θ)=213,  sin(θ)=313,  tan(θ)=32
  3. θ=123.7

7.

Answer.
  1. triangle
  2. cos(θ)=45,  sin(θ)=35,  tan(θ)=34
  3. θ=143.1

9.

Answer.
  1. triangle
  2. cos(θ)=116,  sin(θ)=56,  tan(θ)=511
  3. θ=123.6

11.

Answer.
  1. triangle
  2. cos(θ)=725,  sin(θ)=2425,  tan(θ)=247
  3. θ=106.3

13.

Answer.
9.9,  170.1

15.

Answer.
22.0, 158.0

17.

Answer.
  1. 72
  2. 282

19.

Answer.
5127.39 sq ft

21.

Answer.
20.41

23.

Answer.
a=27.86

25.

Answer.
b=6.03

27.

Answer.
w=62.10

29.

Answer.
s=15.61 or 57.45

31.

Answer.
  1. triangle
  2. 8.82

33.

Answer.
  1. triangle
  2. 32.57

35.

Answer.
  1. triangle
  2. 16.29

37.

Answer.
  1. triangle
  2. 58.65

39.

Answer.
  1. triangle
  2. 17.40
or
  1. triangle
  2. 80.93

41.

Answer.
  1. triangle
  2. 16.08 mi, 80.4 mph

43.

Answer.
  1. triangle Francine-Delbert-Tree
  2. 72.47

45.

Answer.
  1. triangle Giselle-Hakim-blimp
    353.32
  2. 217.52 m

47.

Answer.
  1. 79.64 m
  2. 35.2
  3. 46.12 m

49.

Answer.
6.1

51.

Answer.
triangle
4.2

53.

Answer.
triangle
22.25 ft

55.

Answer.
79,332.6 AU

57.

Answer.
  1. OW bisects the central angle at O, and the inscribed angle θ is half the central angle at O.
  2. sinθ=s2r
  3. r=s2sin(θ)
  4. d=ssin(θ)

4 Trigonometric Functions
4.1 Angles and Rotation
Homework 4.1

1.

Answer.
  1. 216
  2. 108
  3. 480
  4. 960

3.

Answer.
  1. 18
  2. 56
  3. 32
  4. 76

5.

Answer.
  1. 23
  2. 53

7.

Answer.
60

9.

Answer.
60

11.

Answer.
14

13.

Answer.
400 and 320 (Answers vary.)

15.

Answer.
575 and 145 (Answers vary.)

17.

Answer.
665 and 55 (Answers vary.)

19.

Answer.
295

21.

Answer.
70

23.

Answer.
315

25.

Answer.
  1. 36.9, 143.1
  2. angles on grid

27.

Answer.
  1. 72.5, 287.5
  2. graph

29.

Answer.
80
angles

31.

Answer.
36
angles

33.

Answer.
63
angles

35.

Answer.
165, 95, 345
angles

37.

Answer.
140, 220, 320
angles

39.

Answer.
112, 248, 292
angles

41.

Answer.
0.9205

43.

Answer.
0.7193

45.

Answer.
4.705

47.

Answer.
0.7193

49.

Answer.
  1. 120
  2. 135
  3. 150
  4. 210
  5. 225
  6. 240
  7. 300
  8. 315
  9. 330

51.

Answer.
  1. angles
  2. sin(120)=32, cos(120)=12, tan(120)=3,
    sin(240)=32, cos(240)=12, tan(240)=3,
    sin(300)=32, cos(300)=12, tan(300)=3

53.

Answer.
  1. angles
  2. sin(135)=12, cos(135)=12, (tan135)=1,
    sin(225)=12, cos(225)=12, tan(225)=1,
    sin(315)=12, cos(315)=12, tan(315)=1

55.

Answer.
  1. III and IV
  2. II and III
  3. I and III

57.

Answer.
  1. 0 and 180
  2. 90 and 270

59.

Answer.
105

61.

Answer.
264

63.

Answer.
313

65.

Answer.
Sides of similar triangles are proportional.

4.2 Graphs of Trigonometric Functions
Homework 4.2

1.

Answer.
sine graph

3.

Answer.
cosine graph

5.

Answer.
  1. (225,12)
  2. (135,12)
  3. (90,1)
  4. (45,12)
  5. (180,0)
  6. (315,12)

7.

Answer.
  1. (240,12)
  2. (210,32)
  3. (60,12)
  4. (30,32)
  5. (120,12)
  6. (270,0)

9.

Answer.
  1. θ 0 90 180 270 360
    f(θ) 0 1 0 1 0
    cosine graph
  2. θ 0 90 180 270 360
    f(θ) 1 0 1 0 1
    cosine graph

11.

Answer.
72

13.

Answer.
221

15.

Answer.
2

17.

Answer.
212

19.

Answer.
sine graph

21.

Answer.
graph

23.

Answer.
  1. θ 81 82 83 84 85 86 87 88 89
    tan(θ) 6.314 7.115 8.144 9.514 11.43 14.301 19.081 28.636 57.29
  2. tan(θ) approaches 
  3. θ 99 98 97 96 95 94 93 92 91
    tan(θ) 6.314 7.115 8.144 9.514 11.43 14.301 19.081 28.636 57.29
  4. tan(θ) approaches 
  5. The calculator gives an error message because tan(90) is undefined.

25.

Answer.
y=6sin(θ)

27.

Answer.
y=cos(θ)5

29.

Answer.
y=sin(4θ)

31.

Answer.
graph of y = 3 cos theta

33.

Answer.
graph of 3 + sin theta

35.

Answer.
graph of cos 3 theta

37.

Answer.
A(0,3),  B(135,32),  C(300,32)

39.

Answer.
P(112.5,1),  Q(180,0),  R(337.5,1)

41.

Answer.
X(45,3+12),  Y(90,3),  Z(300,2)

43.

Answer.
amp=4, period =360, midline: y=3

45.

Answer.
amp=5, period =180, midline: y=0

47.

Answer.
amp=3, period =120, midline: y=4

49.

Answer.
  1. amp =1, period =90, midline: y=0
  2. y=sin(4θ)

51.

Answer.
  1. amp =1, period =360, midline: y=3
  2. y=3+cos(θ)

53.

Answer.
  1. amp =4, period =360, midline: y=2
  2. y=2+4sin(θ)

55.

Answer.
  1. amp =2, period =120, midline: y=2
  2. y=2+2cos(3θ)

57.

Answer.
y=4+6sin(3θ) (Answers vary)

59.

Answer.
y=3+2cos(θ) (Answers vary)

61.

Answer.
y=12cos(2θ) (Answers vary)

63.

Answer.
y=2+5cos(θ)

65.

Answer.
y=4sin(θ)

4.3 Using Trigonometric Functions
Homework 4.3

1.

Answer.
36.9, 143.1

3.

Answer.
72.5, 287.5

5.

Answer.
191.5, 348.5

7.

Answer.
154.2, 205.8

9.

Answer.
83, 263

11.

Answer.
23, 337

13.

Answer.
265, 275

15.

Answer.
156, 204

17.

Answer.
246, 294

19.

Answer.
149, 329

21.

Answer.
  1. (0.94,0.34)
  2. (1.88,0.68)

23.

Answer.
  1. (0.94,0.34)
  2. (1.88,0.68)

25.

Answer.
(42,42)

27.

Answer.
(10,103)

29.

Answer.
(1532,152)

31.

Answer.
(1.25,5.87)

33.

Answer.
(5.70,11.86)

35.

Answer.
(9.46,3.26)

37.

Answer.
  1. angle
  2. 15.3 mi east, 21 mi north

39.

Answer.
  1. angle
  2. 91.9 km west, 77.1 km south

41.

Answer.
  1. angle
  2. 30.9 km west, 8.3 km north

43.

Answer.
51.34

45.

Answer.
159.44

47.

Answer.
y+5=(tan28)(x3)  or  y+5=0.532(x3)

49.

Answer.
y12=(tan112)(x+8)  or  y12=2.475(x+8)

51.

Answer.
not periodic

53.

Answer.
Periodic with period 4

55.

Answer.
  1. piecewise linear graph
  2. 10 minutes

57.

Answer.
  1. piercewise linear graph
  2. 1 week

59.

Answer.
  1. sinusoidal graph
  2. period 1 sec, midline y=12, amp 10 inches

61.

Answer.
  1. sinusoidal graph
  2. period 1 year, midline y=3500, amp 2500

63.

Answer.
  1. sinusoidal graph
  2. period 1 year, midline y=51, amp 21

65.

Answer.
a. IV b. III c. II d. I

67.

Answer.
two graphs

69.

Answer.
  1. Emotional high: Oct 5 and Nov 3, low: Oct 19; Physical high: Sep 30 and Oct 23, low: Oct 12 and Nov 4; Intellectual high: Oct 10, low: Oct 26
  2. Emotional: 28 days, physical: 23 days, intellectual: 32 days
  3. 5152 days

71.

Answer.
  1. periodic, period 8
  2. 4, midline: y=3
  3. k=8
  4. a=3, b=7

73.

Answer.
  1. systolic 120 mm Hg, diastolic 80 mm Hg, pulse pressure 40 mm Hg.
  2. 9313
  3. 72 beats per minute

75.

Answer.
  1. 69 hours.
  2. 2.2 to 3.5
  3. The larger dip corresponds to when the brighter star is eclipsed, the smaller dip corresponds to when the dimmer star is eclipsed.

4.4 Chapter 4 Summary and Review
Chapter 4 Review Problems

1.

Answer.
12

3.

Answer.
  1. 150, 210
  2. 240, 120
  3. 160, 560
  4. 20, 340

5.

Answer.
  1. I, 60; 120, 240, 300
  2. IV, 25; 155, 205, 335
  3. II, 80; 80, 260, 280
  4. III, 70; 70, 110, 290

7.

Answer.
  1. θ 30 60 90 120 150 180 210 240 270 300 330 360
    f(θ) 30 60 90 60 30 0 30 60 90 60 30 0
  2. graph of referance angle vs angle

9.

Answer.
210, 330

11.

Answer.
120, 240

13.

Answer.
45, 225

15.

Answer.
23, 337

17.

Answer.
72, 252

19.

Answer.
163, 277

21.

Answer.
221.81, 318.19

23.

