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  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Try It

7.1 Angles

1.
Graph of a 240-degree angle with a counterclockwise rotation.
2.

3π 2 3π 2

3.

−135° −135°

4.

7π 10 7π 10

5.

α=150° α=150°

6.

β=60° β=60°

7.

7π 6 7π 6

8.

215π 18 =37.525 units 215π 18 =37.525 units

9.

1.88

10.

3π 2 3π 2 rad/s

11.

1655 kilometers per hour

7.2 Right Triangle Trigonometry

1.

7 25 7 25

2.

sin t = 33 65 , cos t = 56 65 , tan t = 33 56 , sec t = 65 56 , csc t = 65 33 , cot t = 56 33 sin t = 33 65 , cos t = 56 65 , tan t = 33 56 , sec t = 65 56 , csc t = 65 33 , cot t = 56 33

3.

sin( π 4 ) = 2 2 ,cos( π 4 )= 2 2 ,tan( π 4 )=1, sec( π 4 ) = 2 ,csc( π 4 )= 2 ,cot( π 4 )=1 sin( π 4 ) = 2 2 ,cos( π 4 )= 2 2 ,tan( π 4 )=1, sec( π 4 ) = 2 ,csc( π 4 )= 2 ,cot( π 4 )=1

4.

2

5.

adjacent=10;opposite=10 3 ; adjacent=10;opposite=10 3 ; missing angle is π 6 π 6

6.

About 52 ft

7.3 Unit Circle

1.

cos(t)= 2 2 ,sin(t)= 2 2 cos(t)= 2 2 ,sin(t)= 2 2

2.

cos(π)=1,sin(π)=0 cos(π)=1,sin(π)=0

3.

sin(t)= 7 25 sin(t)= 7 25

4.

approximately 0.866025403

5.

π 3 π 3

6.
  1. cos(315°)= 2 2 , sin(315°)= 2 2 cos(315°)= 2 2 , sin(315°)= 2 2
  2. cos( π 6 )= 3 2 ,sin( π 6 )= 1 2 cos( π 6 )= 3 2 ,sin( π 6 )= 1 2
7.

( 1 2 , 3 2 ) ( 1 2 , 3 2 )

7.4 The Other Trigonometric Functions

1.

sint= 2 2 ,cost= 2 2 ,tant=1,sect= 2 ,csct= 2 ,cott=1 sint= 2 2 ,cost= 2 2 ,tant=1,sect= 2 ,csct= 2 ,cott=1

2.

sin π 3 = 3 2 cos π 3 = 1 2 tan π 3 = 3 sec π 3 =2 csc π 3 = 2 3 3 cot π 3 = 3 3 sin π 3 = 3 2 cos π 3 = 1 2 tan π 3 = 3 sec π 3 =2 csc π 3 = 2 3 3 cot π 3 = 3 3

3.

sin( 7π 4 )= 2 2 ,cos( 7π 4 )= 2 2 ,tan( 7π 4 )=1, sec( 7π 4 )= 2 ,csc( 7π 4 )= 2 ,cot( 7π 4 )=1 sin( 7π 4 )= 2 2 ,cos( 7π 4 )= 2 2 ,tan( 7π 4 )=1, sec( 7π 4 )= 2 ,csc( 7π 4 )= 2 ,cot( 7π 4 )=1

4.

3 3

5.

2 2

6.

sint sint

7.

cost= 8 17 , sint= 15 17 , tant= 15 8 csct= 17 15 , cott= 8 15 cost= 8 17 , sint= 15 17 , tant= 15 8 csct= 17 15 , cott= 8 15

8.

sint=1,cost=0,tant=Undefined sect=Undefined,csct=1,cott=0 sint=1,cost=0,tant=Undefined sect=Undefined,csct=1,cott=0

9.

sect= 2 ,csct= 2 ,tant=1,cott=1 sect= 2 ,csct= 2 ,tant=1,cott=1

10.

2.414 2.414

7.1 Section Exercises

1.


Graph of a circle with an angle inscribed, showing the initial side, terminal side, and vertex.
3.

Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

5.

Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

7.


Graph of a circle with an angle inscribed.
9.


Graph of a circle with a 135 degree angle inscribed.
11.


Graph of a circle with a 2pi/3 radians angle inscribed.
13.


Graph of a circle with 5pi/6 radians angle inscribed.
15.


Graph of a circle with a –pi/10 radians angle inscribed.
17.

240° 240°


Graph of a circle showing the equivalence of two angles.
19.

4π 3 4π 3


Graph of a circle showing the equivalence of two angles.
21.

2π 3 2π 3


Graph of a circle showing the equivalence of two angles.
23.

7π 2 11.00  in 2 7π 2 11.00  in 2

25.

