Series: Pre-Calculus Review

Need more practice problems

08/03/2009

~ tansley

The information is fantastic. However - practice makes perfect - so it would be nice to see a few examples.

Great lesson, but could use more real application

07/25/2009

~ sharon5

When would you have a real problem where you would be looking for a position of a spring? I know that is what my students are going to be asking.

Easy way to identify even and odd functions!

07/20/2009

~ brittanie

Great online video tutorial on how to verify if a trig function like sin or cos is even or odd. This question comes up on many college prep assessment tests and it's great to have these videos available for easy review online.

clear and easy inverse sine and cosine

06/04/2009

~ joyherber

This was great. My son was stuck on some homework and it had been years since I had studied sine and cosine. Once he watched the video and reread the assignment it all be came so simple. Thanks.

FANTASTIC!

06/01/2009

~ ht1979

This lesson walks you through several different proofs of trig identities - he does a good job at going step by step and so I didn't get lost anywhere along the way. Great lesson!

Very Well done

05/30/2009

~ isioma

Still a little confused but not because of tutorial.

SOOO EXCITED ABOUT THIS...

04/03/2009

~ Yaa

This video was incredible! I was studying for ACTs when this question came up about odd functions...which I don't think I've ever encountered yet. But this is straight-forward and worthwhile to watch. Thanks for making this tutorial - simply can't thank you enough!!! I'll definitely be looking out for some more tutorials :D

Below are the descriptions for each of the lessons included in the series:

• Pre-Calculus: Solving Trig Equations by Factoring

  Professor Burger teaches how to solve more complicated equations (tanx * sin^2x = tan x) involving trigonometric functions in this lesson. Solving these types of problems involve use of trig identities, factoring, etc and how to find all of the viable solutions for these types of problems. In the problem listed above, Professor Burger will show you how to factor the equation in order to help simplify and then solve it. Professor Burger also gives a warning about cancelling out in equations that involve trig functions. By canceling, you risk missing valid solutions and solution sets.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  About Professor Edward Burger:
  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Finding Coterminal Angles

  Coterminal angles are angles that have their terminating rays in the exact same location. By definition, any two angles whose difference is some multiple of 360 degrees. In this lesson, Professor Burger will show you what coterminal angles look like, how to determine if two angles are coterminal using simple math and then he will review several examples of coterminal angles. He will also demonstrate visually how coterminal angles behave and why the definition is appropriate for these types of angles.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Fundamental Trigonometric Identities

  In this lesson, Professor Burger will reveal and explain several basic trigonometric identity proofs. He will begin by reviewing the definitions of sine, cosine, and tangent. From these definititions, he will prove tanx = sinx/cosx. Then, he uses the Pythagorean Theorem to show you the proofs for 3 more trigonometric identities: cos^2 + sin^2 = 1, 1+ tan^2 = sec^2, and 1 + cot^2 = csc^2. Finally, Professor Burger will tell you which of these identities and proofs you need to memorize and which you can derive simply and don't need to fret about memorizing in advance of your test.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Evaluating Inverse Trig Functions

  Professor Burger shows you how to evaluate inverse trigonometric functions, which include arc sine, arc cosine, arc tangent, etc. He reminds you that inverse functions are asking "What is the angle whose function is X?" Thus, the output for any inverse trig function should be an angle for which, if you apply the indicated trig function, you get the indicated value as a result. Then he walks through finding the inverse of sine, the inverse of cosine, and the inverse of tangent. Finally, Prof Burger shows you how to interpret the presence of a negative sign and how to evaluate inverse trig functions using a calculator (indicated in radians).

  This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Adding Vectors & Multiplying Scalars

  Professor Burger shows you how to add and subtract vectors and use scalar multiplication to elongate or shrink vectors while maintaining their direction angle. The magnitude of a vector can be altered with scalar multiplication. A scalar is simply a number (positive or negative or a fraction) used to multiply a vector by, with the vector keeping its same direction and changing magnitude. Vectors can also be added and subtracted by simply adding or subtracting the components. It is also simple to find the answer graphically by creating a parallelogram with the two vectors, which Professor Burger demonstrates.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Trig Equations with Coefficients

