Skip to Content
OpenStax Logo
Precalculus

A | Basic Functions and Identities

PrecalculusA | Basic Functions and Identities
  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

Graphs of the Parent Functions

Three graphs side-by-side. From left to right, graph of the identify function, square function, and square root function. All three graphs extend from -4 to 4 on each axis.
Figure A1
Three graphs side-by-side. From left to right, graph of the cubic function, cube root function, and reciprocal function. All three graphs extend from -4 to 4 on each axis.
Figure A2
Three graphs side-by-side. From left to right, graph of the absolute value function, exponential function, and natural logarithm function. All three graphs extend from -4 to 4 on each axis.
Figure A3

Graphs of the Trigonometric Functions

Three graphs of trigonometric functions side-by-side. From left to right, graph of the sine function, cosine function, and tangent function. Graphs of the sine and cosine functions extend from negative two pi to two pi on the x-axis and two to negative two on the y-axis. Graph of tangent extends from negative pi to pi on the x-axis and four to negative 4 on the y-axis.
Figure A4
Three graphs of trigonometric functions side-by-side. From left to right, graph of the cosecant function, secant function, and cotangent function. Graphs of the cosecant function and secant function extend from negative two pi to two pi on the x-axis and ten to negative ten on the y-axis. Graph of cotangent extends from negative two pi to two pi on the x-axis and twenty-five to negative twenty-five on the y-axis.
Figure A5
Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse sine function, inverse cosine function, and inverse tangent function. Graphs of the inverse sine and inverse tangent extend from negative pi over two to pi over two on the x-axis and pi over two to negative pi over two on the y-axis. Graph of inverse cosine extends from negative pi over two to pi on the x-axis and pi to negative pi over two on the y-axis.
Figure A6
Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse cosecant function, inverse secant function, and inverse cotangent function.
Figure A7

Trigonometric Identities

Pythagorean Identities cos 2 θ+ sin 2 θ=1 1+ tan 2 θ= sec 2 θ 1+ cot 2 θ= csc 2 θ cos 2 θ+ sin 2 θ=1 1+ tan 2 θ= sec 2 θ 1+ cot 2 θ= csc 2 θ
Even-Odd Identities cos(−θ)=cos θ sec(−θ)=sec θ sin(−θ)=−sin θ tan(−θ)=−tan θ csc(−θ)=−csc θ cot(−θ)=−cot θ cos(−θ)=cos θ sec(−θ)=sec θ sin(−θ)=−sin θ tan(−θ)=−tan θ csc(−θ)=−csc θ cot(−θ)=−cot θ
Cofunction Identities cos θ=sin( π 2 −θ ) sin θ=cos( π 2 −θ ) tan θ=cot( π 2 −θ ) cot θ=tan( π 2 −θ ) sec θ=csc( π 2 −θ ) csc θ=sec( π 2 −θ ) cos θ=sin( π 2 −θ ) sin θ=cos( π 2 −θ ) tan θ=cot( π 2 −θ ) cot θ=tan( π 2 −θ ) sec θ=csc( π 2 −θ ) csc θ=sec( π 2 −θ )
Fundamental Identities tan θ= sin θ cos θ sec θ= 1 cos θ csc θ= 1 sin θ cot θ= 1 tan θ = cos θ sin θ tan θ= sin θ cos θ sec θ= 1 cos θ csc θ= 1 sin θ cot θ= 1 tan θ = cos θ sin θ
Sum and Difference Identities cos(α+β)=cos α cos β−sin α sin β cos(α−β)=cos α cos β+sin α sin β sin(α+β)=sin α cos β+cos α sin β sin(α−β)=sin α cos β−cos α sin β tan(α+β)= tan α+tan β 1−tan α tan β tan(α−β)= tan α−tan β 1+tan α tan β cos(α+β)=cos α cos β−sin α sin β cos(α−β)=cos α cos β+sin α sin β sin(α+β)=sin α cos β+cos α sin β sin(α−β)=sin α cos β−cos α sin β tan(α+β)= tan α+tan β 1−tan α tan β tan(α−β)= tan α−tan β 1+tan α tan β
Double-Angle Formulas sin(2θ)=2 sin θ cos θ cos(2θ)= cos 2 θ− sin 2 θ cos(2θ)=1−2  sin 2 θ cos(2θ)=2  cos 2 θ−1 tan(2θ)= 2 tan θ 1− tan 2 θ sin(2θ)=2 sin θ cos θ cos(2θ)= cos 2 θ− sin 2 θ cos(2θ)=1−2  sin 2 θ cos(2θ)=2  cos 2 θ−1 tan(2θ)= 2 tan θ 1− tan 2 θ
Half-Angle Formulas sin  α 2 =± 1−cos α 2 cos  α 2 =± 1+cos α 2 tan  α 2 =± 1−cos α 1+cos α tan  α 2 = sin α 1+cos α tan  α 2 = 1−cos α sin α sin  α 2 =± 1−cos α 2 cos  α 2 =± 1+cos α 2 tan  α 2 =± 1−cos α 1+cos α tan  α 2 = sin α 1+cos α tan  α 2 = 1−cos α sin α
Reduction Formulas sin 2 θ= 1−cos( 2θ ) 2 cos 2 θ= 1+cos( 2θ ) 2 tan 2 θ= 1−cos( 2θ ) 1+cos( 2θ ) sin 2 θ= 1−cos( 2θ ) 2 cos 2 θ= 1+cos( 2θ ) 2 tan 2 θ= 1−cos( 2θ ) 1+cos( 2θ )
Product-to-Sum Formulas cos α cos β= 1 2 [ cos(α−β)+cos(α+β) ] sin α cos β= 1 2 [ sin(α+β)+sin(α−β) ] sin α sin β= 1 2 [ cos(α−β)−cos(α+β) ] cos α sin β= 1 2 [ sin(α+β)−sin(α−β) ] cos α cos β= 1 2 [ cos(α−β)+cos(α+β) ] sin α cos β= 1 2 [ sin(α+β)+sin(α−β) ] sin α sin β= 1 2 [ cos(α−β)−cos(α+β) ] cos α sin β= 1 2 [ sin(α+β)−sin(α−β) ]
Sum-to-Product Formulas sin α+sin β=2 sin( α+β 2 ) cos( α−β 2 ) sin α−sin β=2 sin( α−β 2 ) cos( α+β 2 ) cos α−cos β=−2 sin( α+β 2 ) sin( α−β 2 ) cos α+cos β=2 cos( α+β 2 ) cos( α−β 2 ) sin α+sin β=2 sin( α+β 2 ) cos( α−β 2 ) sin α−sin β=2 sin( α−β 2 ) cos( α+β 2 ) cos α−cos β=−2 sin( α+β 2 ) sin( α−β 2 ) cos α+cos β=2 cos( α+β 2 ) cos( α−β 2 )
Law of Sines sin α a = sin β b = sin γ c a sin α = b sin β = c sin γ sin α a = sin β b = sin γ c a sin α = b sin β = c sin γ
Law of Cosines a 2 = b 2 + c 2 −2bc cos α b 2 = a 2 + c 2 −2ac cos β c 2 = a 2 + b 2 −2ab cos γ a 2 = b 2 + c 2 −2bc cos α b 2 = a 2 + c 2 −2ac cos β c 2 = a 2 + b 2 −2ab cos γ
Table A1
PreviousNext
Order a print copy
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.