Answer.
123.69, 303.69

25.

Answer.
128.68, 231.32

27.

Answer.
(9.74,2.25)

29.

Answer.
(0.28,8.00)

31.

Answer.
(2.84,0.98)

33.

Answer.
south: 1.74 mi, west: 9.85 mi

35.

Answer.
y=4+7sin(180θ)
sinusoidal graph

37.

Answer.
y=17+7sinθ
sinusoidal graph

39.

Answer.
32

41.

Answer.
0

43.

Answer.
y=1.5cos(θ3), M(90,334),N(180,34)

45.

Answer.
y=3+3sin2θ, A(45,6),B(120,3332)

47.

Answer.
  1. periodic graph
  2. 24 hours

49.

Answer.
  1. periodic graph
  2. 20 sec

51.

Answer.
  1. y = 4 + 2 cos theta
  2. amp: 2, period: 360, midline: y=4

53.

Answer.
  1. y = 1.5 + 3.5 sin (2 theta)
  2. amp: 3.5, period: 180, midline: y=1.5

55.

Answer.
30

57.

Answer.
92.05

59.

Answer.
y=x+2

61.

Answer.
y=3x+334

63.

Answer.
cosine and tangent graphs
The θ-intercepts of cosθ occur at the vertical asymptotes of tanθ.

5 Equations and Identities
5.1 Algebra with Trigonometric Ratios
Homework 5.1

1.

Answer.
2

3.

Answer.
12

5.

Answer.
6

7.

Answer.
12

9.

Answer.
4

11.

Answer.
2

13.

Answer.
1

15.

Answer.
0

17.

Answer.
  1. 0.7660
  2. 0.8164
  3. 0.7660

19.

Answer.
  1. 0.6691
  2. 1.8271
  3. 0.6691

21.

Answer.
  1. 1
  2. 1
  3. 1

23.

Answer.
  1. 2x2x
  2. 2cos2(θ)cos(θ)

25.

Answer.
  1. 4SC
  2. 4sin(θ)cos(θ)

27.

Answer.
  1. 5C2S3
  2. 5cos2(θ)sin3(θ)

29.

Answer.
2cos(t)+2cos(t)sin(t); 0.6360

31.

Answer.
tan(θ)tan(ϕ); 56.91

33.

Answer.
2sin(x)cos(x)2sin(2x); 0

35.

Answer.
No

37.

Answer.
No

39.

Answer.
Yes

41.

Answer.
No

43.

Answer.
No

45.

Answer.
  1. 2x2x
  2. 2sin2(A)sin(A)

47.

Answer.
  1. ab3a2
  2. tan(A)tan(B)3tan2(A)

49.

Answer.
  1. 2C2+C1
  2. 2cos2(ϕ)+cos(ϕ)1

51.

Answer.
  1. a2b2
  2. cos2(θ)cos2(ϕ)

53.

Answer.
  1. 12T+T2
  2. 12tan(θ)+tan2(θ)

55.

Answer.
  1. T44
  2. tan4(θ)4

57.

Answer.
  1. 3(3m+5n)
  2. 3(3cos(α)+5cos(β))

59.

Answer.
  1. 5r(r2q)
  2. 5tan(C)(tan(C)2tan(B))

61.

Answer.
  1. (3C+1)(3C1)
  2. (3cos(β)+1)(3cos(β)1)

63.

Answer.
  1. 2T2(3T4)
  2. 2tan2(A)(3tan(A)4)

65.

Answer.
  1. (t5)(t+4)
  2. (tan(θ)5)(tan(θ)+4)

67.

Answer.
  1. (3c1)(c+1)
  2. (3cos(B)1)(cos(B)+1)

69.

Answer.
  1. (6S+1)(S1)
  2. (6sin(α)+1)(sin(α)1)

5.2 Solving Equations
Homework 5.2

1.

Answer.
70

3.

Answer.
40

5.

Answer.
I: 18; II: 162; III: 198; IV: 342

7.

Answer.
I: 52; II: 128; III: 232; IV: 308

9.

Answer.
  1. 0, 4, 2, 0, 4
  2. 1 or 2

11.

Answer.
  1. 1, 3+12, 2, 3+12
  2. 45

13.

Answer.
  1. 0, 222, 132, 1
  2. 270

15.

Answer.
x=5, 3

17.

Answer.
x=3, 1, 2

19.

Answer.
θ=30  or  θ=210

21.

Answer.
θ=60  or  θ=300

23.

Answer.
θ=210  or  θ=330

25.

Answer.
θ=225  or  θ=315

27.

Answer.
θ=0  or  θ=180

29.

Answer.
θ=60, θ=120, θ=240,  or  θ=300

31.

Answer.
θ=45, θ=135, θ=225,  or  θ=315

33.

Answer.
θ=104.04  or  θ=284.04

35.

Answer.
θ=53.13  or  θ=306.87

37.

Answer.
θ=188.21  or  θ=351.79

39.

Answer.
A=135  or  A=315

41.

Answer.
ϕ=210  or  ϕ=330

43.

Answer.
B=90 or B=270

45.

Answer.
θ=210  or  θ=330

47.

Answer.
t=202  or  t=338

49.

Answer.
B=22 or B=202

51.

Answer.
ϕ=146  or  ϕ=214

53.

Answer.
θ=54.74, θ=125.26, θ=234.74,  or  θ=305.26

55.

Answer.
θ=0,  θ=180,  θ=191.54,  or  θ=348.46

57.

Answer.
θ=60,  θ=180, or  θ=300

59.

Answer.
θ=26.57,  θ=161.57,  θ=206.57, or  θ=341.57

61.

Answer.
θ=78.69,  θ=108.43,  θ=258.69, or  θ=288.43

63.

Answer.
θ=0

65.

Answer.
17.22

67.

Answer.
35.66

5.3 Trigonometric Identities
Homework 5.3

1.

Answer.
not an identity

3.

Answer.
not an identity

5.

Answer.
identity

7.

Answer.
not an identity

9.

Answer.
not an identity

11.

Answer.
not an identity

13.

Answer.
identity

15.

Answer.
identity

17.

Answer.
(1+sin(w))(1sin(w))=1sin2(w)=cos2(w)

19.

Answer.
(cos(θ)sin(θ))2=cos2(θ)2cos(θ)sin(θ)+sin2(θ)=(cos2(θ)+sin2(θ))2sin(θ)cos(θ)=12sin(θ)cos(θ)

21.

Answer.
tan(θ)cos(θ)=sin(θ)cos(θ)cos(θ)=sin(θ)

23.

Answer.
cos4(x)sin4(x)=(cos2(x)sin2(x))(cos2(x)+sin2(x))=(cos2(x)sin2(x))(1)=cos2(x)sin2(x)

25.

Answer.
sin(u)1+cos(u)1cos(u)1cos(u)=sin(u)(1cos(u))1cos2(u)=sin(u)(1cos(u))sin2(u)=1cos(u)sin(u)

27.

Answer.
1

29.

Answer.
1

31.

Answer.
sin2(A)

33.

Answer.
tan2(z)

35.

Answer.
3

37.

Answer.
1

39.

Answer.
6

41.

Answer.
cos(2θ)

43.

Answer.
cos(θ)

45.

Answer.
sin(2t)

47.

Answer.
1+2sin(θ)+sin2(θ)

49.

Answer.
3cos2(ϕ)2

51.

Answer.
θ=90, θ=180, θ=270

53.

Answer.
θ=90, θ=210, θ=330

55.

Answer.
θ=210, θ=330

57.

Answer.
θ=18.43, θ=198.43

59.

Answer.
sin(A)=513, tan(A)=512

61.

Answer.
cos(ϕ)=437, tan(ϕ)=143

63.

Answer.
sin(θ)=15,  cos(θ)=25

65.

Answer.
sin(θ)=35,  cos(θ)=45

67.

Answer.
sin(θ)=32,  cos(θ)=12,  tan(θ)=3

69.

Answer.
sin(β)=25,  cos(β)=15,  tan(β)=2

71.

Answer.
sin(C)=15, cos(C)=25, tan(C)=12or  sin(C)=15, cos(C)=25, tan(C)=12

73.

Answer.
tan(α)1+tan(α)=sin(α)cos(α)1+sin(α)cos(α)cos(α)cos(α)=sin(α)sin(α)+cos(α)

75.

Answer.
1+tan2(β)1tan2(β)=1cos2(β)1sin2(β)cos2(β)cos2(β)cos2(β)=1cos2(β)sin2(β)

77.

Answer.
angle
  1. By the distance formula, x2+y2=r, or x2+y2=r2.
  2. x2r2+y2r2=1
  3. (xr)2+(yr)2=1
  4. (cos(θ))2+(sin(θ))2=1

5.4 Chapter 5 Summary and Review
Chapter 5 Review Problems

1.

Answer.
342

3.

Answer.
16

5.

Answer.
  1. 0.8660
  2. 0.9848; No

7.

Answer.
  1. 1.4821
  2. 1.4821; Yes

9.

Answer.
5sin(x)2sin(x)cos(y)cos(y)

11.

Answer.
2tan(θ)10tan2(θ)

13.

Answer.
Not equivalent

15.

Answer.
Equivalent

17.

Answer.
2cos2α+cosα6

19.

Answer.
tan2(ϕ)2tan(ϕ)cos(ϕ)+cos2(ϕ)

21.

Answer.
6(2sin(3x)sin(2x))

23.

Answer.
(1+3tan(θ))(13tan(θ))

25.

Answer.
cos(α)+sin(α)

27.

Answer.
32

29.

Answer.
3tan(C)+2tan(C)2

31.

Answer.
51.32, 308.68

33.

Answer.
90,  270,  120,  240

35.

Answer.
90,  210,  330

37.

Answer.
30,  150,  210,  330

39.

Answer.
0,  120,  240

41.

Answer.
57.99, 237.99

43.

Answer.
90, 270

45.

Answer.
33.17

47.

Answer.
Identity

49.

Answer.
Not an identity

51.

Answer.
Not an identity

53.

Answer.
Identity

55.

Answer.
1cos2(α)tan(α)=sin2(α)cos(α)sin(α)=sin(α)cos(α)

57.