  Now that you have learned how to solve simple trigonometric equations, Professor Burger will show you how to solve trig equations that have a coefficient in the argument (e.g. sine of 2X versus just sine of X). These are also called multiple angle equations. In evaluating these trigonometry equations, you will generally only be asked to find solutions in one or two periods of the functions, so you will not have an infinite number of solutions (most often, you'll solve for solutions between 0 and 2*pi). Sample problems from this lesson include (2*sin 3*theta) = 1 and tan^2(2X) = 1 and sin2x*tan2x + sin2x = 0.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Graphing Period, Amplitude, Shifts

  Professor Burger shows you how to use all of the tools at your disposal to effectively graph complicated trigonometric functions involving sine and cosine. He will show you how to recognize changes in period, amplitude, and vertical and phase shifts in the equation and how to correctly incorporate them into your trig function graph. He will also show you a three-step process of translating the equation, graphing the intermediate steps, and finailzing the graph. The examples you will use are y = -2sin(x- Pi/4)+1 and y = 2cos(Pi*x)-2. These equations both involve complications like those listed above (as indicated by their added constants and coefficients).

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Complex Numbers - Trig or Polar Form

  This lesson instructs you on how to convert complex numbers into trig form (also known as polar form). Complex numbers, written in the form (a + bi), are an extension of the real numbers obtained by adjoining an imaginary unit, denoted by i, which is the square root of negative 1. To convert complex numbers into trigonometric or polar form, Professor Burger first walks you through sketching a graph of the number and drawing a right triangle. From that, he shows you how to use the trig properties to find the unknown values and the modulus. Then, you plug these falues into the trig form and determine the angle. To illustrate this method, Professor Burger walks you through an example in which he converts (-(3^1/2), +i) to polar or trigonometric form.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Inverse Trig Function Equations

  An inverse function asks the question ""What is the angle whose function is X."" In this lesson, you will learn to solve equations that include an inverse function (arc sine, arc cosine, arc tangent, etc). Professor Burger first shows you how to untangle the equation, re-writing it so that you can understand for what you are solving. He will also show you examples when there may be an infinite numbers of solutions, and how you will need to correctly denote this answer. Finally, he suggests that you check your answers by graphing, and shows you how. This lesson will include several examples of evaluating problems involving arc sin, arc cos, etc. You will begin by seeing how to approach and solve a problem like 'inverse cosine of cosine x = pi/4' While it would seem that the cosine and inverse cosine here would cancel, you will learn in this lesson why this is not the case and how you can correctly solve for the answer.

  This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: The Law of Sines

  Trigonometric functions (sin, cos, tan, etc) originally arose from the ratios of the sides of right triangles. But we can still use sine to evaluate the sines of angles that are in a triangle but not in a right triangle, using the Law of Sines. The Law of Sines states that [(sin a)/A] = [(sin b)/B] = [(sin c)/C], where a is the angle opposite side A (and so on for b/B and c/C). Sometimes the angles, a, b, and c, in this equation are denoted by the Greek symbols for alpha, beta, and gamma. Professor Burger shows you how to think about and use this this law by working through a number of different examples. This law lays the foundation for proving properties about triangles that don't have a right angle, including the calculation of the lengths of their sides and the measures of their angles.

  This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: The Law of Cosines

  In this lesson, Professor Burger begins with a review of the Law of Sines. He then introduces the Law of Cosines, which extends the Pythagorean Theorem to triangles that are not right, allowing us to solve for any angle in the triangle. The Law of Cosines, also called the Al-Kashi Law and the Cosine Formula and the Cosine Rule, states that, for the angle you are solving for, opposite side ^2 = (sum of the squares of adjacent sides) - 2 * (product of the adjacent sides) (cos of the desired angle). The law of cosines is most useful when computing the third side of a triangle when two sides and their enclosed angle are known (SAS) or when computing the angles of a triangle in which all sides are known (SSS).

  This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Convert between Degrees and Radians

  In this lesson, Professor Burger teaches the basics of degrees and radians as they relate to the measurement of angles. He will cover how these terms are related (e.g. 360 degrees= 2*pi radians), how they are different from each other and why they are used in different situations . This lesson should answer and assortment of questions, including: Why do we use 360 degrees? Is there a better way to measure angles? Radian measurement is a different way to measure angles, and is the method of angle measurement used in trigonometric functions. You will learn how to measure angles using radians instead of degrees, how to convert from degrees to radians, and what you should memorize to simplify this conversion. Additionally, Professor Burger will explain the rationale behind using radians in place of degrees at times (mostly with trigonometric functions). Finally, you will review several examples of this conversion.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Word Problems with Sine or Cosine

