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Algebra and trigonometry
OpenStax
@
Preface
0.1
About openstax
0.2
About openstax resources
0.3
Customization
0.4
Errata
0.5
Format
0.6
Coverage and scope
0.7
Development overview
0.8
Accuracy of the content
0.9
Pedagogical foundations and features
0.10
Learning objectives
0.11
Narrative text
0.12
Examples
0.13
Figures
0.14
Supporting features
0.15
Section exercises
0.16
Chapter review features
0.17
Additional resources
0.18
Student and instructor resources
0.19
Partner resources
0.20
About the authors
0.21
Lead author, senior content expert
0.22
Contributing authors
0.23
Reviewers
1
Proofs, identities, and toolkit functions
1.1
Appendix
1.2
Important proofs and derivations
1.3
Trigonometric identities
1.4
Toolkit functions
1.5
Trigonometric functions
Prerequisites
Introduction to prerequisites
1
Real numbers: algebra essentials
1.1
Classifying a real number
1.2
Irrational numbers
1.3
Real numbers
1.4
Sets of numbers as subsets
1.5
Performing calculations using the order of operations
1.6
Using properties of real numbers
1.7
Commutative properties
1.8
Associative properties
1.9
Distributive property
1.10
Identity properties
1.11
Inverse properties
1.12
Evaluating algebraic expressions
1.13
Formulas
1.14
Simplifying algebraic expressions
1.15
Key concepts
1.16
Verbal
1.17
Numeric
1.18
Algebraic
1.19
Real-world applications
1.20
Technology
1.21
Extensions
2
Exponents and scientific notation
2.1
Using the product rule of exponents
2.2
Using the quotient rule of exponents
2.3
Using the power rule of exponents
2.4
Using the zero exponent rule of exponents
2.5
Using the negative rule of exponents
2.6
Finding the power of a product
2.7
Finding the power of a quotient
2.8
Simplifying exponential expressions
2.9
Using scientific notation
2.10
Converting from scientific to standard notation
2.11
Using scientific notation in applications
2.12
Key equations
2.13
Key concepts
2.14
Section exercises
2.15
Verbal
2.16
Numeric
2.17
Algebraic
2.18
Real-world applications
2.19
Technology
2.20
Extensions
3
Radicals and rational exponents
3.1
Evaluating square roots
3.2
Using the product rule to simplify square roots
3.3
Using the quotient rule to simplify square roots
3.4
Adding and subtracting square roots
3.5
Rationalizing denominators
3.6
Using rational roots
3.7
Understanding n Th roots
3.8
Using rational exponents
3.9
Key concepts
3.10
Section exercises
3.11
Verbal
3.12
Numeric
3.13
Algebraic
3.14
Real-world applications
3.15
Extensions
4
Polynomials
4.1
Identifying the degree and leading coefficient of polynomials
4.2
Adding and subtracting polynomials
4.3
Multiplying polynomials
4.4
Multiplying polynomials using the distributive property
4.5
Using foil to multiply binomials
4.6
Perfect square trinomials
4.7
Difference of squares
4.8
Performing operations with polynomials of several variables
4.9
Key equations
4.10
Key concepts
4.11
Section exercises
4.12
Verbal
4.13
Algebraic
4.14
Real-world applications
4.15
Extensions
5
Factoring polynomials
5.1
Factoring the greatest common factor of a polynomial
5.2
Factoring a trinomial with leading coefficient 1
5.3
Factoring by grouping
5.4
Factoring a perfect square trinomial
5.5
Factoring a difference of squares
5.6
Factoring the sum and difference of cubes
5.7
Factoring expressions with fractional or negative exponents
5.8
Key equations
5.9
Verbal
5.10
Algebraic
5.11
Real-world applications
5.12
Extensions
6
Rational expressions
6.1
Simplifying rational expressions
6.2
Multiplying rational expressions
6.3
Dividing rational expressions
6.4
Adding and subtracting rational expressions
6.5
Simplifying complex rational expressions
6.6
Key concepts
6.7
Section exercises
6.8
Verbal
6.9
Algebraic
6.10
Real-world applications
6.11
Extensions
6.12
Chapter review exercises
6.13
Real Numbers: Algebra Essentials
6.14
Exponents and Scientific Notation
6.15
Radicals and Rational Expressions
6.16
Polynomials
6.17
Factoring Polynomials
6.18
Rational Expressions
6.19
Chapter practice test
Equations and inequalities
Introduction to equations and inequalities
1
The rectangular coordinate systems and graphs
1.1
Plotting ordered pairs in the cartesian coordinate system
1.2
Graphing equations by plotting points
1.3
Graphing equations with a graphing utility
1.4
Finding x- Intercepts and y- Intercepts
1.5
Using the distance formula
1.6
Using the midpoint formula
1.7
Key concepts
1.8
Section exercises
1.9
Verbal
1.10
Algebraic
1.11
Graphical
1.12
Numeric
1.13
Technology
1.14
Extensions
1.15
Real-world applications
2
Linear equations in one variable
2.1
Solving linear equations in one variable
2.2
Solving a rational equation
2.3
Finding a linear equation
2.4
The slope of a line
2.5
The point-slope formula
2.6
Standard form of a line
2.7
Vertical and horizontal lines
2.8
Determining whether graphs of lines are parallel or perpendicular
2.9
Writing the equations of lines parallel or perpendicular to a given
2.10
Key concepts
2.11
Section exercises
2.12
Verbal
2.13
Algebraic
2.14
Graphical
2.15
Numeric
2.16
Technology
2.17
Extensions
2.18
Real-world applications
3
Models and applications
3.1
Setting up a linear equation to solve a real-world application
3.2
Using a formula to solve a real-world application
3.3
Key concepts
3.4
Section exercises
3.5
Verbal
3.6
Real-world applications
4
Complex numbers
4.1