Answer.
sin(θ)cos(θ)sin(θ)cos(θ)sin(θ)sin(θ)cos(θ)=sin(θ)sin(θ)cos2(θ)sin2(θ)=sin(θ)(1cos2(θ))sin2(θ)=sin(θ)sin2(θ)sin2(θ)=sin(θ)

59.

Answer.
1sin(θ)cos(θ)

61.

Answer.
1

63.

Answer.
0

65.

Answer.
1

67.

Answer.
1cos2(β)

69.

Answer.
sin(x)

71.

Answer.
sin(β)=685, cos(β)=785, tan(β)=67

73.

Answer.
sin(α)=215, cos(α)=25, tan(α)=212

75.

Answer.
0, 180, 270

77.

Answer.
135, 315

79.

Answer.
0, 60, 180, 300

81.

Answer.
0, 180

6 Radians
6.1 Arclength and Radians
Homework 6.1

1.

Answer.
Radians 0 π4 π2 3π4 π 5π4 3π2 7π4 2π
Degrees 0 45 90 135 180 225 270 315 360
circle

3.

Answer.
  1. 120=2π3radians
  2. 240=4π3radians
  3. 480=8π3radians
  4. 600=10π3radians

5.

Answer.
  1. 45=π4radians
  2. 135=3π4radians
  3. 225=5π4radians
  4. 315=7π4radians

7.

Answer.
circle

9.

Answer.
  1. 0.52
  2. 2.62
  3. 3.67
  4. 5.76

11.

Answer.
circle

13.

Answer.
2.09

15.

Answer.
2.62

17.

Answer.
0.52

19.

Answer.
2.36

21.

Answer.
  1. II
  2. IV
  3. IV
  4. I

23.

Answer.
  1. III
  2. II
  3. I
  4. IV

25.

Answer.
Radians π6 π4 π3
Degrees 30 45 60

27.

Answer.
Radians 7π6 5π4 4π3
Degrees 210 225 240

29.

Answer.
  1. 1.31
  2. 4.12
  3. 5.71

31.

Answer.
  1. 45.8
  2. 200.5
  3. 292.2

33.

Answer.
5.86 in

35.

Answer.
4.13 m

37.

Answer.
160.42

39.

Answer.
  1. 5π6
  2. 32.72 ft

41.

Answer.
867 radians 6.84

43.

Answer.
  1. 33,000π103,672.6 in
  2. 33,000π103.672.6 in per min

45.

Answer.
170π534.1 m per min

47.

Answer.
unit circle
(0.2,0.98),  (0.2,0.98)

49.

Answer.
unit circle
(0.94,0.35),  (0.94,0.35)

51.

Answer.
unit circle
(32,12),  (32,12)

53.

Answer.
  1. circle
  2. θ 1 2 3 4 5 6
    s 4 8 12 16 20 24
  3. linear graph of arclength vs angle
    m=4
  4. Arclength doubles; arclength triples

55.

Answer.
  1. π10 radians per min
  2. 10π9 radians per sec

57.

Answer.
  1. θ2π
  2. 38, 56, 712

59.

Answer.
32.5 cm2

6.2 The Circular Functions
Homework 6.2

1.

Answer.
0000 a b c d
t π4 3π4 5π4 7π4
x 12 12 12 12
y 12 12 12 12

3.

Answer.
0000 a b c d
t π3 2π3 4π3 5π3
x 12 12 12 12
y 32 32 32 32

5.

Answer.
  1. sin(0.4)0.39, cos(0.4)0.92, tan(0.4)0.42
  2. sin(1.2)0.93, cos(1.2)0.36, tan(1.2)2.6
  3. sin(2)0.91, cos(2)0.42, tan(2)2.2

7.

Answer.
  1. sin(2.8)0.33, cos(2.8)0.94, tan(2.8)0.36
  2. sin(3.5)0.35, cos(3.5)0.94, tan(3.5)0.37
  3. sin(5)0.96, cos(5)0.28, tan(5)3.3

9.

Answer.
t1.27 or t5

11.

Answer.
t3.92 or t5.5

13.

Answer.
t2.72 or t5.87

15.

Answer.
II

17.

Answer.
II

19.

Answer.
III

21.

Answer.
Negative

23.

Answer.
Positive

25.

Answer.
Positive

27.

Answer.
sin(3.5), sin(0.5), sin(2.5), sin(1.5)

29.

Answer.
cos(3),cos(4), cos(2), cos(5)

31.

Answer.
January 1: 4:24, April 1: 6:45, July 1: 8:02, October 1: 5:55

33.

Answer.
1.34

35.

Answer.
0.84

37.

Answer.
0.02

39.

Answer.
112π

41.

Answer.
13π

43.

Answer.
14π

45.

Answer.
  1. 5π6,  7π6,  11π6
    circle
  2. 3π4,  5π4,  7π4
    circle
  3. 2π3,  4π3,  5π3
    circle

47.

Answer.
 θ     sin(θ)       cos(θ)       tan(θ)   
7π6 12 32 13
5π4 12 12 1
4π3 32 12 3

49.

Answer.
14

51.

Answer.
3+33

53.

Answer.
3634

55.

Answer.
(cos(2.5),sin(2.5))(0.8,0.6)

57.

Answer.
(cos(8.5),sin(8.5))(0.6,0.8)

59.

Answer.
cos(5)0.28 mi east, sin(5)0.96 mi north, or about 0.96 mi south

61.

Answer.
1.75

63.

Answer.
5.8

65.

Answer.
3.84

67.

Answer.
  1. circle
    Intersections: (12,12) and (12,12)
  2. (cos(π4),sin(π4)) and (cos(5π4),sin(5π4))

69.

Answer.
  1. graph
    m=38
  2. tan1(38)0.3588

71.

Answer.
y2=3(x4)

73.

Answer.
y+8=(tan(2.4))((x5) or y+8=0.916(x5)

75.

Answer.
Any point (x,y) on the terminal side of θ satisfies cos(θ)=xr,  sin(θ)=yr. For the point P where r=1,  cos(θ)=x,  sin(θ)=y. The arc of length t is spanned by an angle θ in standard position. Because arclength is rθ and r=1,  t=θ, so x=cos(t),  y=sin(t).

77.

Answer.
The two right triangles shown are similar, so their sides are proportional. The hypotenuse of the large triangle is r times the hypotenuse of the small triangle, so the two legs of the large triangle must be r times the legs of the small triangle. Thus, because the coordinates of the vertex on the unit circle are (cos(θ),sin(θ)), the coordinates of P must be (rcos(θ),rsin(θ)).

79.

Answer.
71 m west, 587 m north

6.3 Graphs of the Circular Functions
Homework 6.3

1.

Answer.
  1. θ 0 π12 π6 π4 π3 5π12 π2 7π12 2π3 3π4 5π6 11π12 π
    cos(θ) 1 0.97 0.87 0.71 0.50 0.26 0 0.26 0.50 0.71 0.87 0.97 1
  2. cosine graph

3.

Answer.
sine graph
sine graph

5.

Answer.
  1. sine graph
  2. Domain: (,), range: [1,1]

7.

Answer.
  1. tangent graph
  2. Domain: xnπ2, n an odd integer, range: (,)

9.

Answer.
  1. x0.7 or x2.4
  2. x0.36 or x2.78

11.

Answer.
  1. x2 or x4.3
  2. x2.5 or x3.79

13.

Answer.
x1.3 or x4.5

15.

Answer.
x2.7 or x5.8

17.

Answer.
x1.4 or x4.5

19.

Answer.
x2.2 or x5.3

21.

Answer.
I: 0.5, II: 2.7, III: 3.6, IV: 5.8

23.

Answer.
I: 0.6, II: 2.6, III: 3.7, IV: 5.7

25.

Answer.
I: 1.3, II: 1.8, III: 4.5, IV: 4.9

27.

Answer.
t0.74 or t5.55

29.

Answer.
t1.01 or t4.15

31.

Answer.
x3.94 or x5.48

33.

Answer.
t=3π2

35.

Answer.
x=π4  or  x=5π4

37.

Answer.
z=π3  or  z=5π3

39.

Answer.
s=2π3  or  s=5π3

41.

Answer.
t=5π4  or  t=7π4

43.

Answer.
x=5π6  or  x=7π6

45.

Answer.
  1. 0.78
  2. 1.12

47.

Answer.
  1. 0.26
  2. 1.28

49.

Answer.
  1. 0.9
  2. No solution

51.

Answer.
  1. 12
  2. 0.9

53.

Answer.
62

55.

Answer.
43

57.

Answer.
6

59.

Answer.
b-c.
sinusoidal graph
d. t10 and t20    e. t7.5 to t22

61.

Answer.
b-c.
sinusoidal graph
d. High: day 204, 105; low: day 25, 66 e. d128 to d281

63.

Answer.
  1. 0.8, 0.6, 43
  2. 0.8, 0.6, 43
  3. 0.8, 0.6, 43

65.

Answer.
  1. 0.92, 0.39, 9239
  2. 0.92, 0.39, 9239
  3. 0.92, 0.39, 9239

67.

Answer.
three trig graphs

69.

Answer.
three trig graphs

71.

Answer.
  1. parabola
  2. Domain: (,), range: (,9]

73.

Answer.
  1. graph of 2 - 1/(x^2)
  2. Domain: x0, range: (,2)

75.

Answer.
  1. graph of translated square root
  2. Domain: [6,), range: [0,)

77.

Answer.
  1. semicircle
  2. Domain: [2,2], range: [2,0]

79.

Answer.
  1. x 0 π2 π 3π2 2π
    cos(x) 1 0 1 0 1
    cosine graph
  2. Domain: (,), Range: [1,1]

6.4 Chapter 6 Summary and Review
Chapter 6 Review Problems

1.

Answer.
  1. 5π12
  2. 7π6
  3. 17π9

3.

Answer.
  1. 0.47
  2. 2.48
  3. 3.80

5.

Answer.
  1. 150
  2. 54
  3. 230

7.

Answer.
  1. 114.59
  2. 206.26
  3. 45.84

9.

Answer.
  1. 4π3
  2. 7π6
  3. 9π4

11.

Answer.
  1. 18
  2. 516
  3. 76

13.

Answer.
  1. II
  2. I
  3. IV

15.

Answer.
  1. 0.006, 2.17, 0.0379
  2. 0.0379

17.

