Learn advanced trigonometry and core concepts in probability and statistics. Encounter objects from higher math including complex numbers, vectors, and matrices.

- Sketch graphs of rational functions by identifying asymptotes.
- Leverage factoring and sign tables as a strategy to solve equations and inequalities involving polynomials and rational functions.

- Apply trigonometric identities to simplify trigonometric expressions.
- Solve trigonometric equations by leveraging properties of trigonometric functions and extending previous knowledge of linear and quadratic equations.

- Draw connections between algebraic and geometric properties of ellipses and hyperbolas.

- Extend prior knowledge of rectangular curves to parametric and polar curves.

- Manipulate complex numbers algebraically, visualize them geometrically in the complex plane, and connect the algebraic and geometric interpretations through Euler’s formula.

- Generalize prior intuitions about arithmetic to vectors and matrices in higher-dimensional space.
- Interpret matrices as linear transformations of points in the coordinate plane.

- Define probability using the formal language of sets.
- Apply combinatorial techniques to compute probabilities in real-world modeling contexts.
- Understand independent events both conceptually and quantitatively from the perspective of conditional probability.
- Define probability mass functions and apply binomial and geometric distributions in modeling contexts.

- Understand the utility of statistical sampling and be able to identify sources of bias.
- Compute statistical measures of the center and spread of a data set.
- Fit a linear regression to a data set and interpret the coefficients.
- Construct an appropriate regression model for a data set and use it to make predictions.

1.

Sequences and Series
17 topics

1.1. Arithmetic Series

1.1.1. | Introduction to Sigma Notation | |

1.1.2. | Expressing an Arithmetic Series in Sigma Notation | |

1.1.3. | Finding the Sum of an Arithmetic Series | |

1.1.4. | Finding the First Term of an Arithmetic Series | |

1.1.5. | Calculating the Number of Terms in an Arithmetic Series | |

1.1.6. | Modeling With Arithmetic Series |

1.2. Finite Geometric Series

1.2.1. | The Sum of a Finite Geometric Series | |

1.2.2. | The Sum of the First N Terms of a Geometric Series | |

1.2.3. | Writing Geometric Series in Sigma Notation | |

1.2.4. | Finding the Sum of a Geometric Series Given in Sigma Notation | |

1.2.5. | Solving Geometric Series Problems Using Exponential Equations and Inequalities | |

1.2.6. | Modeling With Geometric Series | |

1.2.7. | Modeling Financial Problems Using Geometric Series |

1.3. The Binomial Theorem

1.3.1. | Pascal's Triangle and the Binomial Coefficients | |

1.3.2. | Expanding a Binomial Using Binomial Coefficients | |

1.3.3. | The Special Case of the Binomial Theorem | |

1.3.4. | Approximating Values Using the Binomial Theorem |

2.

Rational Functions
16 topics

2.4. Rational Expressions and Equations

2.4.1. | Closure Properties of Polynomials | |

2.4.2. | Closure Properties of Rational Expressions | |

2.4.3. | Rational Equations With Three Terms | |

2.4.4. | Advanced Rational Equations | |

2.4.5. | Further Advanced Rational Equations |

2.5. Rational Functions

2.5.1. | Finding Roots of Rational Functions | |

2.5.2. | Vertical Asymptotes of Rational Functions | |

2.5.3. | Locating Holes in Rational Functions | |

2.5.4. | Horizontal Asymptotes of Rational Functions | |

2.5.5. | End Behavior of Rational Functions | |

2.5.6. | Infinite Limits of Rational Functions | |

2.5.7. | Infinite Limits of Rational Functions: Advanced Cases | |

2.5.8. | The Domain and Range of a Rational Function | |

2.5.9. | Identifying a Rational Function From a Graph | |

2.5.10. | Identifying a Rational Function From a Graph Containing Holes | |

2.5.11. | Identifying the Graph of a Rational Function |

3.