Expressing square roots of negative numbers as multiples of i
4.2
Plotting a complex number on the complex plane
4.3
Adding and subtracting complex numbers
4.4
Multiplying complex numbers
4.5
Multiplying a complex number by a real number
4.6
Multiplying complex numbers together
4.7
Dividing complex numbers
4.8
Simplifying powers of i
4.9
Key concepts
4.10
Section exercises
4.11
Verbal
4.12
Algebraic
4.13
Graphical
4.14
Numeric
4.15
Technology
4.16
Extensions
5
Quadratic equations
5.1
Solving quadratic equations by factoring
5.2
Solving quadratics with a leading coefficient of 1
5.3
Solving a quadratic equation by factoring when the leading coefficient
5.4
Using the square root property
5.5
Completing the square
5.6
Using the quadratic formula
5.7
The discriminant
5.8
Using the pythagorean theorem
5.9
Key equations
5.10
Key concepts
5.11
Section exercises
5.12
Verbal
5.13
Algebraic
5.14
Technology
5.15
Extensions
5.16
Real-world applications
6
Other types of equations
6.1
Solving equations involving rational exponents
6.2
Solving equations using factoring
6.3
Solving radical equations
6.4
Solving an absolute value equation
6.5
Solving other types of equations
6.6
Solving equations in quadratic form
6.7
Solving rational equations resulting in a quadratic
6.8
Key concepts
6.9
Section exercises
6.10
Verbal
6.11
Algebraic
6.12
Extensions
6.13
Real-world applications
7
Linear inequalities and absolute value inequalities
7.1
Using interval notation
7.2
Using the properties of inequalities
7.3
Solving inequalities in one variable algebraically
7.4
Understanding compound inequalities
7.5
Solving absolute value inequalities
7.6
Key concepts
7.7
Section exercises
7.8
Verbal
7.9
Algebraic
7.10
Graphical
7.11
Numeric
7.12
Technology
7.13
Extensions
7.14
Real-world applications
7.15
Chapter review exercises
7.16
The Rectangular Coordinate Systems and Graphs
7.17
Linear Equations in One Variable
7.18
Models and Applications
7.19
Complex Numbers
7.20
Quadratic Equations
7.21
Other Types of Equations
7.22
Linear Inequalities and Absolute Value Inequalities
7.23
Chapter practice test
Functions
Introduction to functions
1
Functions and function notation
1.1
Determining whether a relation represents a function
1.2
Using function notation
1.3
Representing functions using tables
1.4
Finding input and output values of a function
1.5
Evaluation of functions in algebraic forms
1.6
Evaluating functions expressed in formulas
1.7
Evaluating a function given in tabular form
1.8
Finding function values from a graph
1.9
Determining whether a function is one-to-one
1.10
Using the vertical line test
1.11
Using the horizontal line test
1.12
Identifying basic toolkit functions
1.13
Key equations
1.14
Key concepts
1.15
Section exercises
1.16
Verbal
1.17
Algebraic
1.18
Graphical
1.19
Numeric
1.20
Technology
1.21
Real-world applications
2
Domain and range
2.1
Finding the domain of a function defined by an equation
2.2
Using notations to specify domain and range
2.3
Finding domain and range from graphs
2.4
Finding domains and ranges of the toolkit functions
2.5
Graphing piecewise-defined functions
2.6
Key concepts
2.7
Section exercises
2.8
Verbal
2.9
Algebraic
2.10
Graphical
2.11
Numeric
2.12
Technology
2.13
Extension
2.14
Real-world applications
3
Rates of change and behavior of graphs
3.1
Finding the average rate of change of a function
3.2
Using a graph to determine where a function is increasing, decreasing,
3.3
Analyzing the toolkit functions for increasing or decreasing intervals
3.4
Use a graph to locate the absolute maximum and absolute minimum
3.5
Key equations
3.6
Key concepts
3.7
Section exercises
3.8
Verbal
3.9
Algebraic
3.10
Graphical
3.11
Numeric
3.12
Technology
3.13
Extension
3.14
Real-world applications
4
Composition of functions
4.1
Combining functions using algebraic operations
4.2
Create a function by composition of functions
4.3
Evaluating composite functions
4.4
Evaluating composite functions using tables
4.5
Evaluating composite functions using graphs
4.6
Evaluating composite functions using formulas
4.7
Finding the domain of a composite function
4.8
Decomposing a composite function into its component functions
4.9
Key equation
4.10
Key concepts
4.11
Section exercises
4.12
Verbal
4.13
Algebraic
4.14
Graphical
4.15
Numeric
4.16
Extensions
4.17
Real-world applications
5
Transformation of functions
5.1
Graphing functions using vertical and horizontal shifts
5.2
Identifying vertical shifts
5.3
Identifying horizontal shifts
5.4
Combining vertical and horizontal shifts
5.5
Graphing functions using reflections about the axes
5.6
Determining even and odd functions
5.7
Graphing functions using stretches and compressions
5.8
Vertical stretches and compressions
5.9
Horizontal stretches and compressions
5.10
Performing a sequence of transformations
5.11
Key equations
5.12
Key concepts
5.13
Section exercises
5.14
Verbal
5.15
Algebraic
5.16
Graphical
5.17
Numeric
6
Absolute value functions
6.1
Understanding absolute value
6.2
Graphing an absolute value function
6.3
Solving an absolute value equation
6.4
Key concepts
6.5
Section exercises
6.6
Verbal
6.7
Algebraic
6.8
Graphical
6.9
Technology
6.10
Extensions
6.11
Real-world applications
7
Inverse functions
7.1
Verifying that two functions are inverse functions
7.2
Finding domain and range of inverse functions
7.3
Finding and evaluating inverse functions
7.4