Answer.
6885 mph

19.

Answer.
  1. 0
  2. 83
  3. 12

21.

Answer.
  1. (0.5,0.8)
  2. (0.4,0.9)
  3. (1.0,0.1)

23.

Answer.
  1. (rcos(α),rsin(α))
  2. (rcos(α),rsin(α))
  3. (rcos(α),rsin(α))
  4. (rcos(α),rsin(α))

25.

Answer.
6π

27.

Answer.
>

29.

Answer.
<

31.

Answer.
9.86

33.

Answer.
1.33

35.

Answer.
  1. π6
  2. π4
  3. 3π8
  4. 5π12

37.

Answer.
  1. 0.34
  2. 0.76
  3. 1.25
  4. 1.5

39.

Answer.
158.2

41.

Answer.
graph of cosine and sine

43.

Answer.
  1. sinusoidal graph
    mid: y=5, amp: 3, period: π
  2. sinusoidal curve and horizontal line
    0.86, 2.28, 4.00, 5.42

45.

Answer.
  1. sinusoidal graph
    mid: y=10, amp: 4.8, period: 2π
  2. sinusoidal graph and horizontal line
    1.93, 4.2

47.

Answer.
5π12, 17π12

49.

Answer.
π3, 2π3

51.

Answer.
π

53.

Answer.
1.37, 4.51

55.

Answer.
6.02, 3.40

57.

Answer.
0.32, 5.97

59.

Answer.
  1. 1.21, 5.07
  2. 0.9394

61.

Answer.
  1. 0.40, 2.74
  2. 0.3827

63.

Answer.
parabola
Dom: all real numbers, Rge: y4

65.

Answer.
semicircle
Dom: 4s4, Rge: 4y0

67.

Answer.
  1. x2+y2=1
  2. (cos(t),sin(t))
  3. cos2(t)+sin2(t)=1
  4. Yes

7 Circular Functions
7.1 Transformations of Graphs
Homework 7-1

1.

Answer.
amplitude 2, period 2π, midline y=3

3.

Answer.
amplitude 1, period π2, midline y=0

5.

Answer.
amplitude 5, period 6π, midline y=0

7.

Answer.
amplitude 1, period 2, midline y=1

9.

Answer.
transformations of sine graph

11.

Answer.
transformations of cosine

13.

Answer.
transformations of sine

15.

Answer.
cosine transformations

17.

Answer.
y=2sin(x)

19.

Answer.
y=2cos(x)

21.

Answer.
y=0.75cos(x)

23.

Answer.
  1. amplitude 2, period 2π3, midline y=0
  2. y=2sin(3x)

25.

Answer.
  1. amplitude 3, period 2π, midline y=0
  2. y=3sin(x2)

27.

Answer.
  1. amplitude 0.5, period 4π, midline y=3.5
  2. y=0.5cos(x2)+3.5

29.

Answer.
  1. amplitude 2, period 4, midline y=1
  2. y=1+2sin(πx2)

31.

Answer.
  1. t 2t cos(2t) 5cos(2t) 25cos(2t)
    0 0 1 5 3
    π4 π2 0 0 2
    π2 π 1 5 7
    3π4 3π2 0 0 2
    π 2π 1 5 3
  2. sinusoidal graph

33.

Answer.
  1. t t2 cos(t2) 3cos(t2) 1+3cos(t2)
    0 0 1 3 4
    π π2 0 0 1
    2π π 1 3 2
    3π 3π2 0 0 1
    4π 2π 1 3 4
  2. sinusoidal graph

35.

Answer.
  1. t t3 sin(t3) 2sin(t3) 3+2sin(t3)
    0 0 0 0 3
    3π2 π2 1 2 1
    3π π 0 0 3
    9π2 3π2 1 2 5
    6π 2π 0 0 3
  2. sinusoidal graph

37.

Answer.
sinusoidal graph

39.

Answer.
sinusoidal graph

41.

Answer.
sinusoidal graph

43.

Answer.
sinusoidal graph

45.

Answer.
  1. sinusoidal graph
  2. W(t)=12+8cos(πt6)

47.

Answer.
  1. sinusoidal graph
  2. h=10+14cos(πt5)

49.

Answer.
H=122.4cos(πt6)

51.

Answer.
y=155cos(120πt)

53.

Answer.
  1. x π4 π8 0 π8 π4
    tan2x undef 1 0 1 undef
    transformed tangent function
  2. period π2, midline y=0

55.

Answer.
  1. x π6 π12 0 π12 π6
    4+2tan3x undef 2 0 6 undef
    transformed tangent graph
  2. period π3, midline y=4

57.

Answer.
  1. x 2π π 0 π 2π
    3tan(x4) undef 4 0 2 undef
    transformed tangent graph
  2. period 4π, midline y=3

59.

Answer.
π12,  5π12,  7π12,  11π12,  13π12,  17π12,  19π12,  23π12

61.

Answer.
7π12,  11π12,  19π12,  23π12

63.

Answer.
π12,  5π12,  3π4,  13π12,  17π12,  7π4

65.

Answer.
1.83, 2.88, 4.97, 6.02

67.

Answer.
4.19

69.

Answer.
0.28, 1.81, 2.37, 3.91, 4.47, 6.00

7.2 The General Sinusoidal Function
Homework 7-2

1.

Answer.
  1. x π 5π6 2π3 π2 π3 π6 0 π6 π3 π2 2π3 5π6 π
    f(x) 0 12 32 1 32 12 0 12 32 1 32 12 0
    g(x) 32 12 0 12 32 1 32 12 0 12 32 1 32
  2. sine and translated sine
  3. π3 to the right
  4. 5π6
  5. 2π3, π3

3.

Answer.
  1. x π 3π4 π2 π4 0 π4 π2 3π4 π
    f(x) 0 1 undef 1 0 1 undef 1 0
    g(x) 1 undef 1 0 1 undef 1 0 1
  2. translated tangent function
  3. π4 to the left
  4. π, 0, π
  5. π4, 3π4

5.

Answer.
  1. amplitude 2, shift π6 to the left
  2. x x+π6 cos(x+π6) 2cos(x+π6)
    7π6 π 1 2
    2π3 π2 0 0
    π6 0 1 2
    π3 π2 0 0
    5π6 π 1 2
    4π3 3π2 0 0
    11π6 2π 1 2
  3. sinusoidal graph
  4. π2, 7π6
  5. π3, 4π3

7.

Answer.
  1. f(x)=sin(x+π4)
  2. f(x)=cos(xπ4)

9.

Answer.
  1. f(x)=tan(xπ3)
  2. f(x)=tan(x+2π3)

11.

Answer.
  1. period π, shift π6 to the right
  2. x 2x 2xπ3 cos(2xπ3)
    π6 π3 0 1
    5π12 5π6 π2 0
    2π3 4π3 π 1
    11π12 11π6 3π2 0
    7π6 7π3 2π 1
  3. sinusoidal graph
  4. π6, 7π6
  5. 5π12, 11π12, 13π6, 23π12

13.

Answer.
  1. period 2, shift 13 to the left
  2. x πx πx+π3 sin(πx+π3)
    13 π3 0 0
    16 π6 π2 1
    23 2π3 π 0
    76 7π6 3π2 1
    53 5π3 2π 0
  3. sinusoidal graph
  4. 116, 16
  5. 43, 13, 23, 53

15.

Answer.
  1. midline y=4, period 4π, horizontal shift π3 to the right, amplitude 3
  2. x x2 x2π6 sin(x2π6) 3sin(x2π6)+4
    π3 π6 0 0 4
    4π3 2π3 π2 1 7
    7π3 7π6 π 0 3
    10π3 5π3 3π2 1 1
    13π3 13π6 2π 0 4
  3. sinusoidal graph
  4. no solution for 0x2π
  5. π3

17.

Answer.
y=2sin(2π3(x+4))+5
sinusoidal graph

19.

Answer.
y=5cos(πx180)+12
sinusoidal graph

21.

Answer.
  1. f(x)=3sin(x+2π3)
  2. f(x)=3cos(x+π6)

23.

Answer.
  1. f(x)=2sin(2(xπ4))
  2. f(x)=2cos(2x)

25.

Answer.
  1. f(x)=4sin[14(x7π3)]
  2. f(x)=4cos[14(xπ3)]

27.

Answer.
  1. midline T=35.35, period 12, amplitude 36.95
  2. T(m)=36.95cos(π6m)+35.35
  3. sinusoidal graph

29.

Answer.
  1. midline h=1.4, period 2π0.5112.32, amplitude 1.4
  2. sinusoidal graph
  3. high 11:10 am, low 5:19 pm

31.

Answer.
  1. amplitude 3.2, period 2, midline y=2
  2. f(t)=2+3.2cos(πt)

33.

Answer.
  1. amplitude 5, period 1, midline y=0
  2. H(x)=5sin(2πx)+5

7.3 Solving Equations
Homework 7-3

1.

Answer.
3π8,  7π8,  11π8,  15π8

3.

Answer.
0,  π2,  π,  3π2,  2π

5.

Answer.
2π9,  4π9,  8π9,  10π9,  14π9,  16π9

7.

Answer.
π12,  5π12,  13π12,  17π12

9.

Answer.
π18,  7π18,  13π18,  19π18,  25π18,  31π18

11.

Answer.
0.491, 2.651, 3.632, 5.792

13.

Answer.
0.540,  1.325,  2.110,  2.896,  3.681,  4.467,  5.252,  6.037

15.

Answer.
1.114,  2.027,  3.209,  4.122,  5.303,  6.216

17.

Answer.
0.702,  2.440,  3.843,  5.582

19.

Answer.
0,  1,  2,  3,  4,  5,  6

21.

Answer.
π6,  2π3,  7π6,  5π3

23.

Answer.
5π12,  7π12,  13π12,  5π4,  7π4,  23π12

25.

Answer.
3π2

27.

Answer.
76,  116,  196,  236,  316,  356

29.

Answer.
1.14,  1.62,  3.23,  3.72,  5.24,  5.81

31.

Answer.
0.44,  1.44,  2.44,  3.44,  4.44,  5.44

33.

Answer.
0.01,  3.39,  6.01

35.

Answer.
0.564,  1.182,  2.658,  3.276,  4.752,  5.371

37.

