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Algebra and Trigonometry

Review Exercises

Algebra and TrigonometryReview Exercises
  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

Graphs of the Sine and Cosine Functions

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

1.

f( x )=3cosx+3 f( x )=3cosx+3

2.

f( x )= 1 4 sinx f( x )= 1 4 sinx

3.

f( x )=3cos( x+ π 6 ) f( x )=3cos( x+ π 6 )

4.

f( x )=2sin( x 2π 3 ) f( x )=2sin( x 2π 3 )

5.

f( x )=3sin( x π 4 )4 f( x )=3sin( x π 4 )4

6.

f( x )=2( cos( x 4π 3 )+1 ) f( x )=2( cos( x 4π 3 )+1 )

7.

f( x )=6sin( 3x π 6 )1 f( x )=6sin( 3x π 6 )1

8.

f( x )=100sin( 50x20 ) f( x )=100sin( 50x20 )

Graphs of the Other Trigonometric Functions

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

9.

f( x )=tanx4 f( x )=tanx4

10.

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

11.

f( x )=3tan( 4x )2 f( x )=3tan( 4x )2

12.

f( x )=0.2cos( 0.1x )+0.3 f( x )=0.2cos( 0.1x )+0.3

For the following exercises, graph two full periods. Identify the period, the phase shift, the amplitude, and asymptotes.

13.

f( x )= 1 3 secx f( x )= 1 3 secx

14.

f( x )=3cotx f( x )=3cotx

15.

f( x )=4csc( 5x ) f( x )=4csc( 5x )

16.

f( x )=8sec( 1 4 x ) f( x )=8sec( 1 4 x )

17.

f( x )= 2 3 csc( 1 2 x ) f( x )= 2 3 csc( 1 2 x )

18.

f( x )=csc( 2x+π ) f( x )=csc( 2x+π )

For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: y=12,000+8,000sin( 0.628x ), y=12,000+8,000sin( 0.628x ), where the domain is the years since 1980 and the range is the population of the city.

19.

What is the largest and smallest population the city may have?

20.

Graph the function on the domain of [ 0,40 ] [ 0,40 ] .

21.

What are the amplitude, period, and phase shift for the function?

22.

Over this domain, when does the population reach 18,000? 13,000?

23.

What is the predicted population in 2007? 2010?

For the following exercises, suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.

24.

Suppose the graph of the displacement function is shown in Figure 1, where the values on the x-axis represent the time in seconds and the y-axis represents the displacement in inches. Give the equation that models the vertical displacement of the weight on the spring.

A graph of a consine function over one period. Graphed on the domain of [0,10]. Range is [-5,5].
Figure 1
25.

At time = 0, what is the displacement of the weight?

26.

At what time does the displacement from the equilibrium point equal zero?

27.

What is the time required for the weight to return to its initial height of 5 inches? In other words, what is the period for the displacement function?

Inverse Trigonometric Functions

For the following exercises, find the exact value without the aid of a calculator.

28.

sin 1 ( 1 ) sin 1 ( 1 )

29.

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

30.

tan −1 ( −1 ) tan −1 ( −1 )

31.

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

32.

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

33.

sin 1 ( cos( π 6 ) ) sin 1 ( cos( π 6 ) )

34.

cos 1 ( tan( 3π 4 ) ) cos 1 ( tan( 3π 4 ) )

35.

sin( sec 1 ( 3 5 ) ) sin( sec 1 ( 3 5 ) )

36.

cot( sin 1 ( 3 5 ) ) cot( sin 1 ( 3 5 ) )

37.

tan( cos 1 ( 5 13 ) ) tan( cos 1 ( 5 13 ) )

38.

sin( cos 1 ( x x+1 ) ) sin( cos 1 ( x x+1 ) )

39.

Graph f( x )=cosx f( x )=cosx and f( x )=secx f( x )=secx on the interval [ 0,2π ) [ 0,2π ) and explain any observations.

40.

Graph f(x)=sinx f(x)=sinx and f( x )=cscx f( x )=cscx and explain any observations.

41.

Graph the function f( x )= x 1 x 3 3! + x 5 5! x 7 7! f( x )= x 1 x 3 3! + x 5 5! x 7 7! on the interval [ 1,1 ] [ 1,1 ] and compare the graph to the graph of f( x )=sinx f( x )=sinx on the same interval. Describe any observations.

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