Inverting tabular functions
7.5
Evaluating the inverse of a function, given a graph of the original
7.6
Finding inverses of functions represented by formulas
7.7
Finding inverse functions and their graphs
7.8
Key concepts
7.9
Section exercises
7.10
Verbal
7.11
Algebraic
7.12
Graphical
7.13
Numeric
7.14
Technology
7.15
Real-world applications
7.16
Chapter review exercises
7.17
Functions and Function Notation
7.18
Domain and Range
7.19
Rates of Change and Behavior of Graphs
7.20
Composition of Functions
7.21
Transformation of Functions
7.22
Absolute Value Functions
7.23
Inverse Functions
7.24
Practice test
Linear functions
Introduction to linear functions
1
Linear functions
1.1
Representing linear functions
1.2
Representing a linear function in word form
1.3
Representing a linear function in function notation
1.4
Representing a linear function in tabular form
1.5
Representing a linear function in graphical form
1.6
Determining whether a linear function is increasing, decreasing, or
1.7
Interpreting slope as a rate of change
1.8
Writing and interpreting an equation for a linear function
1.9
Modeling real-world problems with linear functions
1.10
Graphing linear functions
1.11
Graphing a function by plotting points
1.12
Graphing a function using y- Intercept and slope
1.13
Graphing a function using transformations
1.14
Vertical stretch or compression
1.15
Vertical shift
1.16
Writing the equation for a function from the graph of a line
1.17
Finding the x -intercept of a line
1.18
Describing horizontal and vertical lines
1.19
Determining whether lines are parallel or perpendicular
1.20
Writing the equation of a line parallel or perpendicular to a given
1.21
Writing equations of parallel lines
1.22
Writing equations of perpendicular lines
1.23
Key concepts
1.24
Section exercises
1.25
Verbal
1.26
Algebraic
1.27
Graphical
1.28
Numeric
1.29
Technology
1.30
Extensions
1.31
Real-world applications
2
Modeling with linear functions
2.1
Building linear models from verbal descriptions
2.2
Using a given intercept to build a model
2.3
Using a given input and output to build a model
2.4
Using a diagram to build a model
2.5
Modeling a set of data with linear functions
2.6
Key concepts
2.7
Section exercises
2.8
Verbal
2.9
Algebraic
2.10
Graphical
2.11
Numeric
2.12
Real-world applications
3
Fitting linear models to data
3.1
Drawing and interpreting scatter plots
3.2
Finding the line of best fit
3.3
Recognizing interpolation or extrapolation
3.4
Finding the line of best fit using a graphing utility
3.5
Distinguishing between linear and nonlinear models
3.6
Fitting a regression line to a set of data
3.7
Key concepts
3.8
Section exercises
3.9
Verbal
3.10
Algebraic
3.11
Graphical
3.12
Numeric
3.13
Technology
3.14
Extensions
3.15
Real-world applications
3.16
Chapter review exercises
3.17
Linear Functions
3.18
Modeling with Linear Functions
3.19
Fitting Linear Models to Data
3.20
Chapter practice test
Polynomial and rational functions
Introduction to polynomial and rational functions
1
Quadratic functions
1.1
Recognizing characteristics of parabolas
1.2
Understanding how the graphs of parabolas are related to their quadratic
1.3
Finding the domain and range of a quadratic function
1.4
Determining the maximum and minimum values of quadratic functions
1.5
Finding the x - and y -intercepts of a quadratic function
1.6
Rewriting quadratics in standard form
1.7
Key equations
1.8
Key concepts
1.9
Section exercises
1.10
Verbal
1.11
Algebraic
1.12
Graphical
1.13
Numeric
1.14
Technology
1.15
Extensions
1.16
Real-world applications
2
Power functions and polynomial functions
2.1
Identifying power functions
2.2
Identifying end behavior of power functions
2.3
Identifying polynomial functions
2.4
Identifying the degree and leading coefficient of a polynomial function
2.5
Identifying end behavior of polynomial functions
2.6
Identifying local behavior of polynomial functions
2.7
Comparing smooth and continuous graphs
2.8
Key equations
2.9
Key concepts
2.10
Section exercises
2.11
Verbal
2.12
Algebraic
2.13
Graphical
2.14
Numeric
2.15
Technology
2.16
Extensions
2.17
Real-world applications
3
Graphs of polynomial functions
3.1
Recognizing characteristics of graphs of polynomial functions
3.2
Using factoring to find zeros of polynomial functions
3.3
Identifying zeros and their multiplicities
3.4
Determining end behavior
3.5
Understanding the relationship between degree and turning points
3.6
Graphing polynomial functions
3.7
Using the intermediate value theorem
3.8
Writing formulas for polynomial functions
3.9
Using local and global extrema
3.10
Key concepts
3.11
Section exercises
3.12
Verbal
3.13
Algebraic
3.14
Graphical
3.15
Technology
3.16
Extensions
3.17
Real-world applications
4
Dividing polynomials
4.1
Using long division to divide polynomials
4.2
Using synthetic division to divide polynomials
4.3
Using polynomial division to solve application problems
4.4
Key equations
4.5
Key concepts
4.6
Section exercises
4.7
Verbal
4.8
Algebraic
4.9
Graphical
4.10
Technology
4.11
Extensions
4.12
Real-world applications
5
Zeros of polynomial functions
5.1
Evaluating a polynomial using the remainder theorem
5.2
Using the factor theorem to solve a polynomial equation
5.3
Using the rational zero theorem to find rational zeros
5.4
Finding the zeros of polynomial functions
5.5
Using the fundamental theorem of algebra
5.6
