Solution manual for trigonometry 9th edition by larson

a The interval >4,f denotes the set of all real numbers greater than or equal to 4.. a The interval >5, 2 denotes the set of all real numbers greater than or equal to 5 and less than

8 5, 7, 73, 0, 3.12, ,54 3, 12, 5

(a) Natural numbers: 12, 5 (b) Whole numbers: 0, 12, 5 (c) Integers: 7, 0, 3, 12, 5(d) Rational numbers:  7, 73, 0, 3.12, ,54 3, 12, 5(e) Irrational numbers: 5

9 2.01, 0.666 , 13, 0.010110111 , 1, 6

(a) Natural numbers: 1 (b) Whole numbers: 1 (c) Integers: 13, 1,  6(d) Rational numbers: 2.01, 0.666 , 13, 1,  6(e) Irrational numbers: 0.010110111

25, 17,  , 9, 3.12, S, 7, 11.1, 13(a) Natural numbers: 25, 9, 7, 13(b) Whole numbers: 25, 9, 7, 13

(c) Integers: 25, 17, 9, 7, 13(d) Rational numbers:

12 5

25, 17,  , 9, 3.12, 7, 11.1, 13(e) Irrational numbers: 12S

(c) The interval is unbounded

INSTRUCTOR USE ONLY

(b)

(c) The interval is unbounded

19 (a) The interval >4,f denotes the set of all real 

numbers greater than or equal to 4

(b)

(c) The interval is unbounded

20 (a) f, 2denotes the set of all real numbers less

than 2

(b)

(c) The interval is unbounded

21 (a) The inequality 2  x  2denotes the set of all

real numbers greater than 2 and less than 2

(b)

(c) The interval is bounded

22 (a) The inequality 0 x d denotes the set of all real 6

numbers greater than zero and less than or equal to 6

(b)

(c) The interval is bounded

23 (a) The interval >5, 2 denotes the set of all real

numbers greater than or equal to 5 and less than 2

(b)

(c) The interval is bounded

24 (a) The interval 1, 2@denotes the set of all real

numbers greater than  and less than or equal to 2 1(b)

(c) The interval is bounded

64 9 7x

(a) 9 7 3 9 21 30(b) 9 7 3 9 21 12

65 x2  5x 4(a) 2

1 5 1 4 1 5 4 10

        (b) 2



(a) 1 1 2

1 1 0



Division by zero is undefined

68 x  3  x 3 0 Additive Inverse Property

69 2x  3 2˜x  2˜ 3 Distributive Property

70 z  2  0 z  2 Additive Identity Property

71

3 3 Associative Property of Multiplication

3 Commutative Property of Multiplication

x y x y

x y

˜

72 1 1

7 7 12 7 7 12 Associative Property of Multiplication

1 12 Multiplicative Inverse Property

12 Multiplicative Identity Property

˜

INSTRUCTOR USE ONLY

The expression is positive

is neither negative nor irrational For example, a price

of 1.80 or $ 2 would not be reasonable The types

of real numbers to describe a range of lengths are nonnegative and rational because lengths are often not in whole unit amounts but in parts, such as 2.3 centimeters

or 3

4inch

79 False Because 0 is nonnegative but not positive, not

every nonnegative number is positive

80 False Two numbers with different signs will always

have a product less than zero

81 (a)

(b) (i) As n approaches 0, the value of 5 n increases

without bound (approaches infinity)

(ii) As n increases without bound (approaches

infinity), the value of 5 n approaches 0

Section P.2 Solving Equations

x

x x x

x x x

x x

x x x x x

x x

y y y

y y y

y y

y y y y y

y y y y

9

x x

x x x x x

23 24 24

4

24 23 23

9623

x x

x x x x x

x x

x x

x x x x

The second method is easier The fractions are eliminated in the first step

x x

x

x x x

x x

x x x x

x x

x x

x x x x

x x

x x

x x

x x x x

x x

x x

x x x x x

x x

x x x x x

5 05

x x

 

2x 3 0 Ÿ x 

31

2 2

3 4 3 4

9 364

4 2

x x



 r  r

x x x x



 r rr

  The only solution of the equation is x 2

43

2 2



   



 r  r

46

2 2

47

2 2

2

9 18 3

1231

3213213213613

x x

x x

x x x x x x

48

2 2

2 2

x x

x x x x x x

 r

2 2

51 2x2  x 1 0

2

2

42

1 1 4 2 1

2 2

1 1 84

1,

b b ac x

2

2

42

2 2 4 1 2

2 1

2 2 3

1 32

b b ac x

10 10 4 1 22

2 1

10 100 882

10 2 3

5 32

b b ac x

2

2

42

2

2

42

2

2

42

2

2

42

2

2

42

2

2

42

80 80 4 25 61

2 25

80 6400 610050

12 12 4 1 25

2 1

12 2 11

6 112

b b ac y

2

2

42

14 14 4 1 36

2 1

14 52

7 132

b b ac z

2 5.10.976, 0.643

2 0.0050.101 0.0063410.012.137, 18.063

b b ac x

2 3.220.08 369.031

2.995, 2.9716.44

b b ac x

70

2 2

71 2

3 81 Extract square roots

3 9

x x

x x x x x

2

2

11 4 11 4

0 Complete the square

3

x x

x x

x x x x x

3 2 3 2 3 2

3

333

x x x x x



 r  r

1 2

1 Extract square roots

11For 1 :

0 1 No solutionFor 1 :





r        

76.

