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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
altitude
a perpendicular line from one vertex of a triangle to the opposite side, or in the case of an obtuse triangle, to the line containing the opposite side, forming two right triangles
ambiguous case
a scenario in which more than one triangle is a valid solution for a given oblique SSA triangle
Archimedes’ spiral
a polar curve given by r=θ. r=θ. When multiplied by a constant, the equation appears as r=aθ. r=aθ. As r=θ, r=θ, the curve continues to widen in a spiral path over the domain.
argument
the angle associated with a complex number; the angle between the line from the origin to the point and the positive real axis
cardioid
a member of the limaçon family of curves, named for its resemblance to a heart; its equation is given as r=a±bcosθ r=a±bcosθ and r=a±bsinθ, r=a±bsinθ, where a b =1 a b =1
convex limaҫon
a type of one-loop limaçon represented by r=a±bcosθ r=a±bcosθ and r=a±bsinθ r=a±bsinθ such that a b 2 a b 2
De Moivre’s Theorem
formula used to find the nth nthpower or nth roots of a complex number; states that, for a positive integer n, z n n, z n is found by raising the modulus to the nth nthpower and multiplying the angles by n n
dimpled limaҫon
a type of one-loop limaçon represented by r=a±bcosθ r=a±bcosθ and r=a±bsinθ r=a±bsinθ such that 1< a b <2 1< a b <2
dot product
given two vectors, the sum of the product of the horizontal components and the product of the vertical components
Generalized Pythagorean Theorem
an extension of the Law of Cosines; relates the sides of an oblique triangle and is used for SAS and SSS triangles
initial point
the origin of a vector
inner-loop limaçon
a polar curve similar to the cardioid, but with an inner loop; passes through the pole twice; represented by r=a±bcosθ r=a±bcosθ and  r=a±b sinθ   r=a±b sinθ  where a<b a<b
Law of Cosines
states that the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of the included angle
Law of Sines
states that the ratio of the measurement of one angle of a triangle to the length of its opposite side is equal to the remaining two ratios of angle measure to opposite side; any pair of proportions may be used to solve for a missing angle or side
lemniscate
a polar curve resembling a figure 8 and given by the equation r 2 = a 2 cos2θ r 2 = a 2 cos2θ and r 2 = a 2 sin2θ,a0 r 2 = a 2 sin2θ,a0
magnitude
the length of a vector; may represent a quantity such as speed, and is calculated using the Pythagorean Theorem
modulus
the absolute value of a complex number, or the distance from the origin to the point ( x,y ); ( x,y );also called the amplitude
oblique triangle
any triangle that is not a right triangle
one-loop limaҫon
a polar curve represented by r=a±bcosθ r=a±bcosθ and r=a±bsinθ r=a±bsinθ such that a>0,b>0, a>0,b>0, and a b >1; a b >1; may be dimpled or convex; does not pass through the pole
parameter
a variable, often representing time, upon which x xand y yare both dependent
polar axis
on the polar grid, the equivalent of the positive x-axis on the rectangular grid
polar coordinates
on the polar grid, the coordinates of a point labeled ( r,θ ), ( r,θ ),where θ θindicates the angle of rotation from the polar axis and r rrepresents the radius, or the distance of the point from the pole in the direction of θ θ
polar equation
an equation describing a curve on the polar grid.
polar form of a complex number
a complex number expressed in terms of an angle θ θ and its distance from the origin r; r;can be found by using conversion formulas x=rcosθ,y=rsinθ, x=rcosθ,y=rsinθ,and r= x 2 + y 2 r= x 2 + y 2
pole
the origin of the polar grid
resultant
a vector that results from addition or subtraction of two vectors, or from scalar multiplication
rose curve
a polar equation resembling a flower, given by the equations r=acosnθ r=acosnθ and r=asinnθ; r=asinnθ; when n n is even there are 2n 2n petals, and the curve is highly symmetrical; when n n is odd there are n n petals.
scalar
a quantity associated with magnitude but not direction; a constant
scalar multiplication
the product of a constant and each component of a vector
standard position
the placement of a vector with the initial point at ( 0,0 ) ( 0,0 ) and the terminal point (a,b), (a,b), represented by the change in the x-coordinates and the change in the y-coordinates of the original vector
terminal point
the end point of a vector, usually represented by an arrow indicating its direction
unit vector
a vector that begins at the origin and has magnitude of 1; the horizontal unit vector runs along the x-axis and is defined as v 1 = 1,0 v 1 = 1,0 the vertical unit vector runs along the y-axis and is defined as v 2 = 0,1 . v 2 = 0,1 .
vector
a quantity associated with both magnitude and direction, represented as a directed line segment with a starting point (initial point) and an end point (terminal point)
vector addition
the sum of two vectors, found by adding corresponding components
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