  Professor Burger provides a real-world application of periodicity using tidal waves (waves that happen after an earthquake, also called tsunami waves) as an example. In the example, the waves are moving 540 feet/second (or 370 mph) and peaking at a hight of 50 feet every twenty minutes (period of the wave). The question posed is what is the length between each wave? Ocean waves, like sound waves, have a sine curve. You will use this knowledge and the distance formula (D = r*t) to solve a word problem about tsunami waves. These types of sinusoidal waves occur frequently in nature (and often in math word problems). Knowing how to approach and evaluate them is key to being able to solve all of them.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Intro to Sine and Cosine Graphs

  In this lesson, you will examine the graphs of both the following trigonometric functions: sine and cosine. Professor Burger will show you how to graph sin and cos and teach you the acronym ASTC (All Students Take Calculus). Prof Burger also defines and shows you where to look for to evaluate the amplitude, period, and zeros of the sine and cosine graphs and shows you how to find and determine the maximums and mininimums for both sine and cosine functions. Finally, he will compare the graphs of the two functions, demonstrating that they have an identical shape with merely a shift between them to differentiate the two functions from each other. You'll also learn the importance of the π/2 interval in plotting and remembering the trigonometric function graphs of cosine and sine.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Trig to find a Right Triangle angle

  This lesson teaches you to evaluate trigonometric functions to find one (non-right) angle of a right triangle. To do this, Professor Burger will walk through an example in which he presents a right triangle with two known sides and one known angle. From this information, all other angles and sides can be determined using trigonometric functions for the angle (sine, cosine, tangent, cosecant, secant, cotangent, etc). To determine the unknowns, he will apply a range of different formulas (involving the identification and use of opposite and adjacent sides of the various angles). Trigonometric functions are ratios of the sides of a triangle. You will cover an example finding all the trigonometric functions for a right triangle, beginning with finding the hypotenuse using the Pythagorean theorem. Then you will learn how to find the rest of the trig functions for a triangle if you given one function.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Simplifying Using Trig Identities

  Professor Burger demonstrates how to use the fundamental trigonometric identities to simplify complex triognometric expressions. Many seemingly complex expressions can be greatly simplified with simple application of several trigonometric identities. You will practice using expressions such as tanx * cosx and [1+(tan^2)x]/(csc^2)x. In these problems, you will see how substituting 1/cos^2 for sec^2 or 1/sin for csc or sin/cos for tan. By applying the definitions of the different trig functions, you'll often be able to substantially simplify a problem to the point where it will be very easy for you to evaluate and solve it.

  This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Finding Vector Magnitude & Direction

  In this lesson, Professor Burger will show you how to find the magnitude and direction angle of a vector provided in standard form (and how vector notation associated with standard form looks and should be interpreted). He will also review how to depict a vector graphically that we have described by standard form. The magnitude is the length of a vector (reminder: it must be positive), and we use the Pythagorean theorem to calculate the vector's magnitude. The direction angle is always measured counter-clockwise from the positive side of the x-axis. Once we know the vector's length, we can use trigonometric functions to calculate the direction angle of the vector. Last, Professor Burger solves for the magnitude and direction of some of the vectors using a calculator.

  This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Graphing tan, sec, csc, cot

  This lesson introduces the graphs of all the other trigonometric functions (cosecant, secant, tangent, cotangent), using the sine and cosine graphs for points of comparison. Professor Burger shows you how to graph tanx using the identity tanx = sinx/cosx. This graph has asymptotes at all the multiples of Pi/2 and a period of Pi/absolute value of b. Next, you learn to graph secx, which is equal to 1/cosx. This means that secant has an asymptote anywhere cosx = 0. Next, Prof. Burger graphs cosecant, using the identity that cscx = 1/sinx. This graph is identical to secx, but shifted, like the relationship between sin and cos. Finally, you will learn to graph cotx, which is equal to 1/tanx. This means that there will be asymptote where tanx = 0, and zeros where tanx has asymptotes.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Trig Equations and Quadratic Formula

  Sometimes, trigonometric equations cannot be factored. To solve these equations, Professor Burger shows you how to apply the quadratic formula to find solutions to these equations. This is a multi-step process that starts with simplifying the equation. After the equation is simplified, you will be able to solve the quadratic formula and then enter that answer in to solve for X. In the lesson example, Professor Burger uses both a calculator and graphing to ensure he has the correct points. This lesson explains the covered material by walking through sample equations 3sin^2(2x) + sin (2x)-1 = 0. This equation has 2 solutions over the interval from zero to 2*pi. This lesson is loaded with warnings about easy mistakes to make and pitfalls to be wary of when evaluating problems like this.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Trig Functions - Odd, Even, Neither?