Inequalities
18 topics

3.6. Quadratic Inequalities

3.6.1. | Solving Elementary Quadratic Inequalities | |

3.6.2. | Solving Quadratic Inequalities From Graphs | |

3.6.3. | Solving Quadratic Inequalities Using the Graphical Method | |

3.6.4. | Solving Quadratic Inequalities Using the Sign Table Method | |

3.6.5. | Solving Discriminant Problems Using Quadratic Inequalities |

3.7. Polynomial Inequalities

3.7.1. | Inequalities Involving Powers of Binomials | |

3.7.2. | Solving Polynomial Inequalities Using a Graphical Method | |

3.7.3. | Solving Polynomial Inequalities Using Special Factoring Techniques and the Graphical Method | |

3.7.4. | Solving Polynomial Inequalities Using the Sign Table Method |

3.8. Rational Inequalities

3.8.1. | Solving Rational Inequalities | |

3.8.2. | Further Solving of Rational Inequalities |

3.9. Non-Polynomial Inequalities

3.9.1. | Solving Inequalities Involving Radical Functions | |

3.9.2. | Solving Inequalities Involving Exponential Functions | |

3.9.3. | Solving Inequalities Involving Logarithmic Functions | |

3.9.4. | Solving Inequalities Involving Exponential Functions and Polynomials | |

3.9.5. | Solving Inequalities Involving Positive and Negative Factors | |

3.9.6. | Solving Two-Variable Nonlinear Inequalities | |

3.9.7. | Further Solving of Two-Variable Nonlinear Inequalities |

4.

Trigonometry
42 topics

4.10. The Inverse Trigonometric Functions

4.10.1. | Graphing the Inverse Sine Function | |

4.10.2. | Graphing the Inverse Cosine Function | |

4.10.3. | Graphing the Inverse Tangent Function | |

4.10.4. | Evaluating Expressions Containing Inverse Trigonometric Functions | |

4.10.5. | Further Evaluating Expressions Containing Inverse Trigonometric Functions |

4.11. Trigonometric Identities

4.11.1. | Simplifying Expressions Using Basic Trigonometric Identities | |

4.11.2. | Simplifying Expressions Using the Pythagorean Identity | |

4.11.3. | Alternate Forms of the Pythagorean Identity | |

4.11.4. | Simplifying Expressions Using the Secant-Tangent Identity | |

4.11.5. | Alternate Forms of the Secant-Tangent Identity | |

4.11.6. | Simplifying Trigonometric Expressions Using the Cotangent-Cosecant Identity | |

4.11.7. | Simplifying Trigonometric Expressions Using Cofunction Identities |

4.12. The Sum and Difference Formulas

4.12.1. | The Sum and Difference Formulas for Sine | |

4.12.2. | The Sum and Difference Formulas for Cosine | |

4.12.3. | The Sum and Difference Formulas for Tangent | |

4.12.4. | Calculating Trigonometric Ratios Using the Sum Formula for Sine | |

4.12.5. | Calculating Trigonometric Ratios Using the Sum Formula for Cosine | |

4.12.6. | Calculating Trigonometric Ratios Using the Sum Formula for Tangent | |

4.12.7. | Writing Sums of Trigonometric Functions in Amplitude-Phase Form |

4.13. The Double-Angle Formulas

4.13.1. | The Double-Angle Formula for Sine | |

4.13.2. | Verifying Trigonometric Identities Using the Double-Angle Formula for Sine | |

4.13.3. | Using the Double-Angle Formula for Sine With the Pythagorean Theorem | |

4.13.4. | The Double-Angle Formula for Cosine | |

4.13.5. | Verifying Trigonometric Identities Using the Double-Angle Formulas for Cosine | |

4.13.6. | Finding Exact Values of Trigonometric Expressions Using the Double-Angle Formulas for Cosine | |

4.13.7. | Simplifying Expressions Using the Double-Angle Formula for Tangent | |

4.13.8. | Verifying Trigonometric Identities Using the Double-Angle Formula for Tangent |

4.14. Elementary Trigonometric Equations

4.14.1. | Elementary Trigonometric Equations Containing Sine | |

4.14.2. | Elementary Trigonometric Equations Containing Cosine | |

4.14.3. | Elementary Trigonometric Equations Containing Tangent | |

4.14.4. | Elementary Trigonometric Equations Containing Secant | |

4.14.5. | Elementary Trigonometric Equations Containing Cosecant | |

4.14.6. | Elementary Trigonometric Equations Containing Cotangent | |

4.14.7. | General Solutions of Elementary Trigonometric Equations |

4.15. Trigonometric Equations Containing Transformed Functions

4.15.1. | General Solutions of Trigonometric Equations With Transformed Functions | |

4.15.2. | Trigonometric Equations Containing Transformed Sine Functions | |

4.15.3. | Trigonometric Equations Containing Transformed Cosine Functions | |

4.15.4. | Trigonometric Equations Containing Transformed Tangent Functions |

4.16. Advanced Trigonometric Equations

4.16.1. | Solving Trigonometric Equations Using the Sin-Cos-Tan Identity | |

4.16.2. | Solving Trigonometric Equations Using the Zero-Product Property | |

4.16.3. | Quadratic Trigonometric Equations Containing Sine or Cosine | |

4.16.4. | Quadratic Trigonometric Equations Containing Tangent or Cotangent |

5.