Answer.
0.423,  2.977,  4.423

39.

Answer.
1.165, 4.165

41.

Answer.
2.251

43.

Answer.
  1. P(t)=4000cos(π6t)+46,000
  2. t=cos1(14)6π3.48 months (Dec) or t=12cos1(14)6π8.52 months (June)
  3. sinusoidal graph
    P(t) is less than 45,000 between A and B.

45.

Answer.
  1. h(t)=1110cos(π30t)
  2. t=cos(0.7)30π22.40 sec or t=60cos(0.7)30π37.60 sec
  3. sinusoidal graph
    Delbert is above 18 m between A and B.

7.4 Chapter 7 Summary and Review
Review Problems

1.

Answer.
amp: 2, period: 2π3; mid: y=4

3.

Answer.
amp: 2.5, period: 2; mid: y=2

5.

Answer.
sinusoidal graph

7.

Answer.
sinusoidal graph

9.

Answer.
y=3+2sin(x)

11.

Answer.
y=43sin(x4)

13.

Answer.
  1. period: 4π, shift: π3 left
  2. x x2 x2+π6 sin(x2+π6)
    2π3 π3 π6 12
    π3 π6 0 0
    0 0 π6 12
    π6 π12 π4 12
    π3 π6 π3 32
    2π3 π3 π2 1
    π π2 2π3 32
  3. sinusoidal graph
  4. 2π3
  5. π3

15.

Answer.
  1. mid: y=20, period: 0, amp: 5
  2. Fill in the table of values.
    x π30x cos(π30x) 205cos(π30x)
    5 π6 32 2032
    0 0 1 15
    5 π6 32 2032
    10 π3 12 17.5
    15 π2 0 20
    50 π 1 25
  3. sinusoidal graph
  4. 30
  5. 15, 45

17.

Answer.
sinusoidal graph

19.

Answer.
  1. sinusoidal graph
  2. 0.57, 3.07, 3.71

21.

Answer.
y=85.519.5cos(π6t)

23.

Answer.
  1. amp: 3, period: 12, midline: y=15
  2. y=153cos(π6t)

25.

Answer.
7π12, 11π12, 19π12, 23π12

27.

Answer.
0, π4, π2, 3π4, π, 5π4, 7π4, 2π

29.

Answer.
0.066, 1.113, 2.160, 3.207, 4.255, 5.302

31.

Answer.
1.150, 1.991, 4.292, 5.133

33.

Answer.
π24, 5π24, 25π24, 29π24

35.

Answer.
No solution

37.

Answer.
0.375, 1.422, 2.470, 3.517, 4.564, 5.611

39.

Answer.
2.120, 4.880

8 More Functions and Identities
8.1 Sum and Difference Formulas
Homework 8-1

1.

Answer.
angles
x2=x1, y2=y1, and r2=r1. Thus, cos(α)=x2r2=x1r1=cos(α), sin(α)=y2r2=y1r1=sin(α), and tan(α)=y2x2=y1x1=tan(α).

3.

Answer.
(2+6)4
angles

5.

Answer.
cos(0.32x)=0.24, sin(0.32x)=0.97

7.

Answer.
cos(45+45)=cos(90)=0, but cos(45)+cos(45)=12+12=2

9.

Answer.
tan(8729)1.600, but tan(87)tan(29)18.527

11.

Answer.
two sinusoidal graphs
The curves are different.

13.

Answer.
  1. 6365
  2. 1665
  3. 1663

15.

Answer.
  1. 44117
  2. 43

17.

Answer.
  1. 3685
  2. 1384

19.

Answer.
  1. 1665
  2. 6365
  3. 1663
  4. angles

21.

Answer.
cos(15)=6+24, tan(15)=23

23.

Answer.
62+110

25.

Answer.
cos(θ)

27.

Answer.
32cos(t)12sin(t)

29.

Answer.
3tanβ13+tanβ

31.

Answer.
No

33.

Answer.
No

35.

Answer.
1=2(12)(12)

37.

Answer.
12=(32)2(12)2

39.

Answer.
False, but cos(2α)=2(0.32)21

41.

Answer.
False, but 2θ=sin1(h)

43.

Answer.
sin(68)

45.

Answer.
cos(π8)

47.

Answer.
cos(6θ)

49.

Answer.
sin10t

51.

Answer.
tan128

53.

Answer.
cos(4β)

55.

Answer.
  1. 56
  2. 116
  3. 511
  4. 51118
  5. 718
  6. 5117

57.

Answer.
  1. 1w2+1
  2. ww2+1
  3. 1w
  4. 2ww2+1
  5. w21w2+1
  6. 2ww21

59.

Answer.
  1. 513
  2. 120169
  3. 119169
  4. 120119
  5. angles

61.

Answer.
  1. 815
  2. 1517
  3. 817

63.

Answer.
  1. 2sin(θ)cos(θ)+2cos(θ)=0
  2. π2, 5π4, 3π2, 7π4

65.

Answer.
  1. 2cos2(t)5cos(t)+2=0
  2. π3, 5π3

67.

Answer.
  1. 2tan(β)1tan2(β)+2sin(β)=0
  2. 0, π3, π , 5π3

69.

Answer.
  1. 3cos(ϕ)cos(ϕ)=3
  2. π6, 11π6

71.

Answer.
  1. sin(3ϕ)=1
  2. π6 , 5π6 , 3π2

73.

Answer.
  1. cos(θ+90)=sinθ
  2. sin(θ+90)=cosθ

75.

Answer.
  1. cos(π2θ)=cosπ2cos(θ)+sinπ2sinθ=sin(θ)
  2. sin(π2θ)=sinπ2cos(θ)cosπ2sin(θ)=cos(θ)

77.

Answer.
sin(2θ)=sin(θ+θ)=sin(θ)cos(θ)+sin(θ)cos(θ)=2sin(θ)cos(θ)

79.

Answer.
  1. Not an identity.
  2. β=π (many answers possible)

81.

Answer.
Identity

83.

Answer.
  1. Not an identity.
  2. θ=0 (many answers possible)

85.

Answer.
Identity

87.

Answer.
Identity

89.

Answer.
triangle inscribed in rectangle
  1. l1=sin(α),l2=cos(α)
  2. θ1 and β are both complements of ϕ; θ2 and α+β are alternate interior angles
  3. s1=cos(α+β), s2=sin(α+β)
  4. s3=sin(α)sin(β), s4=sin(α)cos(β)
  5. s5=cos(α)cos(β), s6=cos(α)sin(β)
  6. sin(α+β)=sin(α)cos(β)+cos(α)sin(β), cos(α+β)=cos(α)cos(β)+sin(α)sin(β)

91.

Answer.
  1. (AB)2=22cos(αβ)
  2. (AB)2=(cos(α)cos(β))2+(sin(α)sin(β))2
  3. 22cos(αβ)=(cos(α)cos(β))2+(sin(α)sin(β))222cos(αβ)=cos2(α)2cos(α)cos(β)+cos2(β)+000000000+sin2(α)2sin(α)sin(β)+sin2(β)22cos(αβ)=1+12(cos(α)cos(β)sin(α)sin(β))2cos(αβ)=2(cos(α)cos(β)sin(α)sin(β))cos(αβ)=cos(α)cos(β)sin(α)sin(β))

8.2 Inverse Trigonometric Functions
Homework 8-2

1.

Answer.
No inverse: Some horizontal lines intersect the curve in more than one point.

3.

Answer.
Inverse exists: The function is 1-1.

5.

Answer.
graph
No inverse

7.

Answer.
simicircle
No inverse

9.

Answer.
16.5

11.

Answer.
46.4

13.

Answer.
=51.9

15.

Answer.
3π4

17.

Answer.
π6

19.

Answer.
π6

21.

Answer.
triangle
  1. h=500tan(θ)
  2. θ=tan1(h500)
  3. θ=tan1(2), so the angle of elevation is tan1(2)63.4 when the rocket is 1000 yd high.

23.

Answer.
triangle
  1. d=50tanθ
  2. θ=tan1(50d)
  3. θ=tan1(0.25); the bilboard subtends an angle of tan1(0.25)14 at a distance of 200 ft.

25.

Answer.
triangles
  1. α=tan1(1x)
  2. β=tan1(5x)tan1(1x)
  3. β=45tan1(15), so the painting subtends an angle of 45tan1(15)33.7 when Martin is 5 meters from the wall.

27.

Answer.
t=12πω(sin1VV0ϕ)

29.

Answer.
A=sin1(asin(B)b)

31.

Answer.
θ=±cos1(kPR4)

33.

Answer.
25

35.

Answer.
15

37.

Answer.
57

39.

Answer.
1x2x

41.

Answer.
1h2

43.

Answer.
2t4t2+1

45.

Answer.
x 1 32 22 12 0 12 22 32 1
cos1(x) π 5π6 3π4 2π3 π2 π3 π4 π6 0
arccosine

47.

Answer.
x 3 1 13 0 13 1 3
cos1(x) π2 π3 π6 0 π6 π4 π3
arctangent

49.

Answer.
a–b.
transformations of arccos
c. No

51.

Answer.
a.
arctangent
c. No

53.

Answer.
817

55.

Answer.
1665

57.

Answer.
427

59.

Answer.
  1. 6365
  2. 1665
  3. 3365
  4. 5665

61.

Answer.
1

63.

Answer.
  1. 2xx2+1
  2. 1x2

65.

Answer.
sin(2θ)=2x25x225, cos(2θ)=252x225

67.

Answer.
arctan(x3+3x2(x2+9))

69.

Answer.
  1. 1x1
  2. Yes.
  3. All
  4. x<π2 or x>π2

71.

Answer.
  1. Domain: 1x1, range: {π2}
  2. Let θ=sin1(x). Then x=sin(θ)=cos(π2θ) and cos1(x)=π2θ. So  sin1(x)+cos1(x)=θ+(π2θ)=π2 .

73.

Answer.
  1. θ2
  2. t=sin(θ)
  3. 12arcsin(t)

8.3 The Reciprocal Functions
Homework 8-3

1.

Answer.
2.203

3.

Answer.
0.466

5.

Answer.
5.883

7.

Answer.
1.203

9.

Answer.
2

11.

Answer.
1

13.