Using the linear factorization theorem to find polynomials with given
5.7
Using descartes’ rule of signs
5.8
Solving real-world applications
5.9
Key concepts
5.10
Section exercises
5.11
Verbal
5.12
Algebraic
5.13
Graphical
5.14
Numeric
5.15
Technology
5.16
Extensions
5.17
Real-world applications
6
Rational functions
6.1
Using arrow notation
6.2
Local behavior of f ( x ) = 1 x
6.3
End behavior of f ( x ) = 1 x
6.4
Solving applied problems involving rational functions
6.5
Finding the domains of rational functions
6.6
Identifying vertical asymptotes of rational functions
6.7
Vertical asymptotes
6.8
Removable discontinuities
6.9
Identifying horizontal asymptotes of rational functions
6.10
Graphing rational functions
6.11
Writing rational functions
6.12
Key equations
6.13
Key concepts
6.14
Section exercises
6.15
Verbal
6.16
Algebraic
6.17
Graphical
6.18
Numeric
6.19
Technology
6.20
Extensions
6.21
Real-world applications
7
Inverses and radical functions
7.1
Finding the inverse of a polynomial function
7.2
Restricting the domain to find the inverse of a polynomial function
7.3
Solving applications of radical functions
7.4
Solving applications of radical functions
7.5
Determining the domain of a radical function composed with other functions
7.6
Finding inverses of rational functions
7.7
Key concepts
7.8
Section exercises
7.9
Verbal
7.10
Algebraic
7.11
Graphical
7.12
Technology
7.13
Extensions
7.14
Real-world applications
8
Modeling using variation
8.1
Solving direct variation problems
8.2
Solving inverse variation problems
8.3
Solving problems involving joint variation
8.4
Key equations
8.5
Key concepts
8.6
Section exercises
8.7
Verbal
8.8
Algebraic
8.9
Numeric
8.10
Technology
8.11
Extensions
8.12
Real-world applications
8.13
Chapter review exercises
8.14
Quadratic Functions
8.15
Power Functions and Polynomial Functions
8.16
Graphs of Polynomial Functions
8.17
Dividing Polynomials
8.18
Zeros of Polynomial Functions
8.19
Rational Functions
8.20
Inverses and Radical Functions
8.21
Modeling Using Variation
8.22
Chapter test
Exponential and logarithmic functions
Introduction to exponential and logarithmic functions
1
Exponential functions
1.1
Identifying exponential functions
1.2
Defining an exponential function
1.3
Evaluating exponential functions
1.4
Defining exponential growth
1.5
Finding equations of exponential functions
1.6
Applying the compound-interest formula
1.7
Evaluating functions with base e
1.8
Investigating continuous growth
1.9
Key equations
1.10
Key concepts
1.11
Section exercises
1.12
Verbal
1.13
Algebraic
1.14
Numeric
1.15
Technology
1.16
Extensions
1.17
Real-world applications
2
Graphs of exponential functions
2.1
Graphing exponential functions
2.2
Graphing transformations of exponential functions
2.3
Graphing a vertical shift
2.4
Graphing a horizontal shift
2.5
Graphing a stretch or compression
2.6
Graphing reflections
2.7
Summarizing translations of the exponential function
2.8
Key equations
2.9
Key concepts
2.10
Section exercises
2.11
Verbal
2.12
Algebraic
2.13
Graphical
2.14
Numeric
2.15
Technology
2.16
Extensions
3
Logarithmic functions
3.1
Converting from logarithmic to exponential form
3.2
Converting from exponential to logarithmic form
3.3
Evaluating logarithms
3.4
Using common logarithms
3.5
Using natural logarithms
3.6
Key equations
3.7
Key concepts
3.8
Section exercises
3.9
Verbal
3.10
Algebraic
3.11
Numeric
3.12
Technology
3.13
Extensions
3.14
Real-world applications
4
Graphs of logarithmic functions
4.1
Finding the domain of a logarithmic function
4.2
Graphing logarithmic functions
4.3
Graphing transformations of logarithmic functions
4.4
Graphing a horizontal shift of f ( x ) = log b ( x )
4.5
Graphing a vertical shift of y = log b ( x )
4.6
Graphing stretches and compressions of y = log b ( x )
4.7
Graphing reflections of f ( x ) = log b ( x )
4.8
Summarizing translations of the logarithmic function
4.9
Key equations
4.10
Key concepts
4.11
Section exercises
4.12
Verbal
4.13
Algebraic
4.14
Graphical
4.15
Technology
4.16
Extensions
5
Logarithmic properties
5.1
Using the product rule for logarithms
5.2
Using the quotient rule for logarithms
5.3
Using the power rule for logarithms
5.4
Expanding logarithmic expressions
5.5
Condensing logarithmic expressions
5.6
Using the change-of-base formula for logarithms
5.7
Key equations
5.8
Key concepts
5.9
Section exercises
5.10
Verbal
5.11
Algebraic
5.12
Numeric
5.13
Extensions
6
Exponential and logarithmic equations
6.1
Using like bases to solve exponential equations
6.2
Rewriting equations so all powers have the same base
6.3
Solving exponential equations using logarithms
6.4
Equations containing e
6.5
Extraneous solutions
6.6
Using the definition of a logarithm to solve logarithmic equations
6.7
Using the one-to-one property of logarithms to solve logarithmic equations
6.8
Solving applied problems using exponential and logarithmic equations
6.9
Key equations
6.10
Key concepts
6.11
Section exercises
6.12
Verbal
6.13
Algebraic
6.14
Graphical
6.15
Technology
6.16
Extensions
7
Exponential and logarithmic models
7.1
Modeling exponential growth and decay
7.2
Half-life
7.3
Radiocarbon dating
7.4
Calculating doubling time
7.5
Using newton’s law of cooling
7.6
Using logistic growth models
7.7
Choosing an appropriate model for data
7.8
Expressing an exponential model in base e
7.9
Key equations
7.10
Key concepts
7.11
Section exercises
7.12