Vectors
29 topics

5.17. Introduction to Vectors

5.17.1. | Representing Given Information as a Vector | |

5.17.2. | The Triangle Law for the Addition and Subtraction of Two Vectors | |

5.17.3. | Calculating the Magnitude of a Vector From Given Information | |

5.17.4. | Problem Solving Using Vector Diagrams | |

5.17.5. | Parallel Vectors | |

5.17.6. | Unit Vectors | |

5.17.7. | Linear Combinations of Vectors and Their Properties | |

5.17.8. | Describing the Position Vector of a Point Using Known Vectors |

5.18. Vectors in 2D Cartesian Coordinates

5.18.1. | Two-Dimensional Vectors Expressed in Component Form | |

5.18.2. | Addition and Scalar Multiplication of Cartesian Vectors in 2D | |

5.18.3. | Calculating the Magnitude of Cartesian Vectors in 2D | |

5.18.4. | Calculating the Direction of Cartesian Vectors in 2D | |

5.18.5. | Calculating the Components of Cartesian Vectors in 2D | |

5.18.6. | Velocity and Acceleration for Plane Motion | |

5.18.7. | Calculating Displacement for Plane Motion |

5.19. Vectors in 3D Cartesian Coordinates

5.19.1. | Three-Dimensional Vectors Expressed in Component Form | |

5.19.2. | Addition and Scalar Multiplication of Cartesian Vectors in 3D | |

5.19.3. | Calculating the Magnitude of Cartesian Vectors in 3D |

5.20. The Dot Product

5.20.1. | Calculating the Dot Product Using Angle and Magnitude | |

5.20.2. | Calculating the Dot Product Using Components | |

5.20.3. | The Angle Between Two Vectors | |

5.20.4. | Calculating a Scalar Projection | |

5.20.5. | Calculating a Vector Projection |

5.21. The Cross Product

5.21.1. | Calculating the Cross Product of Two Vectors Using the Definition | |

5.21.2. | Calculating the Cross Product Using Determinants | |

5.21.3. | Finding Areas Using the Cross Product | |

5.21.4. | The Scalar Triple Product | |

5.21.5. | Volumes of Parallelepipeds | |

5.21.6. | Finding Volumes of Tetrahedrons and Pyramids Using Vector Products |

6.

