Skip to Content
OpenStax Logo
Precalculus

Review Exercises

PrecalculusReview Exercises
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  14. A | Basic Functions and Identities
  15. 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
  16. Index

Solving Trigonometric Equations with Identities

For the following exercises, find all solutions exactly that exist on the interval [ 0,2π ). [ 0,2π ).

1.

csc 2 t=3 csc 2 t=3

2.

cos 2 x= 1 4 cos 2 x= 1 4

3.

2sinθ=1 2sinθ=1

4.

tanxsinx+sin( x )=0 tanxsinx+sin( x )=0

5.

9sinω2=4 sin 2 ω 9sinω2=4 sin 2 ω

6.

12tan(ω)= tan 2 (ω) 12tan(ω)= tan 2 (ω)

For the following exercises, use basic identities to simplify the expression.

7.

secxcosx+cosx 1 secx secxcosx+cosx 1 secx

8.

sin 3 x+ cos 2 xsinx sin 3 x+ cos 2 xsinx

For the following exercises, determine if the given identities are equivalent.

9.

sin 2 x+ sec 2 x1= ( 1 cos 2 x )( 1+ cos 2 x ) cos 2 x sin 2 x+ sec 2 x1= ( 1 cos 2 x )( 1+ cos 2 x ) cos 2 x

10.

tan 3 x csc 2 x cot 2 xcosxsinx=1 tan 3 x csc 2 x cot 2 xcosxsinx=1

Sum and Difference Identities

For the following exercises, find the exact value.

11.

tan( 7π 12 ) tan( 7π 12 )

12.

cos( 25π 12 ) cos( 25π 12 )

13.

sin( 70 )cos( 25 )cos( 70 )sin( 25 ) sin( 70 )cos( 25 )cos( 70 )sin( 25 )

14.

cos( 83 )cos( 23 )+sin( 83 )sin( 23 ) cos( 83 )cos( 23 )+sin( 83 )sin( 23 )

For the following exercises, prove the identity.

15.

cos( 4x )cos( 3x )cosx= sin 2 x4 cos 2 x sin 2 x cos( 4x )cos( 3x )cosx= sin 2 x4 cos 2 x sin 2 x

16.

cos(3x) cos 3 x=cosx sin 2 xsinxsin(2x) cos(3x) cos 3 x=cosx sin 2 xsinxsin(2x)

For the following exercise, simplify the expression.

17.

tan( 1 2 x )+tan( 1 8 x ) 1tan( 1 8 x )tan( 1 2 x ) tan( 1 2 x )+tan( 1 8 x ) 1tan( 1 8 x )tan( 1 2 x )

For the following exercises, find the exact value.

18.

cos( sin 1 ( 0 ) cos 1 ( 1 2 ) ) cos( sin 1 ( 0 ) cos 1 ( 1 2 ) )

19.

tan( sin 1 ( 0 )+ sin 1 ( 1 2 ) ) tan( sin 1 ( 0 )+ sin 1 ( 1 2 ) )

Double-Angle, Half-Angle, and Reduction Formulas

For the following exercises, find the exact value.

20.

Find sin( 2θ ), cos( 2θ ), sin( 2θ ), cos( 2θ ), and tan( 2θ ) tan( 2θ ) given cosθ= 1 3 cosθ= 1 3 and θ θ is in the interval [ π 2 ,π ]. [ π 2 ,π ].

21.

Find sin( 2θ ), cos( 2θ ), sin( 2θ ), cos( 2θ ), and tan( 2θ ) tan( 2θ ) given secθ= 5 3 secθ= 5 3 and θ θ is in the interval [ π 2 ,π ]. [ π 2 ,π ].

22.

sin( 7π 8 ) sin( 7π 8 )

23.

sec( 3π 8 ) sec( 3π 8 )

For the following exercises, use Figure 1 to find the desired quantities.

Image of a right triangle. The base is 24, the height is unknown, and the hypotenuse is 25. The angle opposite the base is labeled alpha, and the remaining acute angle is labeled beta.
Figure 1
24.

sin(2β),cos(2β),tan(2β),sin(2α),cos(2α), and tan(2α) sin(2β),cos(2β),tan(2β),sin(2α),cos(2α), and tan(2α)

25.

sin( β 2 ),cos( β 2 ),tan( β 2 ),sin( α 2 ),cos( α 2 ), and tan( α 2 ) sin( β 2 ),cos( β 2 ),tan( β 2 ),sin( α 2 ),cos( α 2 ), and tan( α 2 )

For the following exercises, prove the identity.

26.