  In this lesson, Professor Burger teaches you how to determine if a function is even, odd, or neither. He begins by defining even and odd functions and graphing them. A function is even if the function of negative x is equal to the function of x. The graph of an even function is symetric across the y-axis. A function is odd if the function of negative x is equal to the negative function of x. The graph of an odd function is symetric around the origin. After defining these, Professor Burger identifies whether sin and cos are even or odd, and then shows several more examples, including tan x, sin (2x), (sin x)/x, and x cos x. Lastly, Professor Burger describes and illustrates what a function looks like that is neither odd nor even. In this case, it is not symmetric to the Y axis or the origin.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Using Double-Angle Identities

  Double-angle identities allow you to simplify trigonometric equations with a 2 as the coefficient. (similar formulae exist for trig functions with 1/2 or 3 as the coefficient). In this lesson, Professor Burger uses the equation cos2x = sinx as an example. If this equation were simply cos x = sinx, we could divide to re-write the formula as sinx/cosx = tan x = 0, but in this case, we have a coefficient in advance of one of the arguments, which is why we need to use the double-angle formulas. After using the double-angle formulas in the provided example to simplify, you can further simplify these equations using trig identities (like the Pythagorean identity) and factoring. These tools will help you to solve many trig equations. The duble angle identities for sine, cosine, tangent and cotangent are: sin2x = 2sinxcosx, cos2x = cos^2x-sin^2x, tan 2x = 2tanx/(1-tan^2x), and cot2x = (cot^2x-1)/2cotx.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Using Sum and Difference Identities

  Using the sum and difference identities, Professor Burger shows you how to solve a trig function for an unknown angle. For an example, he uses the sin(15 degrees). You can use known angle values for 45 degree and 30 degrees and the formula for the difference of two sines to find the solution. The formula for the difference of two sines is sin(x1 - x2) = sinx1cosx2 - cosx1sinx2. Hence, if you know what the sin and cos values of 30 and 45 degrees are, you should be able to plub them into this formula to arrive at the sine value of 15 degrees. The beauty of the sum and difference formulas for trig functions is that they allow us to decompose a problem we don't know the answer to into component parts to which we do know the answer, thus solving the original problem.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Graph Sine, Cosine with Phase Shifts

  Now that you have learned how to graph the sine and cosine functions, Professor Burger asks the question ""How does changing the x-value affect the graph?"" He shows you how adding or subtracting to the x-value can actually change graphs of the sine and cosine functions, a process called translation. Professor Burger also warns you about classic mistake #8, reminding you that adding and subtracting to the x-value actually creates the opposite effect when graphed (adding to X moves the graph in the negative direction). Finally, Professor Burger shows you how to simplify the equation y = 3sin(x + Pi/2) using translation. The key lies in the fact that adding or subtracting pi/2 or 2*pi to a sine or cosine function means there are some shortcuts that you can take to determine what the graph of the function looks like (e.g. the graph of sine of (x+pi/2) is the same as the graph of cosine and the same as the graph of sine of (x+2*pi)).

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Solving Trigonometric Equations

  This lesson will teach you how to solve equations involving trigonometric functions. Professor Burger shows you two ways to look at these equations, on a graph, and using reference angles. You will learn to rephrase the equation to determine what it is really asking, ""What value of X makes the function of X = n?"" You will learn how to write your answer to indicate infinitely many solutions and the step-by-step process of solving the equations. Examples of problems covered in this lesson involve trig functions, roots, fractions, variables and coefficients, including problems like cos x = 1/2 and sinx = -(2^(1/2)/2). You'll also learn when and why most trig problems like these have multiple (or infinite) solutions and how to correctly identify and denote these solution sets.