Answer.
233

15.

Answer.
2

17.

Answer.
θ 0 π6 π4 π3 π2 2π3 3π4 5π6 π
sec(θ) 1 233 2 2 undefined 2 2 233 1
csc(θ) undefined 2 2 233 1 233 2 2 undefined
cot(θ) undefined 3 1 33 0 33 1 3 undefined

19.

Answer.
  1. 0.980
  2. 1.020
  3. 1.369
  4. 1.020
  5. 0.284
  6. 1.020

21.

Answer.
sin(θ)=45,  cos(θ)=35,  tan(θ)=43,  sec(θ)=53,  csc(θ)=54,  cot(θ)=34

23.

Answer.
sin(θ)=441,  cos(θ)=541,  tan(θ)=45,  sec(θ)=415,  csc(θ)=414,  cot(θ)=54

25.

Answer.
sin(θ)=574,  cos(θ)=774,  tan(θ)=57,  sec(θ)=747,  csc(θ)=745,  cot(θ)=75

27.

Answer.
sin(θ)=58,  cos(θ)=398,  tan(θ)=539,  sec(θ)=839,  csc(θ)=85,  cot(θ)=395

29.

Answer.
  1. d=hcsc(θ)
  2. 155.572 miles

31.

Answer.
  1. 0.78 sec
  2. l=8t2sin(2θ)

33.

Answer.
sin(θ)=7x2+49,  cos(θ)=xx2+49,  tan(θ)=7x,  sec(θ)=x2+49x,  csc(θ)=x2+497,  cot(θ)=x7

35.

Answer.
sin(θ)=S,  cos(θ)=1S2,  tan(θ)=S1S2,  sec(θ)=11S2,  csc(θ)=1S,  cot(θ)=1S2S

37.

Answer.
sin(θ)=9a23,  cos(θ)=a3,  tan(θ)=9a2a,  sec(θ)=3a,  csc(θ)=39a2,  cot(θ)=a9a2

39.

Answer.
AC, OA, BD, OD, OE, EF

41.

Answer.
angle
 sin(θ)=32,  cos(θ)=12,  tan(θ)=3,  sec(θ)=2,  csc(θ)=233,  cot(θ)=33

43.

Answer.
triangle
sin(α)=13,  cos(α)=223,  tan(α)=24,  sec(α)=324,  csc(α)=3,  cot(α)=22

45.

Answer.
angle
sin(γ)=417, cos(γ)=117, tan(γ)=4, sec(γ)=17, csc(γ)=174, cot(γ)=14

47.

Answer.
433+22

49.

Answer.
33

51.

Answer.
463+103

53.

Answer.
x 0 π4 π2 3π4 π 5π4 3π2 7π4 2π
sec(x) 1 2 undefined 2 1 2 undefined 2 1
secant

55.

Answer.
sine and cosecant

57.

Answer.
x 0 π4 π2 3π4 π 5π4 3π2 7π4 2π
cot(x) undefined 1 0 1 undefined 1 0 1 undefined
cotangent

59.

Answer.
csc(x)cot(x)=1sin(x)cos(x)sin(x)=1sin(x)÷cos(x)sin(x)=1sin(x)sin(x)cos(x)=1cos(x)=sec(x)

61.

Answer.
sec(x)cot(x)csc(x)=1cos(x)cos(x)sin(x)1sin(x)=1sin(x)1sin(x)=1

63.

Answer.
tan(x)csc(x)=sin(x)cos(x)1sin(x)=1cos(x)=sec(x)

65.

Answer.
π6, 5π6

67.

Answer.
3π4, 5π4

69.

Answer.
5π6, 11π6

71.

Answer.
55

73.

Answer.
a242

75.

Answer.
w21w

77.

Answer.
sec(s)=54,  csc(s)=53,  cot(s)=43

79.

Answer.
sec(s)=11w2,  csc(s)=1w,  cot(s)=1w2w

81.

Answer.
sin(θ)cos2(θ)

83.

Answer.
sec(t)

85.

Answer.
1sin(β)cos(β)

87.

Answer.
cos(x)

89.

Answer.
cos2(θ)+sin2(θ)=1cos2(θ)cos2(θ)+sin2(θ)cos2(θ)=1cos2(θ)1+tan2(θ)=sec2(θ)

91.

Answer.
  1. csc(θ)=26
  2. sin(θ)=2626, cos(θ)=52626, tan(θ)=15, sec(θ)=265

93.

Answer.
cos(t)=±1sin2(t),  tan(t)=±sin(t)1sin2(t),  sec(t)=±11sin2(t),  csc(t)=1sin(t),  cot(t)=±1sin2(t)sin(t)

95.

Answer.
asin(A)=bsin(B)=csin(C)a1sin(A)=b1sin(B)=c1sin(C)acsc(A)=bcsc(B)=ccsc(C)

8.4 Chapter Summary and Review
Review Problems

1.

Answer.
False

3.

Answer.
True

5.

Answer.
False

7.

Answer.
False

9.

Answer.
22152

11.

Answer.
  1. 533332
  2. 53335(33+11)

13.

Answer.
1

15.

Answer.
tan(t)+313tan(t)

17.

Answer.
  1. 45
  2. 35
  3. 43
  4. 2425
  5. 725
  6. 247

19.

Answer.
sin(9x)

21.

Answer.
tan(2ϕ2)

23.

Answer.
sin(8θ)

25.

Answer.
  1. 12sin2(θ)sin(θ)=1
  2. 0, π, 7π6, 11π6

27.

Answer.
graph
No

29.

Answer.
  1. π3
  2. 2π3

31.

Answer.
  1. tan1(52.8x)
  2. 69.25, 27.83

33.

Answer.
θ=sin1(vy+gtv0)

35.

Answer.
23

37.

Answer.
14t2

39.

Answer.
Because |sin(θ)|1, sin1(t) is undefined for |t|>1. If x0, then either |x|>1 or |1x|>1. If x=0, then 1x is undefined.

41.

Answer.
  1. 2.203
  2. 3.236
  3. 0.466

43.

Answer.
sin(θ)=13313,  cos(θ)=12313,  tan(θ)=1312,  sec(θ)=31312,  csc(θ)=31313,  cot(θ)=1213

45.

Answer.
sin(θ)=13,  cos(θ)=223,  tan(θ)=122,  sec(θ)=322,  csc(θ)=3,  cot(θ)=22

47.

Answer.
sin(θ)=9106,  cos(θ)=5106,  tan(θ)=95,  sec(θ)=1065,  csc(θ)=1069,  cot(θ)=59

49.

Answer.
sin(α)=116,  cos(α)=56,  tan(α)=115,  sec(α)=65, csc(α)=611,  cot(α)=511

51.

Answer.
sin(θ)=s4,  cos(θ)=16s24,  tan(θ)=s16s2,  sec(θ)=416s2,  csc(θ)=4s,  cot(θ)=16s2s

53.

Answer.
sin(θ)=ww2+144,  cos(θ)=12w2+144,  tan(θ)=w12,  sec(θ)=w2+14412,  csc(θ)=w2+144w,  cot(θ)=12w

55.

Answer.
sin(α)=k2,  cos(α)=4k22,  tan(α)=k4k2,  sec(α)=24k2,  csc(α)=2k,  cot(α)=4k2k

57.

Answer.
sin(θ)=0.3, cos(θ)=0.4, tan(θ)=0.75, sec(θ)=2.5, csc(θ)3.33, cotθ()1.33

59.

Answer.
8

61.

Answer.
2

63.

Answer.
θ2.8, θ0.30

65.

Answer.
y=csc(x) or y=cot(x)

67.

Answer.
y=sec(x)

69.

Answer.
y=sec(x) or y=csc(x)

71.

Answer.
f(x)=sin(x)1

73.

Answer.
G(x)=tan(x)1

75.

Answer.
cos2(x)

77.

Answer.
cos2(B)

79.

Answer.
csc(θ)

81.

Answer.
3tan(θ)sin(θ)

83.

Answer.
  1. AC=tan(α), DC=tan(β), AD=tan(α)tan(β)
  2. They are right triangles that share B.
  3. A=F, B is the complement of A, and FDC is the complement of F.
  4. CFCD=tan(α), so CF=tan(α)tan(β)
  5. They are right triangles with A=F.
  6. EBD=αβ, so tan(αβ)=oppadj=DEBE;  DEBE and ADBF are ratios of corresponding sides of similar triangles; AD=tan(α)tan(β) by part (a), BF=BC+CF=1+tan(α)tan(β) by part (d).

85.

Answer.
d=25csc(112), α=45, a19.07, b10.54

9 Vectors
9.1 Geometric Form
Homework 9-1

1.

Answer.
position vector

3.

Answer.
velocity vector

5.

Answer.
velocity vector

7.

Answer.
A and E

9.

Answer.
H and K

11.

Answer.
vecoor on grid

13.

Answer.
vector on grid

15.

Answer.
triangle

17.

Answer.
vectors on grid

19.

Answer.
vectors on grid
A=13, θ=33.7

21.

Answer.
vectors on grid
C=1, θ=90

23.

Answer.
vectors on grid
E=5, θ=90

25.

Answer.
vector on
G=4, θ=180

27.

Answer.
v=13, θ=67.38

29.

Answer.
v=859.22, θ=229.4

31.

Answer.
vectors
v+w=32.9, θ=109.3

33.

Answer.
vectors
v+w=11.4, θ=162.4

35.

Answer.
vectors
4.47 mi, 23.4 east of north

37.

Answer.
vectors
129.4 mph, 85.4 west of north

39.

Answer.
  1. vx=10, vy=103, wx=52, wy=52
  2. 19.9 mph, 59 east of north

41.

Answer.
  1. vx1.23, vy3.38, wx0.32, wy2.23
  2. 1.9 km, 54.5 west of north

43.

Answer.
vectors on grid

45.

Answer.
vectors on grid

47.

Answer.
vectors on grid

49.

Answer.
vectors on grid

51.

Answer.
ux=2,  uy=1,  vx=1, vy=3,  Ax=1,  Ay=4;  Ax=uxvx,  Ay=uyvy

9.2 Coordinate Form
Homework 9-2

1.

Answer.
u=3i+2j
  1. 13
  2. 6i+4j
  3. 213

3.