81π 20 12.72  cm 2 81π 20 12.72  cm 2

27.

20° 20°

29.

60° 60°

31.

−75° −75°

33.

π 2 π 2 radians

35.

−3π −3π radians

37.

π π radians

39.

5π 6 5π 6 radians

41.

5.02π 3 5.26 5.02π 3 5.26 miles

43.

25π 9 8.73 25π 9 8.73 centimeters

45.

21π 10 6.60 21π 10 6.60 meters

47.

104.7198 cm2

49.

0.7697 in2

51.

250° 250°

53.

320° 320°

55.

4π 3 4π 3

57.

8π 9 8π 9

59.

1320 1320 rad/min 210.085 210.085 RPM

61.

7 7 in./s, 4.77 RPM , 28.65 28.65 deg/s

63.

1,809,557.37 mm/min=30.16 m/s 1,809,557.37 mm/min=30.16 m/s

65.

5.76 5.76 miles

67.

120° 120°

69.

794 miles per hour

71.

2,234 miles per hour

73.

11.5 inches

7.2 Section Exercises

1.


A right triangle with side opposite, adjacent, and hypotenuse labeled.
3.

The tangent of an angle is the ratio of the opposite side to the adjacent side.

5.

For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

7.

π 6 π 6

9.

π 4 π 4

11.

b= 20 3 3 ,c= 40 3 3 b= 20 3 3 ,c= 40 3 3

13.

a=10,000,c=10,00.5 a=10,000,c=10,00.5

15.

b= 5 3 3 ,c= 10 3 3 b= 5 3 3 ,c= 10 3 3

17.

5 29 29 5 29 29

19.

5 2 5 2

21.

29 2 29 2

23.

5 41 41 5 41 41

25.

5 4 5 4

27.

41 4 41 4

29.

c=14,b=7 3 c=14,b=7 3

31.

a=15,b=15 a=15,b=15

33.

b=9.9970,c=12.2041 b=9.9970,c=12.2041

35.

a=2.0838,b=11.8177 a=2.0838,b=11.8177

37.

a=55.9808,c=57.9555 a=55.9808,c=57.9555

39.

a=46.6790,b=17.9184 a=46.6790,b=17.9184

41.

a=16.4662,c=16.8341 a=16.4662,c=16.8341

43.

188.3159

45.

200.6737

47.

498.3471 ft

49.

1060.09 ft

51.

27.372 ft

53.

22.6506 ft  

55.

368.7633 ft

7.3 Section Exercises

1.

The unit circle is a circle of radius 1 centered at the origin.

3.

Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, t, t, formed by the terminal side of the angle t t and the horizontal axis.

5.

The sine values are equal.

7.

I

9.

IV

11.

3 2 3 2

13.

1 2 1 2

15.

2 2 2 2

17.

0

19.

-1

21.

3 2 3 2

23.

60° 60°

25.

80° 80°

27.

45° 45°

29.

π 3 π 3

31.

π 3 π 3

33.

π 8 π 8

35.

60°, 60°, Quadrant IV, sin(300°)= 3 2 ,cos(300°)= 1 2 sin(300°)= 3 2 ,cos(300°)= 1 2

37.

45°, 45°, Quadrant II, sin( 135° )= 2 2 ,cos(135°)= 2 2 sin( 135° )= 2 2 ,cos(135°)= 2 2

39.

60°,  60°,  Quadrant II, sin( 120° )= 3 2 ,cos(120°)= 1 2 sin( 120° )= 3 2 ,cos(120°)= 1 2

41.

30°, 30°, Quadrant II, sin( 150° )= 1 2 ,cos(150°)= 3 2 sin( 150° )= 1 2 ,cos(150°)= 3 2

43.

π 6 , π 6 , Quadrant III, sin( 7π 6 )= 1 2 ,cos( 7π 6 )= 3 2 sin( 7π 6 )= 1 2 ,cos( 7π 6 )= 3 2

45.

π 4 , π 4 , Quadrant II, sin( 3π 4 )= 2 2 ,cos( 4π 3 )= 2 2 sin( 3π 4 )= 2 2 ,cos( 4π 3 )= 2 2

47.

π 3 , π 3 , Quadrant II, sin( 2π 3 )= 3 2 ,cos( 2π 3 )= 1 2 sin( 2π 3 )= 3 2 ,cos( 2π 3 )= 1 2

49.

π 4 , π 4 , Quadrant IV, sin( 7π 4 )= 2 2 ,cos( 7π 4 )= 2 2 sin( 7π 4 )= 2 2 ,cos( 7π 4 )= 2 2

51.

77 9 77 9

53.

15 4 15 4

55.

( −10, 10 3 ) ( −10, 10 3 )

57.

( –2.778, 15.757 ) ( –2.778, 15.757 )

59.

