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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

For the following exercises, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.

1.

f( x )=0.5sinx f( x )=0.5sinx

2.

f( x )=5cosx f( x )=5cosx

3.

f( x )=5sinx f( x )=5sinx

4.

f( x )=sin( 3x ) f( x )=sin( 3x )

5.

f( x )=cos( x+ π 3 )+1 f( x )=cos( x+ π 3 )+1

6.

f( x )=5sin( 3( x π 6 ) )+4 f( x )=5sin( 3( x π 6 ) )+4

7.

f( x )=3cos( 1 3 x 5π 6 ) f( x )=3cos( 1 3 x 5π 6 )

8.

f( x )=tan( 4x ) f( x )=tan( 4x )

9.

f( x )=2tan( x 7π 6 )+2 f( x )=2tan( x 7π 6 )+2

10.

f( x )=πcos( 3x+π ) f( x )=πcos( 3x+π )

11.

f( x )=5csc( 3x ) f( x )=5csc( 3x )

12.

f( x )=πsec( π 2 x ) f( x )=πsec( π 2 x )

13.

f( x )=2csc( x+ π 4 )3 f( x )=2csc( x+ π 4 )3

For the following exercises, determine the amplitude, period, and midline of the graph, and then find a formula for the function.

14.

Give in terms of a sine function.

A graph of two periods of a sine function, graphed from -2 to 2. Range is [-6,-2], period is 2, and amplitude is 2.
15.

Give in terms of a sine function.

A graph of two periods of a sine function, graphed over -2 to 2. Range is [-2,2], period is 2, and amplitude is 2.
16.

Give in terms of a tangent function.

A graph of two periods of a tangent function, graphed over -3pi/4 to 5pi/4. Vertical asymptotes at x=-pi/4, 3pi/4. Period is pi.

For the following exercises, find the amplitude, period, phase shift, and midline.

17.

y=sin( π 6 x+π )3 y=sin( π 6 x+π )3

18.

y=8sin( 7π 6 x+ 7π 2 )+6 y=8sin( 7π 6 x+ 7π 2 )+6

19.

The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 68°F at midnight and the high and low temperatures during the day are 80°F and 56°F, respectively. Assuming t t is the number of hours since midnight, find a function for the temperature, D, D, in terms of t. t.

20.

Water is pumped into a storage bin and empties according to a periodic rate. The depth of the water is 3 feet at its lowest at 2:00 a.m. and 71 feet at its highest, which occurs every 5 hours. Write a cosine function that models the depth of the water as a function of time, and then graph the function for one period.

For the following exercises, find the period and horizontal shift of each function.

21.

g( x )=3tan( 6x+42 ) g( x )=3tan( 6x+42 )

22.

n( x )=4csc( 5π 3 x 20π 3 ) n( x )=4csc( 5π 3 x 20π 3 )

23.

Write the equation for the graph in Figure 1 in terms of the secant function and give the period and phase shift.

A graph of 2 periods of a secant function, graphed over -2 to 2. The period is 2 and there is no phase shift.
Figure 1
24.

If tanx=3, tanx=3, find tan( x ). tan( x ).

25.

If secx=4, secx=4, find sec( x ). sec( x ).

For the following exercises, graph the functions on the specified window and answer the questions.

26.

Graph m( x )=sin( 2x )+cos( 3x ) m( x )=sin( 2x )+cos( 3x ) on the viewing window [ 10,10 ] [ 10,10 ] by [ 3,3 ]. [ 3,3 ]. Approximate the graph’s period.

27.

Graph n( x )=0.02sin( 50πx ) n( x )=0.02sin( 50πx ) on the following domains in x: x: [ 0,1 ] [ 0,1 ] and [ 0,3 ]. [ 0,3 ]. Suppose this function models sound waves. Why would these views look so different?

28.

Graph f( x )= sinx x f( x )= sinx x on [ 0.5,0.5 ] [ 0.5,0.5 ]and explain any observations.

For the following exercises, let f( x )= 3 5 cos( 6x ). f( x )= 3 5 cos( 6x ).

29.

What is the largest possible value for f( x )? f( x )?

30.

What is the smallest possible value for f( x )? f( x )?

31.

Where is the function increasing on the interval [ 0,2π ]? [ 0,2π ]?

For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift.

32.

Sine curve with amplitude 3, period π 3 , π 3 , and phase shift ( h,k )=( π 4 ,2 ) ( h,k )=( π 4 ,2 )

33.

Cosine curve with amplitude 2, period π 6 , π 6 , and phase shift ( h,k )=( π 4 ,3 ) ( h,k )=( π 4 ,3 )

For the following exercises, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.

34.

f( x )=5cos( 3x )+4sin( 2x ) f( x )=5cos( 3x )+4sin( 2x )

35.

f( x )= e sint f( x )= e sint

For the following exercises, find the exact value.

36.

sin 1 ( 3 2 ) sin 1 ( 3 2 )

37.

tan 1 ( 3 ) tan 1 ( 3 )

38.

cos 1 ( 3 2 ) cos 1 ( 3 2 )

39.

cos 1 ( sin( π ) ) cos 1 ( sin( π ) )

40.

cos 1 ( tan( 7π 4 ) ) cos 1 ( tan( 7π 4 ) )

41.

cos( sin 1 ( 12x ) ) cos( sin 1 ( 12x ) )

42.

cos 1 ( 0.4 ) cos 1 ( 0.4 )

43.

cos( tan 1 ( x 2 ) ) cos( tan 1 ( x 2 ) )

For the following exercises, suppose sint= x x+1 . sint= x x+1 . Evaluate the following expressions.

44.

tant tant

45.

csct csct

46.

Given Figure 2, find the measure of angle θ θ to three decimal places. Answer in radians.

An illustration of a right triangle with angle theta. Opposite the angle theta is a side with length 12, adjacent to the angle theta is a side with length 19.
Figure 2

For the following exercises, determine whether the equation is true or false.

47.

arcsin( sin( 5π 6 ) )= 5π 6 arcsin( sin( 5π 6 ) )= 5π 6

48.

arccos( cos( 5π 6 ) )= 5π 6 arccos( cos( 5π 6 ) )= 5π 6

49.

The grade of a road is 7%. This means that for every horizontal distance of 100 feet on the road, the vertical rise is 7 feet. Find the angle the road makes with the horizontal in radians.

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