Verbal
7.13
Numeric
7.14
Technology
7.15
Extensions
7.16
Real-world applications
8
Fitting exponential models to data
8.1
Building an exponential model from data
8.2
Building a logarithmic model from data
8.3
Building a logistic model from data
8.4
Key concepts
8.5
Section exercises
8.6
Verbal
8.7
Graphical
8.8
Numeric
8.9
Technology
8.10
Extensions
8.11
Chapter review exercises
8.12
Exponential Functions
8.13
Graphs of Exponential Functions
8.14
Logarithmic Functions
8.15
Graphs of Logarithmic Functions
8.16
Logarithmic Properties
8.17
Exponential and Logarithmic Equations
8.18
Exponential and Logarithmic Models
8.19
Fitting Exponential Models to Data
8.20
Practice test
The unit circle: sine and cosine functions
Introduction to the unit circle: sine and cosine functions
1
Angles
1.1
Drawing angles in standard position
1.2
Converting between degrees and radians
1.3
Relating arc lengths to radius
1.4
Using radians
1.5
Identifying special angles measured in radians
1.6
Converting between radians and degrees
1.7
Finding coterminal angles
1.8
Finding coterminal angles measured in radians
1.9
Determining the length of an arc
1.10
Finding the area of a sector of a circle
1.11
Use linear and angular speed to describe motion on a circular path
1.12
Key equations
1.13
Key concepts
1.14
Section exercises
1.15
Verbal
1.16
Graphical
1.17
Algebraic
1.18
Real-world applications
1.19
Extensions
2
Right triangle trigonometry
2.1
Using right triangles to evaluate trigonometric functions
2.2
Reciprocal functions
2.3
Finding trigonometric functions of special angles using side lengths
2.4
Using equal cofunction of complements
2.5
Using trigonometric functions
2.6
Using right triangle trigonometry to solve applied problems
2.7
Key equations
2.8
Key concepts
2.9
Section exercises
2.10
Verbal
2.11
Algebraic
2.12
Graphical
2.13
Technology
2.14
Extensions
2.15
Real-world applications
3
Unit circle
3.1
Finding trigonometric functions using the unit circle
3.2
Defining sine and cosine functions from the unit circle
3.3
Finding sines and cosines of angles on an axis
3.4
The pythagorean identity
3.5
Finding sines and cosines of special angles
3.6
Finding sines and cosines of 45° Angles
3.7
Finding sines and cosines of 30° And 60° Angles
3.8
Using a calculator to find sine and cosine
3.9
Identifying the domain and range of sine and cosine functions
3.10
Finding reference angles
3.11
Using reference angles
3.12
Using reference angles to evaluate trigonometric functions
3.13
Using reference angles to find coordinates
3.14
Key equations
3.15
Key concepts
3.16
Section exercises
3.17
Verbal
3.18
Algebraic
3.19
Numeric
3.20
Graphical
3.21
Technology
3.22
Extensions
3.23
Real-world applications
4
The other trigonometric functions
4.1
Finding exact values of the trigonometric functions secant, cosecant,
4.2
Using reference angles to evaluate tangent, secant, cosecant, and
4.3
Using even and odd trigonometric functions
4.4
Recognizing and using fundamental identities
4.5
Alternate forms of the pythagorean identity
4.6
Evaluating trigonometric functions with a calculator
4.7
Key equations
4.8
Key concepts
4.9
Section exercises
4.10
Verbal
4.11
Algebraic
4.12
Graphical
4.13
Technology
4.14
Extensions
4.15
Real-world applications
4.16
Chapter review exercises
4.17
Angles
4.18
Right Triangle Trigonometry
4.19
Unit Circle
4.20
The Other Trigonometric Functions
4.21
Chapter practice test
Periodic functions
Introduction to periodic functions
1
Graphs of the sine and cosine functions
1.1
Graphing sine and cosine functions
1.2
Investigating sinusoidal functions
1.3
Determining the period of sinusoidal functions
1.4
Determining amplitude
1.5
Analyzing graphs of variations of y = sin x And y = cos x
1.6
Graphing variations of y = sin x And y = cos x
1.7
Using transformations of sine and cosine functions
1.8
Key equations
1.9
Key concepts
1.10
Section exercises
1.11
Verbal
1.12
Graphical
1.13
Algebraic
1.14
Technology
1.15
Real-world applications
2
Graphs of the other trigonometric functions
2.1
Analyzing the graph of y = tan x
2.2
Graphing variations of y = tan x
2.3
Graphing one period of a stretched or compressed tangent function
2.4
Graphing one period of a shifted tangent function
2.5
Analyzing the graphs of y = sec x And y = csc x
2.6
Graphing variations of y = sec x And y = csc x
2.7
Analyzing the graph of y = cot x
2.8
Graphing variations of y = cot x
2.9
Using the graphs of trigonometric functions to solve real-world problems
2.10
Key equations
2.11
Key concepts
2.12
Section exercises
2.13
Verbal
2.14
Algebraic
2.15
Graphical
2.16
Technology
2.17
Real-world applications
3
Inverse trigonometric functions
3.1
Understanding and using the inverse sine, cosine, and tangent functions
3.2
Finding the exact value of expressions involving the inverse sine, cosine
3.3
Using a calculator to evaluate inverse trigonometric functions
3.4
Finding exact values of composite functions with inverse trigonometric
3.5
Evaluating compositions of the form f ( f −1 ( y )) and f −1 ( f ( x )
3.6
Evaluating compositions of the form f −1 ( g ( x ))
3.7
Evaluating compositions of the form f ( g −1 ( x ))
3.8
Key concepts
3.9
Section exercises
3.10
Verbal
3.11
Algebraic
3.12
Extensions
3.13
Graphical
3.14
Real-world applications
3.15
Chapter review exercises
3.16
Graphs of the Sine and Cosine Functions
3.17
Graphs of the Other Trigonometric Functions
