Precalculus II (editable text only)

Content Overview

Text

Angles

• Draw angles in standard position
• Converting Between Degrees and Radians
• Finding Coterminal Angles
• Determining the Length of an Arc
• Use Linear and Angular Speed to Describe Motion on a Circular Path

Unit Circle: Sine and Cosine Functions

• Finding Function Values for the Sine and Cosine
• Use reference angles to evaluate trigonometric functions

The Other Trigonometric Functions

• Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent
• Using Even and Odd Trigonometric Functions
• Recognize and Use Fundamental Identities
• Evaluating Trigonometric Functions with a Calculator

Right Triangle Trigonometry

• Using Right Triangles to Evaluate Trigonometric Functions
• Using Equal Cofunction of Complements
• Using Right Triangle Trigonometry to Solve Applied Problems

Graphs of the Sine and Cosine Functions

• Graph variations of  y=sin( x )  and  y=cos( x )
• Using Transformations of Sine and Cosine Functions

Graphs of the Other Trigonometric Functions

• Analyzing the Graph of y = tan x and Its Variations
• Analyzing the Graphs of y = sec x and y = cscx and Their Variations
• Analyzing the Graph of y = cot x and Its Variations

Inverse Trigonometric Functions

• Understanding and Using the Inverse Sine, Cosine, and Tangent Functions
• Finding the Exact Value of Expressions Involving the Inverse Sine, Cosine, and Tangent Functions
• Using a Calculator to Evaluate Inverse Trigonometric Functions
• Finding Exact Values of Composite Functions with Inverse Trigonometric Functions

Solving Trigonometric Equations Part I

• Verify the fundamental trigonometric identities
• Simplify trigonometric expressions using algebra and the identities

Sum and Difference Identities

• Use sum and difference formulas for cosine
• Use sum and difference formulas for sine
• Use sum and difference formulas for tangent
• Use sum and difference formulas for cofunctions
• Use sum and difference formulas to verify identities

Double-Angle, Half-Angle, and Reduction Formulas

• Using Double-Angle Formulas to Find Exact Values
• Using Double-Angle Formulas to Verify Identities
• Use Reduction Formulas to Simplify an Expression
• Using Half-Angle Formulas to Find Exact Values

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

• Expressing Products as Sums
• Expressing Sums as Products

Solving Trigonometric Equations Part II

• Solving Linear Trigonometric Equations in Sine and Cosine
• Solving Equations Involving a Single Trigonometric Function
• Solving Trigonometric Equations in Quadratic Form
• Solving Trigonometric Equations Using Fundamental Identities
• Solving Right Triangle Problems

Modeling with Trigonometric Equations

• Determining the Amplitude and Period of a Sinusoidal Function
• Finding Equations and Graphing Sinusoidal Functions
• Modeling Periodic Behavior
• Modeling Harmonic Motion Functions

Non-right Triangles: Law of Sines

• Using the Law of Sines to Solve Oblique Triangles
• Finding the Area of an Oblique Triangle Using the Sine Function
• Solving Applied Problems Using the Law of Sines

Non-right Triangles: Law of Cosines

• Using the Law of Cosines to Solve Oblique Triangles
• Solving Applied Problems Using the Law of Cosines
• Using Heron’s Formula to Find the Area of a Triangle

Polar Coordinates

• Plotting Points Using Polar Coordinates
• Converting Between Polar Coordinates to Rectangular Coordinates
• Transforming Equations between Polar and Rectangular Forms
• Identify and Graph Polar Equations by Converting to Rectangular Equations

Polar Coordinates: Graphs

• Testing Polar Equations for Symmetry
• Graphing Polar Equations by Plotting Points
• Graphing Circles and the 5 Classic Polar Curves

Polar Form of Complex Numbers

• Plotting Complex Numbers in the Complex Plane
• Finding the Absolute Value of a Complex Number
• Writing Complex Numbers in Polar Form
• Converting a Complex Number from Polar to Rectangular Form
• Finding Products and Quotients of Complex Numbers in Polar Form
• Finding Powers and Roots of Complex Numbers in Polar Form

Parametric Equations

• Parameterizing a Curve
• Methods for Finding Cartesian and Polar Equations from Curves

Parametric Equations: Graphs

• Graphing Parametric Equations by Plotting Points
• Applications of Parametric Equations

Vectors

• Finding Magnitude and Direction
• Performing Vector Addition and Scalar Multiplication
• Finding the Unit Vector in the Direction of v
• Performing Operations with Vectors in Terms of i and j
• Calculating the Component Form of a Vector: Direction
• Finding the Dot Product of Two Vectors

Systems of Linear Equations: Two Variables

• Solving Systems of Equations by Graphing
• Solving Systems of Equations by Substitution
• Solving Systems of Equations in Two Variables by the Addition Method
• Identifying and Expressing Solutions to Systems of Equations
• Using Systems of Equations to Investigate Profits

Systems of Linear Equations: Three Variables

• Solving Systems of Three Equations in Three Variables
• Inconsistent and Dependent Systems in Three Variables

