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Mathematics Describing the Real World: Precalculus and Trigonometry

Prepare your student or yourself for success with this course on precalculus and trigonometry by the author of one of the most widely used textbooks on the subject.
Mathematics Describing the Real World: Precalculus and Trigonometry is rated 4.6 out of 5 by 81.
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Rated 5 out of 5 by from Easy to follow It's taken me a while, but I finally finished the course today, 06/24/24. Topics were well covered and examples very helpful, Excellent refresher for me and looking forward to calculus.
Date published: 2024-06-24
Rated 2 out of 5 by from skipping too much very similar to a most math teachers, not enough details, didn't explain why they do what they did, feels rushed and not enough examples.
Date published: 2024-03-01
Rated 5 out of 5 by from As close to perfect as you can get I'm a retired physical scientist and university professor, and I took a rather unusual path toward this course. First I took Prof Edwards' multivariable calculus course, and then decided to take his other calculus courses, and finally precalculus. Edwards is so good that doing it this way works just fine. Everything is at the same, easily understood level all the way through. The instructor's style is such that you understand everything he is saying, only rarely needing to pause and review. I have seen lots of university math classes at all levels, and Edwards has no equal for clarity and topic selection. Even though this class was produced over a decade ago, it stands the test of time. There is no need for an update.
Date published: 2023-08-12
Rated 5 out of 5 by from Review of Precalculus Incredibly terrific. I majored in Mathematics and graduated from UCLA in 1965. I decided to review all my college Math and started with this course. It was always a retirement wish to do this review. Dr Edwards is not only a great lecturer but shows such enthusiasm and knowledge of his subject that I’m encouraged to go forward with further Math courses with this program. Jack Lewin
Date published: 2023-01-21
Rated 5 out of 5 by from Did it ALL… I remembered I liked calculus a long time ago, so earlier in the year I took Prof Edwards Calculus 1 and it was great, but I certainly noticed my pre calculus skills were weak. So I planned to take a few pre calc lectures to improve trig in particular. He is such a good teacher, that I did all 36 lectures over 3 months. I did them 10 or 15 minutes at a time and found following along with notes helps. It was all so much better than classes I had long ago, that I really think this should be how math is taught with video and some workshops (unfortunately that did not exist when I was 17). Highly recommend the course and I’m a Bruce Edwards fan. ( PS: It’s so good, its worth improving the workbook).
Date published: 2022-10-14
Rated 5 out of 5 by from great instructional method with excellent content-related examples
Date published: 2022-09-16
Rated 4 out of 5 by from Matrix Row Reduction??? Professor never really ties together row reduction with the matrix form. He laughs that he never wants to do this. But to work through a matrix to get an identity might be helpful. I'll learn this elsewhere. Otherwise, this seems to be a good course. This professor is just like most other professors I've had "they're full of themselves." But I can learn and work around this flaw and the others.
Date published: 2022-07-11
Rated 3 out of 5 by from Kind of disappointed I'd heard wonderful things about Professor Edwards, but I must say I'm somewhat disappointed by this course. I'd taken Algebra I and II and had no trouble with any of the concepts in these two courses, but I'm find that Professor Edwards' explanations are not always illuminating. At times I find myself scratching my head about something he's stated and wondering: "how did he arrive at that conclusion?" I just wish he'd gone into more detail at times. I also found myself looking up some of the topics on YouTube and finding what I felt was a clearer explanation of that topic.
Date published: 2022-06-30
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Finally make sense of the mysteries of precalculus and trigonometry in the company of master educator and award-winning Professor Bruce Edwards. In the 36 intensively illustrated lectures of Mathematics Describing the Real World: Precalculus and Trigonometry, he takes you through all the major topics of a typical precalculus course taught in high school or college. You'll gain new insights into functions, complex numbers, matrices, and much more. The course also comes complete with a workbook featuring a wealth of additional explanations and problems.


Bruce H. Edwards

I love mathematics and tried to communicate this passion to others, regardless of their mathematical backgrounds.