  This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Polar & Rectangular Coordinates

  You will learn how to convert from polar coordinates to rectangular coordinates (or Cartesian coordinates or coordinates in a Cartesian plane), and vice versa in this lesson. First, Professor Burger gives you an overview of polar and rectangular coordinates. Then, you will learn how to convert a polar cordinate (r, Theta) into a rectangular coordinate (x, y), using the equations x = rcosTheta and y = rsinTheta. To convert from rectangular to polar, you will use the equations r = root(x^2 + y^2) and Theta = arctan (y/x). To illustrate the use of all of these formulae, Professor Burger will walk you through the conversion of (3, pi/6) from polar to rectangular coordinates and the conversion of (-1.1) in rectangular coordinates to equivalent polar coordinates.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Word Problems and Trig Equations

  This lesson provides a more real-world application of trigonometric equations using a word problem. Professor Burger walks you through solving the trig word problem about spring motion. The motion of the particular spring in question is described by the function: sin (2T)+ 3^(1/2)*cost (2T), where T is the time in seconds. The problem asks us to solve for all times, T, when the object is located where it started the experiment. As he demonstrates how to solve the word problem, Prof Burger uses many of the trig information he taught in previous lessons, including identities, graphing, and angles. Finally, he reminds you to check your answer to make sure that the solutions are allowable. Additionally, he highlights that you should 'reality check' your answer as it's obviously not possible to have solutions for T that are negative given that this is a real-world example and time should never be negative.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Factoring Trigonometric Expressions

  Just as you can simplify trigonometric expressions using the trig identities, you can often simplify the expressions by factoring, as you would with other types of expressions. Simplifying using the identities and factoring will save you a lot of effort in solving trig problems. In order to recognize opportunities to factor when working trigonometric problems, Professor Burger recommends that you use trigonometry identities to convert trig functions to sin and cos, whenever possible. Some examples you will learn how to simplify include (sin^2) x+ (sin^2)x(cos^2)x and sinx - (cos^2)x - 1 and sin^2x + (2/cscx)+x.

  This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Graph Sine, Cosine with Coefficients

  After learning how to graph the sine and cosine functions, now we will modify the graphs of these functions by adding in coefficients. Professor Burgers shows you a simple, 2-step process to determine the graphs. First, he will teach you about changes in the coefficient of the function. The introduction of a coefficient changes the amplitude of the graphed trigonometric function (sine or cosine). This is difference between AM (amplitude modulation) radio stations; changes in amplitude produce AM radio signals.The amplitude is equal to the absolute value of the coefficient of the trigonometric function. Prof Burger will also show you how changing the coefficient of the independent variable changes the period of the graphed sine or cosine function. This is the difference in FM radio stations (frequency modulation). The period =(2 Pi) / coefficient of X.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Power and Roots of Complex Numbers

  Professor Burger explains how to find the powers and roots of complex numbers. The equation of a complex number is z= r(cosx + isinx). To raise the complex number to a power, n, the equation is z^n = r^n[cos(nx) + isin(nx)]. In general, if you are raising a complex number to the power of n or 1/n (taking the nth root), you will come up with n solutions, as you will always have one solution for each of the degrees of power. When taking the root of a complex number, you will find one solution for each degree of power. To find the nth root of a complex number the equation is n root of z = (n root r) *[cos ((x + 2 Pi K)/n) + 1 sin ((x + 2 Pi K)/n)] where k = 0, 1, 2,...n-1.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Trig to Find Right Triangle sides

  Professor Burger explains how to use trigonometry to find the unknown sides of right Triangles. First, he explains how to evaluate and use trig functions with the help of a calculator. Then he will teach you how to find the hypotenuse of a right triangle using trigonometry, as well as how to find the adjacent side of a right triangle, given the measure of the hypotenuse. Once you have the measures of two sides of a right triangle, you will be able to deduce what the third side is equal to by applying the Pythagorean Theorem. Professor Burger will also highlight that trigonometry uses functions (e.g. you are not multplying by sine, but finding the function, sine, of a number). This is an especially important distinction to remember when manipulating trigonometric expressions and applying trigonometric properties.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

• Pre-Calculus: Find Angle Complements & Supplements

  This trigonometry lesson introduces and explains the terminology behind complementary angles (angles that add up to 90 degrees) and supplementary angles (that add up to 180 degrees). Professor Burger gives you the definition of these two types of angles, shows you what these angles will look like when combined and and shows you how to find a complement angle or a supplement angle to a provide angle with a known degree measure. Professor Burger will also explain whether supplementary angles and complementary angles could be negative.

  Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Precalculus. This course and others are available from Thinkwell, Inc. The full course can be found at: http://www.thinkwell.com/student/product/precalculus. The full course covers angles in degrees and radians, trigonometric functions, trigonometric expressions, trigonometric equations, vectors, complex numbers, and more.

  Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

  He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

  Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

  Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.