[ –1, 1 ] [ –1, 1 ]

61.

sint= 1 2 ,cost= 3 2 sint= 1 2 ,cost= 3 2

63.

sint= 2 2 ,cost= 2 2 sint= 2 2 ,cost= 2 2

65.

sint= 3 2 ,cost= 1 2 sint= 3 2 ,cost= 1 2

67.

sint= 2 2 ,cost= 2 2 sint= 2 2 ,cost= 2 2

69.

sint=0, cost=1 sint=0, cost=1

71.

sint=0.596, cost=0.803 sint=0.596, cost=0.803

73.

sint= 1 2 ,cost= 3 2 sint= 1 2 ,cost= 3 2

75.

sint= 1 2 ,cost= 3 2 sint= 1 2 ,cost= 3 2

77.

sint=0.761,cost=0.649 sint=0.761,cost=0.649

79.

sint=1,cost=0 sint=1,cost=0

81.

−0.1736

83.

0.9511

85.

−0.7071

87.

−0.1392

89.

−0.7660

91.

2 4 2 4

93.

6 4 6 4

95.

2 4 2 4

97.

2 4 2 4

99.

0

101.

( 0,–1 ) ( 0,–1 )

103.

37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

7.4 Section Exercises

1.

Yes, when the reference angle is π 4 π 4 and the terminal side of the angle is in quadrants I and III. Thus, a x= π 4 , 5π 4 , x= π 4 , 5π 4 , the sine and cosine values are equal.

3.

Substitute the sine of the angle in for y y in the Pythagorean Theorem x 2 + y 2 =1. x 2 + y 2 =1. Solve for x x and take the negative solution.

5.

The outputs of tangent and cotangent will repeat every π π units.

7.

2 3 3 2 3 3

9.

3 3

11.

2 2

13.

1

15.

2

17.

3 3 3 3

19.

2 3 3 2 3 3

21.

3 3

23.

2 2

25.

–1

27.

-2

29.

3 3 3 3

31.

2

33.

3 3 3 3

35.

–2

37.

–1

39.

sint= 2 2 3 ,sect=3,csct= 3 2 4 ,tant=2 2 ,cott= 2 4 sint= 2 2 3 ,sect=3,csct= 3 2 4 ,tant=2 2 ,cott= 2 4

41.

sect=2, sect=2, csct= 2 3 3 ,  csct= 2 3 3 ,  tant= 3 ,  tant= 3 ,  cott= 3 3 cott= 3 3

43.

2 2 2 2

45.

3.1

47.

1.4

49.

sint= 2 2 ,cost= 2 2 ,tant=1,cott=1,sect= 2 ,csct= 2 sint= 2 2 ,cost= 2 2 ,tant=1,cott=1,sect= 2 ,csct= 2

51.

sint= 3 2 ,cost= 1 2 ,tant= 3 ,cott= 3 3 ,sect=2,csct= 2 3 3 sint= 3 2 ,cost= 1 2 ,tant= 3 ,cott= 3 3 ,sect=2,csct= 2 3 3

53.

–0.228

55.

–2.414

57.

1.414

59.

1.540

61.

1.556

63.

sin( t )0.79 sin( t )0.79

65.

csct1.16 csct1.16

67.

even

69.

even

71.

sint cost =tant sint cost =tant

73.

13.77 hours, period: 1000π 1000π

75.

3.46 inches

Review Exercises

1.

45° 45°

3.

7π 6 7π 6

5.

10.385 meters

7.

60° 60°

9.

2π 11 2π 11

11.


This is an image of a graph of a circle with a negative angle inscribed.
13.


This is an image of a graph of a circle with an angle inscribed.
15.

1036.73 miles per hour

17.

2 2 2 2

19.

3 3 3 3

21.

72° 72°

23.

a= 10 3 ,c= 2 106 3 a= 10 3 ,c= 2 106 3

25.

6 11 6 11

27.

a= 5 3 2 ,b= 5 2 a= 5 3 2 ,b= 5 2

29.

369.2136 ft

31.

2 2 2 2

33.

60° 60°

35.

3 2 3 2

37.

all real numbers

39.

3 2 3 2

41.

2 3 3 2 3 3

43.

2

45.

–2.5

47.

1 3 1 3

49.

cosine, secant

Practice Test

1.

150° 150°

3.

6.283 centimeters

5.

15° 15°

7.


This is an image of a graph of a circle with an angle inscribed.
9.

3.351 feet per second, 2π 75 2π 75 radians per second

11.

a= 9 2 ,b= 9 3 2 a= 9 2 ,b= 9 3 2

13.

1 2 1 2

15.

real numbers

17.

1

19.

2 2

21.

–0.68

23.

π 3 π 3

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