Matrices
36 topics

6.22. Introduction to Matrices

6.22.1. | Introduction to Matrices | |

6.22.2. | Index Notation for Matrices | |

6.22.3. | Adding and Subtracting Matrices | |

6.22.4. | Properties of Matrix Addition | |

6.22.5. | Scalar Multiplication of Matrices | |

6.22.6. | Zero, Square, Diagonal and Identity Matrices | |

6.22.7. | The Transpose of a Matrix |

6.23. Matrix Multiplication

6.23.1. | Multiplying a Matrix by a Column Vector | |

6.23.2. | Multiplying Square Matrices | |

6.23.3. | Conformability for Matrix Multiplication | |

6.23.4. | Multiplying Matrices | |

6.23.5. | Powers of Matrices | |

6.23.6. | Multiplying a Matrix by the Identity Matrix | |

6.23.7. | Properties of Matrix Multiplication | |

6.23.8. | Representing 2x2 Systems of Equations Using a Matrix Product | |

6.23.9. | Representing 3x3 Systems of Equations Using a Matrix Product |

6.24. Determinants

6.24.1. | The Determinant of a 2x2 Matrix | |

6.24.2. | The Geometric Interpretation of the 2x2 Determinant | |

6.24.3. | The Minors of a 3x3 Matrix | |

6.24.4. | The Determinant of a 3x3 Matrix |

6.25. The Inverse of a Matrix

6.25.1. | Introduction to the Inverse of a Matrix | |

6.25.2. | Calculating the Inverse of a 2x2 Matrix | |

6.25.3. | Calculating the Inverse of a 3x3 Matrix Using the Cofactor Method | |

6.25.4. | Solving 2x2 Systems of Equations Using Inverse Matrices | |

6.25.5. | Solving 3x3 Systems of Equations Using Inverse Matrices |

6.26. Linear Transformations

6.26.1. | Introduction to Linear Transformations | |

6.26.2. | The Standard Matrix of a Linear Transformation | |

6.26.3. | Linear Transformations of Points and Lines in the Plane | |

6.26.4. | Linear Transformations of Objects in the Plane | |

6.26.5. | Dilations and Reflections as Linear Transformations | |

6.26.6. | Shear and Stretch as Linear Transformations | |

6.26.7. | Right-Angle Rotations as Linear Transformations | |

6.26.8. | Rotations as Linear Transformations | |

6.26.9. | Combining Linear Transformations Using 2x2 Matrices | |

6.26.10. | Using Inverse Matrices to Reverse Linear Transformations | |

6.26.11. | Determining the Area Scale Factor of a Linear Transformation |

7.

Conic Sections
20 topics

7.27. Ellipses as Conic Sections

7.27.1. | Introduction to Ellipses | |

7.27.2. | Equations of Ellipses Centered at the Origin | |

7.27.3. | Equations of Ellipses Centered at a General Point | |

7.27.4. | Finding the Center and Axes of Ellipses by Completing the Square | |

7.27.5. | Finding Intercepts of Ellipses | |

7.27.6. | Finding Intersections of Ellipses and Lines | |

7.27.7. | Foci of Ellipses | |

7.27.8. | Vertices and Eccentricity of Ellipses | |

7.27.9. | Directrices of Ellipses | |

7.27.10. | The Area of an Ellipse |

7.28. Hyperbolas as Conic Sections

7.28.1. | Equations of Hyperbolas Centered at the Origin | |

7.28.2. | Equations of Hyperbolas Centered at a General Point | |

7.28.3. | Asymptotes of Hyperbolas Centered at the Origin | |

7.28.4. | Asymptotes of Hyperbolas Centered at a General Point | |

7.28.5. | Finding Intercepts and Intersections of Hyperbolas | |

7.28.6. | Transverse Axes of Hyperbolas | |

7.28.7. | Conjugate Axes of Hyperbolas | |

7.28.8. | Foci of Hyperbolas | |

7.28.9. | Eccentricity and Vertices of Hyperbolas | |

7.28.10. | Directrices of Hyperbolas |

8.

Parametric Equations
7 topics

8.29. Parametric Equations

8.29.1. | Graphing Curves Defined Parametrically | |

8.29.2. | Finding the Cartesian Equation of Curves Defined Parametrically | |

8.29.3. | Finding Intercepts of Curves Defined Parametrically | |

8.29.4. | Finding Intersections of Parametric Curves and Lines | |

8.29.5. | Parametric Equations of Circles | |

8.29.6. | Parametric Equations of Ellipses | |

8.29.7. | Parametric Equations of Parabolas Centered at the Origin |

9.

Polar Equations
6 topics

9.30. Polar Coordinates

9.30.1. | Introduction to Polar Coordinates | |

9.30.2. | Converting from Polar Coordinates to Cartesian Coordinates | |

9.30.3. | Polar Equations of Circles Centered at the Origin | |

9.30.4. | Polar Equations of Radial Lines | |

9.30.5. | Polar Equations of Circles Centered on the Coordinate Axes | |

9.30.6. | Finding Intersections of Polar Curves |

10.