University of Florida

Dr. Bruce H. Edwards is Professor of Mathematics at the University of Florida. Professor Edwards received his B.S. in Mathematics from Stanford University and his Ph.D. in Mathematics from Dartmouth College. After his years at Stanford, he taught mathematics at a university near Bogota, Colombia, as a Peace Corps volunteer. Professor Edwards has won many teaching awards at the University of Florida, including Teacher of the Year in the College of Liberal Arts and Sciences, Liberal Arts and Sciences Student Council Teacher of the Year, and the University of Florida Honors Program Teacher of the Year. He was selected by the Office of Alumni Affairs to be the Distinguished Alumni Professor for 1991-1993. Professor Edwards has taught a variety of mathematics courses at the University of Florida, from first-year calculus to graduate-level classes in algebra and numerical analysis. He has been a frequent speaker at research conferences and meetings of the National Council of Teachers of Mathematics. He has also coauthored a wide range of mathematics textbooks with Professor Ron Larson. Their textbooks have been honored with various awards from the Text and Academic Authors Association.

By This Professor

Prove It: The Art of Mathematical Argument
Understanding Multivariable Calculus: Problems, Solutions, and Tips
Understanding Calculus II: Problems, Solutions, and Tips
Understanding Calculus: Problems, Solutions and Tips
Mathematics Describing the Real World: Precalculus and Trigonometry
Mathematics Describing the Real World: Precalculus and Trigonometry


An Introduction to Precalculus-Functions

01: An Introduction to Precalculus-Functions

Precalculus is important preparation for calculus, but it's also a useful set of skills in its own right, drawing on algebra, trigonometry, and other topics. As an introduction, review the essential concept of the function, try your hand at simple problems, and hear Professor Edwards's recommendations for approaching the course.

31 min
Polynomial Functions and Zeros

02: Polynomial Functions and Zeros

The most common type of algebraic function is a polynomial function. As examples, investigate linear and quadratic functions, probing different techniques for finding roots, or "zeros." A valuable tool in this search is the intermediate value theorem, which identifies real-number roots for polynomial functions.

31 min
Complex Numbers

03: Complex Numbers

Step into the strange and fascinating world of complex numbers, also known as imaginary numbers, where i is defined as the square root of -1. Learn how to calculate and find roots of polynomials using complex numbers, and how certain complex expressions produce beautiful fractal patterns when graphed....

31 min
Rational Functions

04: Rational Functions

Investigate rational functions, which are quotients of polynomials. First, find the domain of the function. Then, learn how to recognize the vertical and horizontal asymptotes, both by graphing and comparing the values of the numerator and denominator. Finally, look at some applications of rational functions.

31 min
Inverse Functions

05: Inverse Functions

Discover how functions can be combined in various ways, including addition, multiplication, and composition. A special case of composition is the inverse function, which has important applications. One way to recognize inverse functions is on a graph, where the function and its inverse form mirror images across the line y = x....

32 min
Solving Inequalities

06: Solving Inequalities

You have already used inequalities to express the set of values in the domain of a function. Now study the notation for inequalities, how to represent inequalities on graphs, and techniques for solving inequalities, including those involving absolute value, which occur frequently in calculus.

31 min
Exponential Functions

07: Exponential Functions

Explore exponential functions-functions that have a base greater than 1 and a variable as the exponent. Survey the properties of exponents, the graphs of exponential functions, and the unique properties of the natural base e. Then sample a typical problem in compound interest....

32 min
Logarithmic Functions

08: Logarithmic Functions

A logarithmic function is the inverse of the exponential function, with all the characteristics of inverse functions covered in Lecture 5. Examine common logarithms (those with base 10) and natural logarithms (those with base e), and study such applications as the "rule of 70" in banking....

30 min
Properties of Logarithms

09: Properties of Logarithms

Learn the secret of converting logarithms to any base. Then review the three major properties of logarithms, which allow simplification or expansion of logarithmic expressions-methods widely used in calculus. Close by focusing on applications, including the pH system in chemistry and the Richter scale in geology.

