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Mathematics Describing the Real World: Precalculus and Trigonometry

Season 1
Finally make sense of the mysteries of precalculus and trigonometry in the company of master educator and award-winning professor Bruce Edwards. In 36 intensively illustrated episodes, 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.
201136 episodesTV-PG
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Episodes

  1. S1 E1 - An Introduction to Precalculus - Functions
    June 16, 2011
    31min
    TV-PG
    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 series. #Science & Mathematics
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  2. S1 E2 - Polynomial Functions and Zeros
    June 16, 2011
    31min
    TV-PG
    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.
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  3. S1 E3 - Complex Numbers
    June 16, 2011
    31min
    TV-PG
    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.
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  4. S1 E4 - Rational Functions
    June 16, 2011
    31min
    ALL
    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.
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  5. S1 E5 - Inverse Functions
    June 16, 2011
    32min
    TV-PG
    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.
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  6. S1 E6 - Solving Inequalities
    June 16, 2011
    31min
    TV-PG
    You've 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.
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  7. S1 E7 - Exponential Functions
    June 16, 2011
    32min
    TV-PG
    Explore exponential functions, which 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.
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  8. S1 E8 - Logarithmic Functions
    June 16, 2011
    30min
    TV-PG
    A logarithmic function is the inverse of the exponential function, with all the characteristics of inverse functions covered earlier. 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.
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  9. S1 E9 - Properties of Logarithms
    June 16, 2011
    32min
    TV-PG
    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 and are widely used in calculus. Close by focusing on applications, including the pH system in chemistry and the Richter scale in geology.
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  10. S1 E10 - Exponential and Logarithmic Equations
    June 16, 2011
    31min
    TV-PG
    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.
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  11. S1 E11 - Exponential and Logarithmic Models
    June 16, 2011
    31min
    TV-PG
    Finish the algebra portion of the series 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.
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  12. S1 E12 - Introduction to Trigonometry and Angles
    June 16, 2011
    30min
    TV-PG
    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.
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  13. S1 E13 - Trigonometric Functions - Right Triangle Definition
    June 16, 2011
    32min
    TV-PG
    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.
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  14. S1 E14 - Trigonometric Functions - Arbitrary Angle Definition
    June 16, 2011
    32min
    TV-PG
    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.
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  15. S1 E15 - Graphs of Sine and Cosine Functions
    June 16, 2011
    32min
    TV-PG
    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.
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  16. S1 E16 - Graphs of Other Trigonometric Functions
    June 16, 2011
    32min
    TV-PG
    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.
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  17. S1 E17 - Inverse Trigonometric Functions
    June 16, 2011
    32min
    TV-PG
    For a given trigonometric function, only a small part of its graph qualifies as an inverse function. 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.
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  18. S1 E18 - Trigonometric Identities
    June 16, 2011
    32min
    TV-PG
    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.
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  19. S1 E19 - Trigonometric Equations
    June 16, 2011
    31min
    TV-PG
    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.
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  20. S1 E20 - Sum and Difference Formulas
    June 16, 2011
    31min
    TV-PG
    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.
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  21. S1 E21 - Law of Sines
    June 16, 2011
    30min
    TV-PG
    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.
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  22. S1 E22 - Law of Cosines
    June 16, 2011
    31min
    TV-PG
    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.
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  23. S1 E23 - Introduction to Vectors
    June 16, 2011
    32min
    TV-PG
    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.
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  24. S1 E24 - Trigonometric Form of a Complex Number
    June 16, 2011
    32min
    TV-PG
    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.
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  25. S1 E25 - Systems of Linear Equations and Matrices
    June 16, 2011
    31min
    TV-PG
    Embark on the first of four episodes 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.
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Extras

Bonus

Mathematics Describing the Real World: Precalculus and Trigonometry
Mathematics Describing the Real World: Precalculus and Trigonometry
1minTV-PG
Finally make sense of the mysteries of precalculus and trigonometry in the company of master educator and award-winning professor Bruce Edwards. In 36 intensively illustrated episodes, 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.
Finally make sense of the mysteries of precalculus and trigonometry in the company of master educator and award-winning professor Bruce Edwards. In 36 intensively illustrated episodes, 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.
Finally make sense of the mysteries of precalculus and trigonometry in the company of master educator and award-winning professor Bruce Edwards. In 36 intensively illustrated episodes, 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.

Details

More info

Subtitles
None available
Producers
The Great Courses
Cast
Bruce H. Edwards
Studio
The Great Courses
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