2cos( 2x ) sin( 2x ) =cotxtanx 2cos( 2x ) sin( 2x ) =cotxtanx

27.

cotxcos(2x)=sin(2x)+cotx cotxcos(2x)=sin(2x)+cotx

For the following exercises, rewrite the expression with no powers.

28.

cos 2 x sin 4 (2x) cos 2 x sin 4 (2x)

29.

tan 2 x sin 3 x tan 2 x sin 3 x

Sum-to-Product and Product-to-Sum Formulas

For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.

30.

cos( π 3 )sin( π 4 ) cos( π 3 )sin( π 4 )

31.

2sin( 2π 3 )sin( 5π 6 ) 2sin( 2π 3 )sin( 5π 6 )

32.

2cos( π 5 )cos( π 3 ) 2cos( π 5 )cos( π 3 )

For the following exercises, evaluate the sum by using a product formula. Write the exact answer.

33.

sin( π 12 )sin( 7π 12 ) sin( π 12 )sin( 7π 12 )

34.

cos( 5π 12 )+cos( 7π 12 ) cos( 5π 12 )+cos( 7π 12 )

For the following exercises, change the functions from a product to a sum or a sum to a product.

35.

sin(9x)cos(3x) sin(9x)cos(3x)

36.

cos(7x)cos(12x) cos(7x)cos(12x)

37.

sin(11x)+sin(2x) sin(11x)+sin(2x)

38.

cos(6x)+cos(5x) cos(6x)+cos(5x)

Solving Trigonometric Equations

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

39.

tanx+1=0 tanx+1=0

40.

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

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

41.

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

42.

cos 2 xcosx1=0 cos 2 xcosx1=0

43.

2 sin 2 x+5sinx+3=0 2 sin 2 x+5sinx+3=0

44.

cosx5sin( 2x )=0 cosx5sin( 2x )=0

45.

1 sec 2 x +2+ sin 2 x+4 cos 2 x=0 1 sec 2 x +2+ sin 2 x+4 cos 2 x=0

For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval [0,2π). [0,2π). Round to four decimal places.

46.

3 cot 2 x+cotx=1 3 cot 2 x+cotx=1

47.

csc 2 x3cscx4=0 csc 2 x3cscx4=0

For the following exercises, graph each side of the equation to find the zeroes on the interval [0,2π). [0,2π).

48.

20 cos 2 x+21cosx+1=0 20 cos 2 x+21cosx+1=0

49.

sec 2 x2secx=15 sec 2 x2secx=15

Modeling with Trigonometric Equations

For the following exercises, graph the points and find a possible formula for the trigonometric values in the given table.

50.
x x 0 0 1 1 2 2 3 3 4 4 5 5
y y 1 1 6 6 11 11 6 6 1 1 6 6
51.
x x y y
0 0 2 2
1 1 1 1
2 2 2 2
3 3 5 5
4 4 2 2
5 5 1 1
52.
x x y y
3 3 3+2 2 3+2 2
2 2 3 3
1 1 2 2 1 2 2 1
0 0 1 1
1 1 32 2 32 2
2 2 1 1
3 3 −12 2 −12 2
53.

A man with his eye level 6 feet above the ground is standing 3 feet away from the base of a 15-foot vertical ladder. If he looks to the top of the ladder, at what angle above horizontal is he looking?

54.

Using the ladder from the previous exercise, if a 6-foot-tall construction worker standing at the top of the ladder looks down at the feet of the man standing at the bottom, what angle from the horizontal is he looking?

For the following exercises, construct functions that model the described behavior.

55.

A population of lemmings varies with a yearly low of 500 in March. If the average yearly population of lemmings is 950, write a function that models the population with respect to t, t, the month.

56.

Daily temperatures in the desert can be very extreme. If the temperature varies from 90°F 90°F to 30°F 30°F and the average daily temperature first occurs at 10 AM, write a function modeling this behavior.

For the following exercises, find the amplitude, frequency, and period of the given equations.

57.

y=3cos(xπ) y=3cos(xπ)

58.

y=−2sin(16xπ) y=−2sin(16xπ)

For the following exercises, model the described behavior and find requested values.

59.

An invasive species of carp is introduced to Lake Freshwater. Initially there are 100 carp in the lake and the population varies by 20 fish seasonally. If by year 5, there are 625 carp, find a function modeling the population of carp with respect to t, t, the number of years from now.

60.

The native fish population of Lake Freshwater averages 2500 fish, varying by 100 fish seasonally. Due to competition for resources from the invasive carp, the native fish population is expected to decrease by 5% each year. Find a function modeling the population of native fish with respect to t, t, the number of years from now. Also determine how many years it will take for the carp to overtake the native fish population.

Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
Citation information

© Feb 10, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.