32 min
Exponential and Logarithmic Equations

10: Exponential and Logarithmic Equations

Practice solving a range of equations involving logarithms and exponents, seeing how logarithms are used to bring exponents "down to earth" for easier calculation. Then try your hand at a problem that models the heights of males and females, analyzing how the models are put together.

31 min
Exponential and Logarithmic Models

11: Exponential and Logarithmic Models

Finish the algebra portion of the course by delving deeper into exponential and logarithmic equations, using them to model real-life phenomena, including population growth, radioactive decay, SAT math scores, the spread of a virus, and the cooling rate of a cup of coffee.

31 min
Introduction to Trigonometry and Angles

12: Introduction to Trigonometry and Angles

Trigonometry is a key topic in applied math and calculus with uses in a wide range of applications. Begin your investigation with the two techniques for measuring angles: degrees and radians. Typically used in calculus, the radian system makes calculations with angles easier.

30 min
Trigonometric Functions-Right Triangle Definition

13: Trigonometric Functions-Right Triangle Definition

The Pythagorean theorem, which deals with the relationship of the sides of a right triangle, is the starting point for the six trigonometric functions. Discover the close connection of sine, cosine, tangent, cosecant, secant, and cotangent, and focus on some simple formulas that are well worth memorizing.

32 min
Trigonometric Functions-Arbitrary Angle Definition

14: Trigonometric Functions-Arbitrary Angle Definition

Trigonometric functions need not be confined to acute angles in right triangles; they apply to virtually any angle. Using the coordinate plane, learn to calculate trigonometric values for arbitrary angles. Also see how a table of common angles and their trigonometric values has wide application.

32 min
Graphs of Sine and Cosine Functions

15: Graphs of Sine and Cosine Functions

The graphs of sine and cosine functions form a distinctive wave-like pattern. Experiment with functions that have additional terms, and see how these change the period, amplitude, and phase of the waves. Such behavior occurs throughout nature and led to the discovery of rapidly rotating stars called pulsars in 1967.

32 min
Graphs of Other Trigonometric Functions

16: Graphs of Other Trigonometric Functions

Continue your study of the graphs of trigonometric functions by looking at the curves made by tangent, cosecant, secant, and cotangent expressions. Then bring several precalculus skills together by using a decaying exponential term in a sine function to model damped harmonic motion.

32 min
Inverse Trigonometric Functions

17: Inverse Trigonometric Functions

For a given trigonometric function, only a small part of its graph qualifies as an inverse function as defined in Lecture 5. However, these inverse trigonometric functions are very important in calculus. Test your skill at identifying and working with them, and try a problem involving a rocket launch.

32 min
Trigonometric Identities

18: Trigonometric Identities

An equation that is true for every possible value of a variable is called an identity. Review several trigonometric identities, seeing how they can be proved by choosing one side of the equation and then simplifying it until a true statement remains. Such identities are crucial for solving complicated trigonometric equations.

32 min
Trigonometric Equations

19: Trigonometric Equations

In calculus, the difficult part is often not the steps of a problem that use calculus but the equation that's left when you're finished, which takes precalculus to solve. Hone your skills for this challenge by identifying all the values of the variable that satisfy a given trigonometric equation.

31 min
Sum and Difference Formulas

20: Sum and Difference Formulas

Study the important formulas for the sum and difference of sines, cosines, and tangents. Then use these tools to get a preview of calculus by finding the slope of a tangent line on the cosine graph. In the process, you discover the derivative of the cosine function.

31 min
Law of Sines

21: Law of Sines

Return to the subject of triangles to investigate the law of sines, which allows the sides and angles of any triangle to be determined, given the value of two angles and one side, or two sides and one opposite angle. Also learn a sine-based formula for the area of a triangle.

30 min
Law of Cosines

22: Law of Cosines

Given three sides of a triangle, can you find the three angles? Use a generalized form of the Pythagorean theorem called the law of cosines to succeed. This formula also allows the determination of all sides and angles of a triangle when you know any two sides and their included angle.