Systems of Nonlinear Equations and Inequalities: Two Variables

• Solving a System of Nonlinear Equations Using Substitution
• Solving a System of Nonlinear Equations Using Elimination
• Graphing Nonlinear Inequalities and Systems of Nonlinear Inequalities

Partial Fractions

• Decomposing P(x) / Q(x), Where Q(x) Has Only Nonrepeated Linear Factors
• Decomposing P(x)/ Q(x), Where Q(x) Has Repeated Linear Factors
• Decomposing P(x) / Q(x), Where Q(x) Has a Nonrepeated Irreducible Quadratic Factor
• Decomposing P(x) / Q(x), When Q(x) Has a Repeated Irreducible Quadratic Factor

Matrices and Matrix Operations

• Finding the Sum and Difference of Two Matrices
• Finding Scalar Multiples of a Matrix
• Finding the Product of Two Matrices

Solving Systems with Gaussian Elimination

• The Augmented Matrix of a System of Equations
• Performing Row Operations on a Matrix
• Solving a System of Linear Equations Using Matrices

Solving Systems with Inverses

• Finding the Inverse of a Matrix
• Solving a System of Linear Equations Using the Inverse of a Matrix

Solving Systems with Cramer's Rule

• Using Cramer’s Rule to Solve a System of Two Equations in Two Variables
• Using Cramer’s Rule to Solve a System of Three Equations in Three Variables
• Understanding Properties of Determinants

The Ellipse

• Writing Equations of Ellipses in Standard Form
• Deriving the Equation of an Ellipse Centered at the Origin
• Writing Equations of Ellipses Not Centered at the Origin
• Graphing Ellipses
• Solving Applied Problems Involving Ellipses

The Hyperbola

• Locating the Vertices and Foci of a Hyperbola
• Deriving the Equation of a Hyperbola Centered at the Origin
• Writing Equations of Hyperbolas in Standard Form
• Graphing Hyperbolas
• Solving Applied Problems Involving Hyperbolas

The Parabola

• Graphing Parabolas with Vertices at the Origin
• Writing Equations of Parabolas in Standard Form
• Graphing Parabolas with Vertices Not at the Origin
• Solving Applied Problems Involving Parabolas

Rotation of Axes

• Identifying Nondegenerate Conics in General Form
• Finding a New Representation of the Given Equation after Rotating through a Given Angle
• Writing Equations of Rotated Conics in Standard Form
• Identifying Conics without Rotating Axes

Conic Sections in Polar Coordinates

• Identifying a Conic in Polar Form
• Graphing the Polar Equations of Conics
• Defining Conics in Terms of a Focus and a Directrix

Sequences and Their Notations

• Writing the Terms of a Sequence Defined by an Explicit Formula
• Investigating Alternating Sequences
• Investigating Explicit Formulas
• Writing the Terms of a Sequence Defined by a Recursive Formula

Arithmetic Sequences

• Finding Common Differences
• Using Formulas for Arithmetic Sequences
• Finding the Number of Terms in a Finite Arithmetic Sequence

Geometric Sequences

• Finding Common Ratios
• Writing Terms of Geometric Sequences
• Solving Application Problems with Geometric Sequences

Series and Their Notations

• Using Summation Notation
• Using the Formula for Arithmetic Series
• Using the Formula for Geometric Series
• Finding Sums of Infinite Series
• Solving Annuity Problems

Counting Principles

• Using the Addition and Multiplication Principles
• Finding the Number of Permutations of n Distinct Objects
• Find the Number of Combinations Using the Formula
• Finding the Number of Subsets of a Set
• Finding the Number of Permutations of n Non-Distinct Objects

Binomial Theorem

• Identifying Binomial Coefficients
• Using the Binomial Theorem
• Using the Binomial Theorem to Find a Single Term

Probability

• Constructing Probability Models
• Computing the Probability of the Union of Two Events
• Computing the Probability of Mutually Exclusive Events
• Using the Complement Rule to Compute Probabilities
• Computing Probability Using Counting Theory

Finding Limits: Numerical and Graphical Approaches

• Understanding Limit Notation
• Understanding Left-Hand Limits and Right-Hand Limits
• Finding a Limit Using a Graph
• Finding a Limit Using a Table

Finding Limits: Properties of Limits

• Finding the Limit of a Sum, a Difference, and a Product
• Finding the Limit of Some Basic Mathematical Expressions

Continuity

• Determining Whether a Function Is Continuous at a Number
• Identifying Discontinuities
• Recognizing Continuous and Discontinuous Real-Number Functions
• Determining the Input Values for Which a Function Is Discontinuous
• Determining Whether a Function Is Continuous

Derivatives

• Finding the Average Rate of Change of a Function
• Understanding the Instantaneous Rate of Change
• Derivatives: Interpretations and Notation
• Finding Derivatives of Rational Functions
• Finding Derivatives of Functions with Roots
• Finding Instantaneous Rates of Change
• Using Graphs to Find Instantaneous Rates of Change
• Finding Points Where a Function’s Derivative Does Not Exist
• Finding an Equation of a Line Tangent to the Graph of a Function