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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, simplify the given expression.

1.

cos( x )sinxcotx+ sin 2 x cos( x )sinxcotx+ sin 2 x

2.

sin(x)cos(2x)sin(x)cos(2x) sin(x)cos(2x)sin(x)cos(2x)

3.

csc( θ )cot( θ )( sec 2 θ1 ) csc( θ )cot( θ )( sec 2 θ1 )

4.

cos 2 ( θ ) sin 2 ( θ )( 1+ cot 2 ( θ ) )( 1+ tan 2 ( θ ) ) cos 2 ( θ ) sin 2 ( θ )( 1+ cot 2 ( θ ) )( 1+ tan 2 ( θ ) )

For the following exercises, find the exact value.

5.

cos( 7π 12 ) cos( 7π 12 )

6.

tan( 3π 8 ) tan( 3π 8 )

7.

tan( sin 1 ( 2 2 )+ tan 1 3 ) tan( sin 1 ( 2 2 )+ tan 1 3 )

8.

2sin( π 4 )sin( π 6 ) 2sin( π 4 )sin( π 6 )

9.

cos( 4π 3 +θ ) cos( 4π 3 +θ )

10.

tan( π 4 +θ ) tan( π 4 +θ )

For the following exercises, simplify each expression. Do not evaluate.

11.

cos 2 (32°) tan 2 (32°) cos 2 (32°) tan 2 (32°)

12.

cot( θ 2 ) cot( θ 2 )

For the following exercises, find all exact solutions to the equation on [0,2π). [0,2π).

13.

cos 2 x sin 2 x1=0 cos 2 x sin 2 x1=0

14.

cos 2 x=cosx4 sin 2 x+2sinx3=0 cos 2 x=cosx4 sin 2 x+2sinx3=0

15.

cos( 2x )+ sin 2 x=0 cos( 2x )+ sin 2 x=0

16.

2 sin 2 xsinx=0 2 sin 2 xsinx=0

17.

Rewrite the expression as a product instead of a sum: cos( 2x )+cos( 8x ). cos( 2x )+cos( 8x ).

For the following exercise, rewrite the product as a sum or difference.

18.

8cos( 15x )sin( 3x ) 8cos( 15x )sin( 3x )

For the following exercise, rewrite the sum or difference as a product.

19.

2( sin( 8θ )sin( 4θ ) ) 2( sin( 8θ )sin( 4θ ) )

20.

Find all solutions of tan(x) 3 =0. tan(x) 3 =0.

21.

Find the solutions of sec 2 x2secx=15 sec 2 x2secx=15 on the interval [ 0,2π ) [ 0,2π ) algebraically; then graph both sides of the equation to determine the answer.

For the following exercises, find all solutions exactly on the interval 0θπ 0θπ

22.

2cos( θ 2 )=1 2cos( θ 2 )=1

23.

3 cot( y )=1 3 cot( y )=1

24.

Find sin( 2θ ),cos( 2θ ), sin( 2θ ),cos( 2θ ), and tan( 2θ ) tan( 2θ ) given cotθ= 3 4 cotθ= 3 4 and θ θ is on the interval [ π 2 ,π ]. [ π 2 ,π ].

25.

Find sin( θ 2 ),cos( θ 2 ), sin( θ 2 ),cos( θ 2 ), and tan( θ 2 ) tan( θ 2 ) given cosθ= 7 25 cosθ= 7 25 and θ θ is in quadrant IV.

26.

Rewrite the expression sin 4 x sin 4 x with no powers greater than 1.

For the following exercises, prove the identity.

27.

tan 3 xtanx sec 2 x=tan( x ) tan 3 xtanx sec 2 x=tan( x )

28.

sin( 3x )cosxsin( 2x )= cos 2 xsinx sin 3 x sin( 3x )cosxsin( 2x )= cos 2 xsinx sin 3 x

29.

sin( 2x ) sinx cos( 2x ) cosx =secx sin( 2x ) sinx cos( 2x ) cosx =secx

30.

Plot the points and find a function of the form y=Acos( Bx+C )+D y=Acos( Bx+C )+D that fits the given data.

x x 0 0 1 1 2 2 3 3 4 4 5 5
y y −2 −2 2 2 −2 −2 2 2 −2 −2 2 2
31.

The displacement h(t) h(t) in centimeters of a mass suspended by a spring is modeled by the function h(t)= 1 4 sin(120πt), h(t)= 1 4 sin(120πt), where t t is measured in seconds. Find the amplitude, period, and frequency of this displacement.

32.

A woman is standing 300 feet away from a 2000-foot building. If she looks to the top of the building, at what angle above horizontal is she looking? A bored worker looks down at her from the 15th floor (1500 feet above her). At what angle is he looking down at her? Round to the nearest tenth of a degree.

33.

Two frequencies of sound are played on an instrument governed by the equation n(t)=8cos(20πt)cos(1000πt). n(t)=8cos(20πt)cos(1000πt). What are the period and frequency of the “fast” and “slow” oscillations? What is the amplitude?

34.

The average monthly snowfall in a small village in the Himalayas is 6 inches, with the low of 1 inch occurring in July. Construct a function that models this behavior. During what period is there more than 10 inches of snowfall?

35.

A spring attached to a ceiling is pulled down 20 cm. After 3 seconds, wherein it completes 6 full periods, the amplitude is only 15 cm. Find the function modeling the position of the spring t t seconds after being released. At what time will the spring come to rest? In this case, use 1 cm amplitude as rest.

36.

Water levels near a glacier currently average 9 feet, varying seasonally by 2 inches above and below the average and reaching their highest point in January. Due to global warming, the glacier has begun melting faster than normal. Every year, the water levels rise by a steady 3 inches. Find a function modeling the depth of the water t t months from now. If the docks are 2 feet above current water levels, at what point will the water first rise above the docks?

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