Answer.
w=6i3j
  1. 35
  2. 6i+3j
  3. 35

5.

Answer.
  1. u+v=2i+5j and u+v=29
  2. u+vu+v

7.

Answer.
  1. vector
    5i+8j
  2. v=89,  θ=122

9.

Answer.
  1. vector
    2ij
  2. v=5,  θ=206.6

11.

Answer.
  1. 18i+12j
  2. v=613,  θ=33.7

13.

Answer.
v=62,  θ=135

15.

Answer.
w=14,  θ=30

17.

Answer.
q=4745,  θ=61.56

19.

Answer.
v=32i32j

21.

Answer.
v6.629i+4.995j

23.

Answer.
i2j
vectors

25.

Answer.
4i+4j
vectors

27.

Answer.
12i+3j

29.

Answer.
2.8i+1.9j

31.

Answer.
3i+7j

33.

Answer.
8i20j

35.

Answer.
14i9j

37.

Answer.
9i+23j

39.

Answer.
1213i+513j

41.

Answer.
12i12j

43.

Answer.
24i+45j

45.

Answer.
1210i+410j

47.

Answer.
  1. vectors
  2. u=2.393i+1.016j,  v=4.242i3.956j
  3. 1.849i2.940j

49.

Answer.
  1. vectors
  2. u=11.97i+32.889j,  v=57.955i+15.529j
  3. 45.98i+17.36j

51.

Answer.
  1. vectors
  2. 1700 m, 28.1 east of south

53.

Answer.
  1. vectors
  2. 21.98 km, 2.27 north of west

55.

Answer.
  1. vectors
  2. 83 mi, 62 east of north

57.

Answer.
  1. 4i5j
  2. 4i+5j

59.

Answer.
  1. i3j
  2. i+3j

61.

Answer.
  1. v=10, 2v=20=210
  2. kv=(ka)2+(kb)2=ka2+b2

9.3 The Dot Product
Homework 9-3

1.

Answer.
3313

3.

Answer.
12

5.

Answer.
25

7.

Answer.
  1. w=(5613i+8413j)+(4813i3213j)
  2. vectors

9.

Answer.
  1. w=(4i4j)+(2i+2j)
  2. vectors

11.

Answer.
22

13.

Answer.
0

15.

Answer.
12

17.

Answer.
318.2

19.

Answer.
not orthogonal

21.

Answer.
orthogonal

23.

Answer.
4.4

25.

Answer.
97.1

27.

Answer.
8

29.

Answer.
10

31.

Answer.
21

33.

Answer.
42i28j

35.

Answer.
4

37.

Answer.
38.57 lbs

39.

Answer.
1289 lbs

41.

Answer.
  1. 12i+12j  and  12i+12j
  2. uv=0
  3. 112 and 52
  4. orthogonal vectors

43.

Answer.
vv=c2+d2

45.

Answer.
kuv=kac+kbd=k(ac+bd)=(akc+bkd)

47.

Answer.
(uv)(u+v)=(ac)(a+c)+(bd)(b+d)=(a2+b2)(c2+d2)

49.

Answer.
a1+b01=a and a0+b11=b

51.

Answer.
  1. Both ii=1 and jj=1 because 11cos0=1; ij=11cos90=0
  2. (ai+bj)(ci+dj)=ac(1)+ad(0)+bc(0)+bd(1)=ac+bd

53.

Answer.
  1. uv2=uu2uv+vv=u2+v22uvcosθ
  2. Let a=u, b=v, c=uv, and C=θ

9.4 Chapter Summary and Review
Review Problems

1.

Answer.
vector
vN=8.45 mph, vE=18.13 mph

3.

Answer.
vector
vN=1127.63 lbs, vE=410.42 lbs

5.

Answer.
A=10.9, θ=236.3

7.

Answer.
i3j

9.

Answer.
  1. vector
    15i+3j
  2. v=15.3, θ=11.3

11.

Answer.
  1. vector
    2i6j
  2. v=6.3 mi, θ=288.4

13.

Answer.
  1. vectors
  2. 7.64 km, θ=30.31

15.

Answer.
  1. vectors
  2. 8.46 mi, θ=155.6

17.

Answer.
  1. F1=200i, F2=602i602j, F3=503i+50j, F4=125i+1253j
  2. 73.25i+181.65j

19.

Answer.
13i+5j

21.

Answer.
7i14j

23.

Answer.
213i+313j

25.

Answer.
629i1529j

27.

Answer.
3.45

29.

Answer.
8.08

31.

Answer.
106.26

10 Polar Coordinates and Complex Numbers
10.1 Polar Coordinates
Homework 10-1

1.

Answer.
polar plot

3.

Answer.
polar plot

5.

Answer.
polar plot

7.

Answer.
polar plot

9.

Answer.
(5,3π4)

11.

Answer.
(1,π)

13.

Answer.
(3,4π3)

15.

Answer.
(2,π12)

17.

Answer.
(3,33)

19.

Answer.
(32,32)

21.

Answer.
(2.15,1.06)

23.

Answer.
(0.14,1.99)

25.

Answer.
(72,π4)

27.

Answer.
(22,11π6)

29.

Answer.
(13,π+tan123)

31.

Answer.
(2,π)

33.

Answer.
  1. (2,11π6)
  2. (2,7π6)

35.

Answer.
  1. (3,0)
  2. (3,π)

37.

Answer.
  1. (2.3,2.06)
  2. (2.3,1.08)

39.

Answer.
polar plot

41.

Answer.
polar grid

43.

Answer.
polar grid

45.

Answer.
r0, π6θπ3

47.

Answer.
r1, π2θπ

49.

Answer.
1r1, 3π4θπ

51.

Answer.
x2+y2=2

53.

Answer.
x2+y2=4x

55.

Answer.
y=1

57.

Answer.
y=2x

59.

Answer.
x2+y2=3x

61.

Answer.
x2=44y

63.

Answer.
2x+y=1

65.

Answer.
r=2sec(θ)

67.

Answer.
2r2=sec(θ)csc(θ)

69.

Answer.
r=4cot(θ)csc(θ)

71.

Answer.
r=4

73.

Answer.
d=(x2x1)2+(y2y1)2=(r2cos(θ2)r1cos(θ1))2+(r2sin(θ2)r1sin(θ1))2=r22cos2(θ2)2r2r1cos(θ2)cos(θ1)+r12cos2(θ1)+r22sin2(θ2)2r2r1sin(θ2)sin(θ1)+r12sin2(θ1)=r22+r122r2r1(cos(θ2)cos(θ1)sin(θ2)sin(θ1))=r12+r222r1r2cos(θ2θ1)

10.2 Polar Graphs
Homework 10-2

1.

Answer.
  1. circles
    k is the radius
  2. x2+y2=1, x2+y2=4, x2+y2=9

3.

Answer.
  1. lines on polar grid
    tank is the slope
  2. y=x3, y=3x, y=3x, y=x3

5.

Answer.
θ 0 π4 π2 3π4 π 5π4 3π2 7π4
r=2 2 2 2 2 2 2 2 2
θ 0 π4 π2 3π4 π 5π4 3π2 7π4
r=2 2 2 2 2 2 2 2 2
The graph of r=2 begins at the right-most point (and proceeds counter-clockwise); the graph of r=2 begins at the left-most point.
polar points on circle

7.

Answer.
  1. circle on polar grid
  2. θ π 5π4 3π2 7π4 2π
    r 4 22 0 22 4
    The graph is traced again.
  3. center: (2,0), radius: 2
  4. (x2)2+y2=4

9.

Answer.
  1. circcles on polar grid
  2. For a>0, a is the radius of a circle centerd on the positive y-axis; for a<0, |a| is the radius of a circle centerd on the negative y-axis.

11.

Answer.
  1. cardioid
    θ 0 π2 π 3π2 2π
    r 1 2 1 0 1
  2. cardioid
    θ 0 π2 π 3π2 2π
    r 1 0 1 2 1
  3. cardioid
    θ 0 π2 π 3π2 2π
    r 1 0 1 2 1
  4. cardioid
    θ 0 π2 π 3π2 2π
    r 1 2 1 0 1

13.

Answer.
  1. limacon
    θ 0 π2 π 3π2 2π
    r 3 2 1 2 3
  2. limacon
    θ 0 π2 π 3π2 2π
    r 1 2 3 2 1
  3. limacon
    θ 0 π2 π 3π2 2π
    r 3 1 1 1 3
  4. limacon
    θ 0 π2 π 3π2 2π
    r 1 1 3 1 1

15.

Answer.
  1. 4-petal rose
    3-petal rose
    8-petal rose
    5 petal rose
    There are n petals if n is odd, and 2n petals if n is even.
  2. n=2: π4,  3π4, 5π4, 7π4;  n=3: π6, 5π6, 3π2; n=4: π8, 3π8, 5π8, 7π8, 9π8, 11π8, 13π8, 15π8; n=5:  π10, π2, 9π10, 13π10, 17π10
  3. 3-petal rose
    3-petal rose
    3-petal rose
    a is the length of the petal.

17.

Answer.
  1. r=±3cos2θ
  2. lemniscate
  3. a is the length of the loop.

19.

Answer.
Archimedean spiral

21.

Answer.
  1. θ 0 π12 π6 π4 π3 5π12 π2 7π12 2π3
    3θ 0 π4 π2 3π4 π 5π4 3π2 7π4 2π
    r 0 22 1 22 0 22 1 22 0
    sinusoidal curve
  2. θ 0 π12 π6 π4 π3 5π12 π2 7π12 2π3
    3θ 0 π4 π2 3π4 π 5π4 3π2 7π4 2π
    r 0 22 1 22 0 22 1 22 0
    three-petal rose

23.

Answer.
  1. θ 0 π4 π2 3π4 π 5π4 3π2 7π4 2π
    y 4 2+2 2 22 0 22 2 2+2 4
    sinusoidal graph
  2. θ 0 π4 π2 3π4 π 5π4 3π2 7π4 2π
    y 4 2+2 2 22 0 22 2 2+2 4
    cardioid

25.

Answer.
circle
circle on polar grid

27.

Answer.
line
line on polar grid

29.

Answer.
circle
circle

31.

Answer.
cardioid
cardioid

33.

Answer.
limaçon
limacon

35.