3.18
Inverse Trigonometric Functions
3.19
Chapter practice test
Trigonometric identities and equations
Introduction to trigonometric identities and equations
1
Solving trigonometric equations with identities
1.1
Verifying the fundamental trigonometric identities
1.2
Using algebra to simplify trigonometric expressions
1.3
Key equations
1.4
Key concepts
1.5
Section exercises
1.6
Verbal
1.7
Algebraic
1.8
Extensions
2
Sum and difference identities
2.1
Using the sum and difference formulas for cosine
2.2
Using the sum and difference formulas for sine
2.3
Using the sum and difference formulas for tangent
2.4
Using sum and difference formulas for cofunctions
2.5
Using the sum and difference formulas to verify identities
2.6
Key equations
2.7
Key concepts
2.8
Section exercises
2.9
Verbal
2.10
Algebraic
2.11
Graphical
2.12
Technology
2.13
Extensions
3
Double-angle, half-angle, and reduction formulas
3.1
Using double-angle formulas to find exact values
3.2
Using double-angle formulas to verify identities
3.3
Use reduction formulas to simplify an expression
3.4
Using half-angle formulas to find exact values
3.5
Key equations
3.6
Key concepts
3.7
Section exercises
3.8
Verbal
3.9
Algebraic
3.10
Technology
3.11
Extensions
4
Sum-to-product and product-to-sum formulas
4.1
Expressing products as sums
4.2
Expressing products as sums for cosine
4.3
Expressing the product of sine and cosine as a sum
4.4
Expressing products of sines in terms of cosine
4.5
Expressing sums as products
4.6
Key equations
4.7
Key concepts
4.8
Section exercises
4.9
Verbal
4.10
Algebraic
4.11
Numeric
4.12
Technology
4.13
Extensions
5
Solving trigonometric equations
5.1
Solving linear trigonometric equations in sine and cosine
5.2
Solving equations involving a single trigonometric function
5.3
Solve trigonometric equations using a calculator
5.4
Solving trigonometric equations in quadratic form
5.5
Solving trigonometric equations using fundamental identities
5.6
Solving trigonometric equations with multiple angles
5.7
Solving right triangle problems
5.8
Key concepts
5.9
Section exercises
5.10
Verbal
5.11
Algebraic
5.12
Graphical
5.13
Technology
5.14
Extensions
5.15
Real-world applications
5.16
Chapter review exercises
5.17
Solving Trigonometric Equations with Identities
5.18
Sum and Difference Identities
5.19
Double-Angle, Half-Angle, and Reduction Formulas
5.20
Sum-to-Product and Product-to-Sum Formulas
5.21
Solving Trigonometric Equations
5.22
Practice test
Further applications of trigonometry
Introduction to further applications of trigonometry
1
Non-right triangles: law of sines
1.1
Using the law of sines to solve oblique triangles
1.2
Using the law of sines to solve ssa triangles
1.3
Finding the area of an oblique triangle using the sine function
1.4
Solving applied problems using the law of sines
1.5
Key equations
1.6
Key concepts
1.7
Section exercises
1.8
Verbal
1.9
Algebraic
1.10
Graphical
1.11
Extensions
1.12
Real-world applications
2
Non-right triangles: law of cosines
2.1
Using the law of cosines to solve oblique triangles
2.2
Solving applied problems using the law of cosines
2.3
Using heron’s formula to find the area of a triangle
2.4
Key equations
2.5
Key concepts
2.6
Section exercises
2.7
Verbal
2.8
Algebraic
2.9
Graphical
2.10
Extensions
2.11
Real-world applications
3
Polar coordinates
3.1
Plotting points using polar coordinates
3.2
Converting from polar coordinates to rectangular coordinates
3.3
Converting from rectangular coordinates to polar coordinates
3.4
Transforming equations between polar and rectangular forms
3.5
Identify and graph polar equations by converting to rectangular equations
3.6
Key equations
3.7
Key concepts
3.8
Section exercises
3.9
Verbal
3.10
Algebraic
3.11
Graphical
3.12
Technology
3.13
Extensions
4
Polar coordinates: graphs
4.1
Testing polar equations for symmetry
4.2
Graphing polar equations by plotting points
4.3
Finding zeros and maxima
4.4
Investigating circles
4.5
Investigating cardioids
4.6
Investigating limaçons
4.7
Investigating lemniscates
4.8
Investigating rose curves
4.9
Investigating the archimedes’ spiral
4.10
Summary of curves
4.11
Key concepts
4.12
Section exercises
4.13
Verbal
4.14
Graphical
4.15
Technology
4.16
Extensions
5
Polar form of complex numbers
5.1
Plotting complex numbers in the complex plane
5.2
Finding the absolute value of a complex number
5.3
Writing complex numbers in polar form
5.4
Converting a complex number from polar to rectangular form
5.5
Finding products of complex numbers in polar form
5.6
Finding quotients of complex numbers in polar form
5.7
Finding powers of complex numbers in polar form
5.8
Finding roots of complex numbers in polar form
5.9
Key concepts
5.10
Section exercises
5.11
Verbal
5.12
Algebraic
5.13
Graphical
5.14
Technology
6
Parametric equations
6.1
Parameterizing a curve
6.2
Eliminating the parameter
6.3
Eliminating the parameter from polynomial, exponential, and logarithmic
6.4
Eliminating the parameter from trigonometric equations
6.5
Finding cartesian equations from curves defined parametrically
6.6
Finding parametric equations for curves defined by rectangular equations
6.7
Key concepts
6.8
Section exercises
6.9
Verbal
6.10
Algebraic
6.11
Technology
6.12
Extensions
7
Parametric equations: graphs
7.1
Graphing parametric equations by plotting points
7.2
Applications of parametric equations
7.3
Key concepts
7.4