31 min
Introduction to Vectors

23: Introduction to Vectors

Vectors symbolize quantities that have both magnitude and direction, such as force, velocity, and acceleration. They are depicted by a directed line segment on a graph. Experiment with finding equivalent vectors, adding vectors, and multiplying vectors by scalars.

32 min
Trigonometric Form of a Complex Number

24: Trigonometric Form of a Complex Number

Apply your trigonometric skills to the abstract realm of complex numbers, seeing how to represent complex numbers in a trigonometric form that allows easy multiplication and division. Also investigate De Moivre's theorem, a shortcut for raising complex numbers to any power.

32 min
Systems of Linear Equations and Matrices

25: Systems of Linear Equations and Matrices

Embark on the first of four lectures on systems of linear equations and matrices. Begin by using the method of substitution to solve a simple system of two equations and two unknowns. Then practice the technique of Gaussian elimination, and get a taste of matrix representation of a linear system.

31 min
Operations with Matrices

26: Operations with Matrices

Deepen your understanding of matrices by learning how to do simple operations: addition, scalar multiplication, and matrix multiplication. After looking at several examples, apply matrix arithmetic to a commonly encountered problem by finding the parabola that passes through three given points.

31 min
Inverses and Determinants of Matrices

27: Inverses and Determinants of Matrices

Get ready for applications involving matrices by exploring two additional concepts: the inverse of a matrix and the determinant. The algorithm for calculating the inverse of a matrix relies on Gaussian elimination, while the determinant is a scalar value associated with every square matrix.

30 min
Applications of Linear Systems and Matrices

28: Applications of Linear Systems and Matrices

Use linear systems and matrices to analyze such questions as these: How can the stopping distance of a car be estimated based on three data points? How does computer graphics perform transformations and rotations? How can traffic flow along a network of roads be modeled?

32 min
Circles and Parabolas

29: Circles and Parabolas

In the first of two lectures on conic sections, examine the properties of circles and parabolas. Learn the formal definition and standard equation for each, and solve a real-life problem involving the reflector found in a typical car headlight.

30 min
Ellipses and Hyperbolas

30: Ellipses and Hyperbolas

Continue your survey of conic sections by looking at ellipses and hyperbolas, studying their standard equations and probing a few of their many applications. For example, calculate the dimensions of the U.S. Capitol's "whispering gallery," an ellipse-shaped room with fascinating acoustical properties.

31 min
Parametric Equations

31: Parametric Equations

How do you model a situation involving three variables, such as a motion problem that introduces time as a third variable in addition to position and velocity? Discover that parametric equations are an efficient technique for solving such problems. In one application, you calculate whether a baseball hit at a certain angle and speed will be a home run.

31 min
Polar Coordinates

32: Polar Coordinates

Take a different mathematical approach to graphing: polar coordinates. With this system, a point's location is specified by its distance from the origin and the angle it makes with the positive x axis. Polar coordinates are surprisingly useful for many applications, including writing the formula for a valentine heart!...

32 min
Sequences and Series

33: Sequences and Series

Get a taste of calculus by probing infinite sequences and series-topics that lead to the concept of limits, the summation notation using the Greek letter sigma, and the solution to such problems as Zeno's famous paradox. Also investigate Fibonacci numbers and an infinite series that produces the number e....

32 min
Counting Principles

34: Counting Principles

Counting problems occur frequently in real life, from the possible batting lineups on a baseball team to the different ways of organizing a committee. Use concepts you've learned in the course to distinguish between permutations and combinations and provide precise counts for each.

30 min
Elementary Probability

35: Elementary Probability

What are your chances of winning the lottery? Of rolling a seven with two dice? Of guessing your ATM PIN number when you've forgotten it? Delve into the rudiments of probability, learning basic vocabulary and formulas so that you know the odds.

30 min
GPS Devices and Looking Forward to Calculus

36: GPS Devices and Looking Forward to Calculus

In a final application, locate a position on the surface of the earth with a two-dimensional version of GPS technology. Then close by finding the tangent line to a parabola, thereby solving a problem in differential calculus and witnessing how precalculus paves the way for the next big mathematical adventure.

31 min