Complex Numbers
27 topics

10.31. The Complex Plane

10.31.1. | The Complex Plane | |

10.31.2. | The Magnitude of a Complex Number | |

10.31.3. | The Argument of a Complex Number | |

10.31.4. | Arithmetic in the Complex Plane | |

10.31.5. | Geometry in the Complex Plane |

10.32. Further Complex Numbers

10.32.1. | The Complex Conjugate | |

10.32.2. | Special Properties of the Complex Conjugate | |

10.32.3. | The Complex Conjugate and the Roots of a Quadratic Equation | |

10.32.4. | Dividing Complex Numbers | |

10.32.5. | Solving Equations by Equating Real and Imaginary Parts | |

10.32.6. | Extending Polynomial Identities to the Complex Numbers |

10.33. Complex Numbers in Polar Form

10.33.1. | The Polar Form of a Complex Number | |

10.33.2. | Products of Complex Numbers Expressed in Polar Form | |

10.33.3. | Quotients of Complex Numbers Expressed in Polar Form | |

10.33.4. | The CIS Notation |

10.34. De Moivre's Theorem

10.34.1. | De Moivre's Theorem | |

10.34.2. | Finding Powers of Complex Numbers Using De Moivre's Theorem | |

10.34.3. | The Power-Reducing Formulas for Sine and Cosine | |

10.34.4. | Euler's Formula | |

10.34.5. | The Roots of Unity | |

10.34.6. | Square Roots of Complex Numbers | |

10.34.7. | Higher Roots of Complex Numbers |

10.35. The Fundamental Theorem of Algebra

10.35.1. | The Fundamental Theorem of Algebra for Quadratic Equations | |

10.35.2. | The Fundamental Theorem of Algebra with Cubic Equations | |

10.35.3. | Solving Cubic Equations With Complex Roots | |

10.35.4. | The Fundamental Theorem of Algebra with Quartic Equations | |

10.35.5. | Solving Quartic Equations With Complex Roots |

11.

Probability & Combinatorics
42 topics

11.36. Introduction to Probability

11.36.1. | Probability From Experimental Data | |

11.36.2. | Sample Spaces and Events in Probability | |

11.36.3. | Single Events in Probability | |

11.36.4. | The Complement of an Event | |

11.36.5. | Introduction to Sets | |

11.36.6. | The Union of Sets | |

11.36.7. | The Intersection of Sets | |

11.36.8. | Venn Diagrams in Probability | |

11.36.9. | Geometric Probability |

11.37. Compound Events in Probability

11.37.1. | Compound Events in Probability From Experimental Data | |

11.37.2. | Computing Probabilities for Compound Events Using Venn Diagrams | |

11.37.3. | Computing Probabilities for Three Events Using Venn Diagrams | |

11.37.4. | The Addition Law of Probability | |

11.37.5. | Mutually Exclusive Events |

11.38. Conditional Probability

11.38.1. | Conditional Probabilities From Venn Diagrams | |

11.38.2. | Conditional Probabilities From Tables | |

11.38.3. | The Multiplication Law for Conditional Probability | |

11.38.4. | The Law of Total Probability | |

11.38.5. | Tree Diagrams for Dependent Events | |

11.38.6. | Tree Diagrams for Dependent Events: Applications | |

11.38.7. | Independent Events | |

11.38.8. | Tree Diagrams for Independent Events |

11.39. Discrete Random Variables

11.39.1. | Probability Mass Functions of Discrete Random Variables | |

11.39.2. | Cumulative Distribution Functions for Discrete Random Variables | |

11.39.3. | Expected Values of Discrete Random Variables | |

11.39.4. | The Binomial Distribution | |

11.39.5. | Modeling With the Binomial Distribution | |

11.39.6. | The Geometric Distribution | |

11.39.7. | Modeling With the Geometric Distribution |

11.40. The Normal Distribution

11.40.1. | The Standard Normal Distribution | |

11.40.2. | The Normal Distribution | |

11.40.3. | Mean and Variance of the Normal Distribution | |

11.40.4. | Percentage Points of the Standard Normal Distribution | |

11.40.5. | Modeling With the Normal Distribution | |

11.40.6. | The Empirical Rule for the Normal Distribution |

11.41. Combinatorics

11.41.1. | The Rule of Sum and the Rule of Product | |

11.41.2. | Factorials | |

11.41.3. | Factorials in Variable Expressions | |

11.41.4. | Ordering Objects | |

11.41.5. | Permutations | |

11.41.6. | Combinations | |

11.41.7. | Computing Probabilities Using Combinatorics |

12.

Statistics
18 topics

12.42. Analyzing Data

12.42.1. | Sampling | |

12.42.2. | The Mean of a Data Set | |

12.42.3. | Variance and Standard Deviation of Population Data | |

12.42.4. | Covariance of Population Data | |

12.42.5. | The Z-Score |

12.43. Correlation

12.43.1. | Scatter Plots | |

12.43.2. | Trend Lines | |

12.43.3. | Making Predictions Using Trend Lines | |

12.43.4. | Interpreting Coefficients of Trend Lines | |

12.43.5. | Linear Correlation | |

12.43.6. | Residuals and Residual Plots | |

12.43.7. | The Linear Correlation Coefficient | |

12.43.8. | Correlation vs. Causation |

12.44. Regression

12.44.1. | Selecting a Regression Model | |

12.44.2. | Linear Regression | |

12.44.3. | Quadratic Regression | |

12.44.4. | Semi-Log Scatter Plots | |

12.44.5. | Exponential Regression |