Section exercises
7.5
Verbal
7.6
Graphical
7.7
Technology
7.8
Extensions
8
Vectors
8.1
A geometric view of vectors
8.2
Finding magnitude and direction
8.3
Performing vector addition and scalar multiplication
8.4
Multiplying by a scalar
8.5
Finding component form
8.6
Finding the unit vector in the direction of v
8.7
Performing operations with vectors in terms of i And j
8.8
Performing operations on vectors in terms of i And j
8.9
Calculating the component form of a vector: direction
8.10
Finding the dot product of two vectors
8.11
Key concepts
8.12
Section exercises
8.13
Verbal
8.14
Algebraic
8.15
Graphical
8.16
Extensions
8.17
Real-world applications
8.18
Chapter review exercises
8.19
Non-right Triangles: Law of Sines
8.20
Non-right Triangles: Law of Cosines
8.21
Polar Coordinates
8.22
Polar Coordinates: Graphs
8.23
Polar Form of Complex Numbers
8.24
Parametric Equations
8.25
Parametric Equations: Graphs
8.26
Vectors
8.27
Practice test
Systems of equations and inequalities
Introduction to systems of equations and inequalities
1
Systems of linear equations: two variables
1.1
Introduction to systems of equations
1.2
Solving systems of equations by graphing
1.3
Solving systems of equations by substitution
1.4
Solving systems of equations in two variables by the addition method
1.5
Identifying inconsistent systems of equations containing two variables
1.6
Expressing the solution of a system of dependent equations containing
1.7
Using systems of equations to investigate profits
1.8
Key concepts
1.9
Section exercises
1.10
Verbal
1.11
Algebraic
1.12
Graphical
1.13
Technology
1.14
Extensions
1.15
Real-world applications
2
Systems of linear equations: three variables
2.1
Solving systems of three equations in three variables
2.2
Identifying inconsistent systems of equations containing three variables
2.3
Expressing the solution of a system of dependent equations containing
2.4
Key concepts
2.5
Section exercises
2.6
Verbal
2.7
Algebraic
2.8
Extensions
2.9
Real-world applications
3
Systems of nonlinear equations and inequalities: two variables
3.1
Solving a system of nonlinear equations using substitution
3.2
Intersection of a parabola and a line
3.3
Intersection of a circle and a line
3.4
Solving a system of nonlinear equations using elimination
3.5
Graphing a nonlinear inequality
3.6
Graphing a system of nonlinear inequalities
3.7
Key concepts
3.8
Section exercises
3.9
Verbal
3.10
Algebraic
3.11
Graphical
3.12
Extensions
3.13
Technology
3.14
Real-world applications
4
Partial fractions
4.1
Decomposing P ( x ) Q ( x ) Where Q(x) Has only nonrepeated linear
4.2
Decomposing P ( x ) Q ( x ) Where Q(x) Has repeated linear factors
4.3
Decomposing P ( x ) Q ( x ) , Where Q(x) Has a nonrepeated irreducible
4.4
Decomposing P ( x ) Q ( x ) When Q(x) Has a repeated irreducible
4.5
Key concepts
4.6
Section exercises
4.7
Verbal
4.8
Algebraic
4.9
Extensions
5
Matrices and matrix operations
5.1
Finding the sum and difference of two matrices
5.2
Describing matrices
5.3
Adding and subtracting matrices
5.4
Finding scalar multiples of a matrix
5.5
Finding the product of two matrices
5.6
Key concepts
5.7
Section exercises
5.8
Verbal
5.9
Algebraic
5.10
Technology
5.11
Extensions
6
Solving systems with gaussian elimination
6.1
Writing the augmented matrix of a system of equations
6.2
Writing a system of equations from an augmented matrix
6.3
Performing row operations on a matrix
6.4
Solving a system of linear equations using matrices
6.5
Key concepts
6.6
Section exercises
6.7
Verbal
6.8
Algebraic
6.9
Extensions
6.10
Real-world applications
7
Solving systems with inverses
7.1
Finding the inverse of a matrix
7.2
Finding the multiplicative inverse using matrix multiplication
7.3
Finding the multiplicative inverse by augmenting with the identity
7.4
Finding the multiplicative inverse of 2×2 matrices using a formula
7.5
Finding the multiplicative inverse of 3×3 matrices
7.6
Solving a system of linear equations using the inverse of a matrix
7.7
Key equations
7.8
Key concepts
7.9
Section exercises
7.10
Verbal
7.11
Algebraic
7.12
Technology
7.13
Extensions
7.14
Real-world applications
8
Solving systems with cramer's rule
8.1
Evaluating the determinant of a 2×2 matrix
8.2
Using cramer’s rule to solve a system of two equations in two variables
8.3
Evaluating the determinant of a 3 × 3 matrix
8.4
Using cramer’s rule to solve a system of three equations in three
8.5
Understanding properties of determinants
8.6
Key concepts
8.7
Section exercises
8.8
Verbal
8.9
Algebraic
8.10
Technology
8.11
Real-world applications
8.12
Review exercises
8.13
Systems of Linear Equations: Two Variables
8.14
Systems of Linear Equations: Three Variables
8.15
Systems of Nonlinear Equations and Inequalities: Two Variables
8.16
Partial Fractions
8.17
Matrices and Matrix Operations
8.18
Solving Systems with Gaussian Elimination
8.19
Solving Systems with Inverses
8.20
Solving Systems with Cramer's Rule
8.21
Practice test
Analytic geometry
Introduction to analytic geometry
1
The ellipse
1.1
Writing equations of ellipses in standard form
1.2
Deriving the equation of an ellipse centered at the origin
1.3
Writing equations of ellipses centered at the origin in standard form
1.4
Writing equations of ellipses not centered at the origin
1.5
Graphing ellipses centered at the origin
1.6
Graphing ellipses not centered at the origin
1.7