2 2

3 97

4 4

x x

x x x

 

  r   r

5 3 5 3

x x x x



10 0

10 1626

x x x x

x x x x x

 

 

   

84. 3

3

124 3

x x x

     





 r  r 

91

3 2 2 3

3

2 2

5 27

5 27

5 27

5 91414

x x x x x x







 r

3 2 2

2 2

INSTRUCTOR USE ONLY

2 2

28.440.43265.8

y x

x x x x

33.150.44973.8

y x

x x x x





| Because the height of the male is about 73.8 inches or

6 feet 2 inches, it is possible the femur belongs to the missing man

99 False—See Example 14 on page 123

102 (a) The formula for volume of the glass cube is

V = Length × Width × Height

The volume of water in the cube is the length × width × height of the water

So, the volume is x˜x˜x 3 x x2  3 (b) Given the equation x x2 3 320 The dimensions of the glass cube can be found by

solving for x

So, the capacity of the cube is equal to 3

V x

103 Equivalent equations are derived from the substitution

principle and simplification techniques They have the same solution(s)

2x 3 and 28 x are equivalent equations 5

Section P.3 The Cartesian Plane and Graphs of Equations

13 3, 4

14 12, 0

15 x ! 0 and y  in Quadrant IV 0

16 x  and 4 y ! in Quadrant II 0

17 x, is in the second Quadrant means that y x y is , 

in Quadrant III

18 x y, ,xy ! means x and y have the same signs 0

This occurs in Quadrant I or III

2 2

3.9 9.5 8.2 2.6

13.4 10.8179.56 116.64296.2

26 (a) The distance between 1, 1 and 9, 1 is 10

The distance between 9, 1 and 9, 4 is 3

The distance between 1, 1 and 9, 4 is

d 

| The plane flies about 192 kilometers

2 2

42 18 50 12

24 382020

2 50545

d   



| The pass is about 45 yards

No, the point is not on the graph

Yes, the point is on the graph

9 4 20

13 20

z

No, the point is not on the graph

2 2

4 1642

x x

x r 2, 0 , 2, 0 



  3, 0

y-intercept:

2 2

0 339

0, 9

y y y



51 y 5x 6

x-intercept:

6 5

0 5 6

6 5

x x x

0 8 3

3 8

x x



 4, 0

2 1 0

x x

y-intercept

55 y 3x  7

x-intercept:

7 3

0 3 7

0 3 70

x x

x x

x

 

  10, 0

y-intercept: 0 10

10 10

y    

0, 10 

INSTRUCTOR USE ONLY

y y

 r

  1, 0

y-intercepts: 2 0 1

1

y y

 r

x x

y y y

x x

-axis symmetry1

y y x

x x

y y

x x

64 42

93 y 500,000 40,000 , 0t d t d 8

94 y 8000 900 , 0t d t d 6

INSTRUCTOR USE ONLY

Because the line is close to the points, the model fits the data well

(b) Graphically: The point 90, 75.4 represents a life expectancy of 75.4 years in 1990 

0.002 0.5 46.6 0.002 90 0.5 90 46.6 75.4

y  t  t    

So, the life expectancy in 1990 was about 75.4 years

(c) Graphically: The point 94.6, 76.0 represents a life expectancy of 76 years during the year 1994 

Algebraically: 2

2 2

0.002 0.5 46.676.0 0.002 0.5 46.6

So, 94.6t or 155.4.t Since 155.4 is not in the domain, the solution is t 94.6, which is the year 1994

(d) When t 115:

2

20.002 0.5 46.6 0.002 115 0.5 115 46.6 77.65

y  t  t   The life expectancy using the model is 77.65 years, which is slightly less than the given projection of 78.9 years

(e) Answers will vary Sample answer: No Because the model is quadratic, the life expectancies begin to decrease

after a certain point

y 414.8 103.7 25.9 11.5 6.5 4.1 2.9 2.1 1.6 1.3 1.0



 r  r

1 2

1 Extract square roots

1 1For :

0 No solutionFor :





r        

76.

... to the missing man

99 False—See Example 14 on page 123

102 (a) The formula for volume of the glass cube is

V = Length × Width × Height

The volume... x2 3 320 The dimensions of the glass cube can be found by

solving for x

So, the capacity of the cube is equal to 3

V