Answer.
rose
four-petal rose

37.

Answer.
rose
rose

39.

Answer.
limaçon
limacon

41.

Answer.
lemniscate
lemniscate

43.

Answer.
circle
circle

45.

Answer.
arcs of a circle
arcs of circle

47.

Answer.
semicircle
semicircle

49.

Answer.
rose
eight-petal rose

51.

Answer.
cardioid
cardioid

53.

Answer.
parabola
parabola

55.

Answer.
ellipse
ellipse

57.

Answer.
hyperbola
hyperbola

59.

Answer.
r=2+2cos(θ)

61.

Answer.
r=3sin(5θ)

63.

Answer.
r=5sin(θ)

65.

Answer.
r=1+2cos(θ)

67.

Answer.
(0,0), (12,π3), (12,5π3)

69.

Answer.
(0,0),  (32,π4),  (32,5π4)

71.

Answer.
(1,π2),  (1,3π2)

73.

Answer.
(4+22,3π4),  (422,7π4)

75.

Answer.
polar plot

77.

Answer.
conchoid

79.

Answer.
strophoid

81.

Answer.
polar plot

83.

Answer.
The curve has n large loops and n small loops.

10.3 Complex Numbers
Homework 10-3

1.

Answer.
  1. 5i4
  2. 4+i
  3. 5626i

3.

Answer.
3±2i

5.

Answer.
16±116i

7.

Answer.
13+4i

9.

Answer.
0.8+3.8i

11.

Answer.
2010i

13.

Answer.
14+34i

15.

Answer.
46+14i3

17.

Answer.
52

19.

Answer.
22i

21.

Answer.
1+4i

23.

Answer.
353+203i

25.

Answer.
2529+1029i

27.

Answer.
3434i

29.

Answer.
23+53i

31.

Answer.
i

33.

Answer.
  1. 1
  2. 1
  3. i
  4. 1

35.

Answer.
  1. 0
  2. 0

37.

Answer.
  1. 0
  2. 0

39.

Answer.
  1. 0
  2. 0

41.

Answer.
4z2+49

43.

Answer.
x2+6x+10

45.

Answer.
v28v+17

47.

Answer.
complex conjugates

49.

Answer.
complex conjugates

51.

Answer.
disk

53.

Answer.
inequality

55.

Answer.
complex numbers as vectors

57.

Answer.
complex numbers as vectors

59.

Answer.
(a+bi)(c+di)=ac+adi+bci+bdi2=(acbd)+(ad+bc)i

61.

Answer.
z1+z2=(a+bi)+(c+di)=(a+c)+(b+d)i=(c+a)+(d+b)i=(c+di)+(a+bi)=z2+z1
z1z2=(a+bi)(c+di)=(acbd)+(ad+bc)i=(cadb)+(da+cb)i=z2z1

63.

Answer.
  1. z+z¯=(a+bi)+(abi)=2a;    zz¯=(a+bi)(abi)=2bi
  2. zz¯=(a+bi)(abi)=a2+b2=|z|2

65.

Answer.
No. Let t=i and z=i. Then w=t+z=ii=0, so |w|=0, but |t|+|z|=|i|+|i|=1+1=2.

67.

Answer.
  1. 25
  2. x24x1=0

69.

Answer.
  1. 4+3i
  2. x28x+25=0

71.

Answer.
x46x3+23x250x+50=0

73.

Answer.
x47x3+20x219x+13=0

10.4 Polar Form for Complex Numbers
Homework 10-4

1.

Answer.
1,  i,  1, i,  1
powers of i

3.

Answer.
1+2i, 2+i
complex numbers

5.

Answer.
3+3i3

7.

Answer.
1+i

9.

Answer.
2.344.21i

11.

Answer.
5.07+10.88i

13.

Answer.
3(cos(π2)+isin(π2)), 3(cos(3π2)+isin(3π2))

15.

Answer.
23(cos(7π6)+isin(7π6)), 23(cos(11π6)+isin(11π6))

17.

Answer.
4.47(cos(2.68)+isin(2.68)),  4.47(cos(5.82)+isin(5.82))

19.

Answer.
8.60(cos(5.78)+isin(5.78)),  8.60(cos(0.51)+isin(0.51))

21.

Answer.
5(cos(0.93)+isin(0.93)),  5(cos(5.36)+isin(5.36)),  5(cos(2.21)+isin(2.21)),  5(cos(4.07)+isin(4.07))

23.

Answer.
If z=r(cos(θ)+isin(θ)), then z¯=r(cos(2πθ)+isin(2πθ)

25.

Answer.
z1z2=2(cos(π6)+isin(π6))=3+i;  z1z2=8(cos(π2)+isin(π2))=8i

27.

Answer.
z1z2=6(cos(9π10)+isin(9π10));   z1z2=32(cos(3π10)+isin(3π10))

29.

Answer.
z1z2=8;  z1z2=12

31.

Answer.
z1z2=42(cos7π12+isin7π12);   z1z2=22(cos13π12+isin13π12)

33.

Answer.
128128i

35.

Answer.
1281283i

37.

Answer.
512+5123i

39.

Answer.
14+14i

41.

Answer.
2868i

43.

Answer.
  1. 3(cosπ4+isinπ4),  3(cos3π4+isin3π4)
  2. 32+32i,  3232i
square roots

45.

Answer.
  1. 2, 2(cos2π5+isin2π5),  2(cos4π5+isin4π5) ,  2(cos6π5+isin6π5), 2(cos8π5+isin8π5)
  2. 2, 0.618+1.9i, 1.618+1.176i, 1.6181.176i, 0.6181.902i
fifth roots of complex number

47.

Answer.
  1. 4(cosπ18+isinπ18),  4(cos13π18+isin13π18),  4(cos25π18+isin25π18)
  2. 1.97+0.347i,  1.286+1.532i,  0.6841.879i
cube roots of complex number

49.

Answer.
|z|=|cos(θ)+isin(θ)|=cos2(θ)+sin2(θ)=1

51.

Answer.
  1. 1, (cos2π3+isin2π3), (cos4π3+isin4π3)
  2. 1, i, 1, i
  3. 1, (cos2π5+isin2π5), (cos4π5+isin4π5), (cos6π5+isin6π5), (cos8π5+isin8π5)
  4. 1, (cosπ3+isinπ3), (cos2π3+isin2π3), 1, (cos4π3+isin4π3), (cos5π3+isin5π3)

53.

Answer.
(ωk)n=1n(cos(n2πkn)+isin(n2πkn))=1(cos2πk+isin2πk)=1

55.

Answer.
81/4(cos(3π8)+isin(3π8)),  81/4(cos(5π8)+isin(5π8)),  81/4(cos(11π8)+isin(11π8)),  81/4(cos(13π8)+isin(13π8))

57.

Answer.
2,  2(cos(π3)+isin(π3)),  2(cos(2π3)+isin(2π3)),  2,  2(cos(4π3)+isin(4π3)),  2(cos(5π3)+isin(5π3))

59.

Answer.
2(cos(π3)+isin(π3)),  2(cos(2π3)+isin(2π3)),  2(cos(4π3)+isin(4π3)),  2(cos(5π3)+isin(5π3))

61.

Answer.
  1. cos2(θ)sin2(θ)+(2sin(θ)cos(θ))i
  2. cos(2θ)+isin(2θ)
  3. sin(2θ)=2sin(θ)cos(θ);  cos(2θ)=cos2(θ)sin2(θ)

63.

Answer.
  1. ba
  2. ab
  3. 1,  π2

65.

Answer.
  1. z1z2=(acbd)+(ad+bc)i
  2. a=rcos(α), b=rsin(α), c=Rcos(β), d=Rsin(β)
  3. (acbd)+(ad+bc)i=(rRcos(α)cos(β)rRsin(α)sin(β)) +(rRcos(α)sin(β)+rRsin(α)cos(β))i
  4. rR(cos(α+β)+isin(α+β))

10.5 Chapter Summary and Review
Review Problems

1.

Answer.
polar plot

3.

Answer.
polar plot

5.

Answer.
(22,22)

7.

Answer.
(0.241,3.391)

9.

Answer.
(32,3π4)

11.

Answer.
(29,tan1(25)+2π)

13.

Answer.
region on polar grid

15.

Answer.
region on polar grid

17.

Answer.
x2+y2=1

19.

Answer.
x2+y2=(2x+6)2

21.

Answer.
rcos(θ)+rsin(θ)=2

23.

Answer.
tan(θ)=r

25.

Answer.
Circle of radius 3 centered at the origin

27.

Answer.
Circle of radius 3 centered at (3,0)

29.

Answer.
r=4

31.

Answer.
r=4cos(θ)

33.

Answer.
(4,π6),  (4,5π6)

35.

Answer.
(22,3π4) and the pole

37.

Answer.
43i

39.

Answer.
2+4i

41.

Answer.
  1. 1
  2. 1

43.

Answer.
  1. 44
  2. 44

45.

Answer.
(2±i)24(2±i)+5=(4±4i1)(8±4i)+5=0

47.

Answer.
z2+4z+5

49.

Answer.
s210s+41

51.

Answer.
complex numbers

53.

Answer.
  1. 17i
  2. x2+2x+50=0

55.

Answer.
  1. 3+2i
  2. x26x+11=0

57.

Answer.
535i

59.

Answer.
5+5i

61.

Answer.
32(cos(7π4)+isin(7π4))

63.

Answer.
5(cos(π)+isin(π))

65.

Answer.
2(cos(4π3)+isin(4π3))

67.

Answer.
z1z2=16(cos(π)+isin(π))=16, z1z2=4(cos(2π3)+isin(2π3))=223i

69.

Answer.
z1z2=52(cos(π3)+isin(π3))=54532i, z1z2=10(cos(5π6)+isin(5π6))=535i

71.

Answer.
1

73.

Answer.
1100

75.

Answer.
  1. complex square roots
  2. 22+22i,  2222i

77.

Answer.
  1. complex cube roots
  2. 3i,  33232i,  33232i

79.

Answer.
3(cosθ+isinθ), for θ=π6,  π2,  5π6,  7π6,  3π2,  11π6

81.

Answer.
2(cos(θ)+isin(θ)), for θ=π3,  2π3,  4π3,  5π3