Solving applied problems involving ellipses
1.8
Key equations
1.9
Key concepts
1.10
Section exercises
1.11
Verbal
1.12
Algebraic
1.13
Graphical
1.14
Extensions
1.15
Real-world applications
2
The hyperbola
2.1
Locating the vertices and foci of a hyperbola
2.2
Deriving the equation of an ellipse centered at the origin
2.3
Writing equations of hyperbolas in standard form
2.4
Hyperbolas centered at the origin
2.5
Hyperbolas not centered at the origin
2.6
Graphing hyperbolas centered at the origin
2.7
Graphing hyperbolas not centered at the origin
2.8
Solving applied problems involving hyperbolas
2.9
Key equations
2.10
Key concepts
2.11
Section exercises
2.12
Verbal
2.13
Algebraic
2.14
Graphical
2.15
Extensions
2.16
Real-world applications
3
The parabola
3.1
Graphing parabolas with vertices at the origin
3.2
Writing equations of parabolas in standard form
3.3
Graphing parabolas with vertices not at the origin
3.4
Solving applied problems involving parabolas
3.5
Key equations
3.6
Key concepts
3.7
Section exercises
3.8
Verbal
3.9
Algebraic
3.10
Graphical
3.11
Extensions
3.12
Real-world applications
4
Rotation of axes
4.1
Identifying nondegenerate conics in general form
4.2
Finding a new representation of the given equation after rotating through
4.3
Writing equations of rotated conics in standard form
4.4
Identifying conics without rotating axes
4.5
Key equations
4.6
Key concepts
4.7
Section exercises
4.8
Verbal
4.9
Algebraic
4.10
Graphical
5
Conic sections in polar coordinates
5.1
Identifying a conic in polar form
5.2
Graphing the polar equations of conics
5.3
Deﬁning conics in terms of a focus and a directrix
5.4
Key concepts
5.5
Section exercises
5.6
Verbal
5.7
Algebraic
5.8
Extensions
5.9
Chapter review exercises
5.10
The Ellipse
5.11
The Hyperbola
5.12
The Parabola
5.13
Rotation of Axes
5.14
Conic Sections in Polar Coordinates
5.15
Practice test
Sequences, probability, and counting theory
Introduction to sequences, probability and counting theory
1
Sequences and their notations
1.1
Writing the terms of a sequence defined by an explicit formula
1.2
Investigating alternating sequences
1.3
Investigating piecewise explicit formulas
1.4
Finding an explicit formula
1.5
Writing the terms of a sequence defined by a recursive formula
1.6
Using factorial notation
1.7
Key equations
1.8
Key concepts
1.9
Section exercises
1.10
Verbal
1.11
Algebraic
1.12
Graphical
1.13
Technology
1.14
Extensions
2
Arithmetic sequences
2.1
Finding common differences
2.2
Writing terms of arithmetic sequences
2.3
Using recursive formulas for arithmetic sequences
2.4
Using explicit formulas for arithmetic sequences
2.5
Finding the number of terms in a finite arithmetic sequence
2.6
Solving application problems with arithmetic sequences
2.7
Key equations
2.8
Key concepts
2.9
Section exercises
2.10
Verbal
2.11
Algebraic
2.12
Graphical
2.13
Technology
2.14
Extensions
3
Geometric sequences
3.1
Finding common ratios
3.2
Writing terms of geometric sequences
3.3
Using recursive formulas for geometric sequences
3.4
Using explicit formulas for geometric sequences
3.5
Solving application problems with geometric sequences
3.6
Key equations
3.7
Key concepts
3.8
Section exercises
3.9
Verbal
3.10
Algebraic
3.11
Graphical
3.12
Extensions
4
Series and their notations
4.1
Using summation notation
4.2
Using the formula for arithmetic series
4.3
Using the formula for geometric series
4.4
Using the formula for the sum of an infinite geometric series
4.5
Determining whether the sum of an infinite geometric series is defined
4.6
Finding sums of infinite series
4.7
Solving annuity problems
4.8
Key equations
4.9
Key concepts
4.10
Section exercises
4.11
Verbal
4.12
Algebraic
4.13
Graphical
4.14
Numeric
4.15
Extensions
4.16
Real-world applications
5
Counting principles
5.1
Using the addition principle
5.2
Using the multiplication principle
5.3
Finding the number of permutations of n Distinct objects
5.4
Finding the number of permutations of n Distinct objects using the
5.5
Finding the number of permutations of n Distinct objects using a formula
5.6
Find the number of combinations using the formula
5.7
Finding the number of subsets of a set
5.8
Finding the number of permutations of n Non-distinct objects
5.9
Key equations
5.10
Key concepts
5.11
Section exercises
5.12
Verbal
5.13
Numeric
5.14
Extensions
5.15
Real-world applications
6
Binomial theorem
6.1
Identifying binomial coefficients
6.2
Using the binomial theorem
6.3
Using the binomial theorem to find a single term
6.4
Key equations
6.5
Key concepts
6.6
Section exercises
6.7
Verbal
6.8
Algebraic
6.9
Graphical
6.10
Extensions
7
Probability
7.1
Constructing probability models
7.2
Computing probabilities of equally likely outcomes
7.3
Computing the probability of the union of two events
7.4
Computing the probability of mutually exclusive events
7.5
Using the complement rule to compute probabilities
7.6
Computing probability using counting theory
7.7
Key equations
7.8
Key concepts
7.9
Section exercises
7.10
Verbal
7.11
Numeric
7.12
Extensions
7.13
Real-world applications
7.14
Chapter review exercises
7.15
Sequences and Their Notation
7.16
Arithmetic Sequences
7.17
Geometric Sequences
7.18
Series and Their Notation
7.19
Counting Principles
7.20
Binomial Theorem
7.21
Probability
7.22
Practice test
Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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