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Trigonometry: Sum & Difference Identities

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About this Lesson

  • Type: Video Tutorial
  • Length: 4:41
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 50 MB
  • Posted: 07/01/2009

This lesson is part of the following series:

Trigonometry: Full Course (152 lessons, $148.50)
Trigonometry: Trigonometric Identities (23 lessons, $26.73)
Trigonometry: The Sum and Difference Identities (3 lessons, $3.96)

Taught by Professor Edward Burger, this lesson comes from a comprehensive course, Trigonometry. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/trigonometry. The full course covers trigonometric functions, trigonometric identities, application of trig, complex numbers, polar coordinates, exponential functions, logarithmic functions, conic sections, 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.

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Using Sum and Difference Identities to Simplify an Expression
Now suppose someone said, "Hey, I want you to evaluate exactly some really complicated expression with just sines and cosines and tangents, whatever." For example, suppose someone said, "I want the exact value of this." Let's take a look at . Hey, what are you going to say? You're going to say, "Well, gee, ? I thought a standard one--I don't even know what that is in degrees. , I have no idea what this is." So there's no hope. There's no hope.
But supposed the person had put a gun to your head and said, "I want the answer and I want it now!" Well, now you're more nervous because of the gun to your head. This is an awful, really tense moment. But then in a moment of inspiration, you suddenly look at this and say, "Hey, wait a minute. I see a sine and then cosine plus--and then the sine of the other angle times the cosine of the first angle. That actually looks like the formula for the sum of angles with a sine in front, the sine of a sum." So actually, this is the answer to the question, what is the sine of the sum? . Let's just check.
Because what does sine do to a sum? Well, remember sine is friendly so it shares the wealth. You get sine of this times cosine of that, and sine is a friendly thing so plus-plus, and then switch the roles. So now we have the sine of this person times the cosine of that person. That's exactly the same thing. These are exactly the same.
But what is that sum? Well, that's sine of--well, is the same thing as , and is what? Well, it's . But is the same thing as . And what's ? Well, that one we know. You can think of the graph or just know that in fact is 1. So this horrendous thing turns out just to be 1. Even though I don't know the value for any one of the individual things, I know when you put it up in this combination, since it equals this, it equals 1. So sometimes when you see something really complicated like this, you may be able to sort of condensify it into just one trig function of a sum or a difference, looking at this formula and detecting what it would be.
Let me show you one more example to sort of drive this home. How about if someone says to you, "Find the value of ." Well, of course, this may appear hopeless at first, but then you immediately say, "Gee, that looks like something to do with the tangent of either a sum or a difference of angles, except that negative sign; this must be the difference of angles." So immediately without even sweating, you're going report, "Well, wait a minute. That's exactly tan 47°-2°." Because remember the formula for the tangent of a difference is the tangent of one minus the tangent of the second, divided by 1--and then you switch the sign to plus, the product of the tangent. So what is that? Well, that's just tan 45°. And what's the tangent of 45°? Well, remember it's just . Sin 45° is . Cos 45° is . So divided by itself is 1. So in fact, tan 45° = 1. So this complicated expression turns out to exactly equal 1. Even though I don't know the value of tan 47° or the value of tan 2°, I know exactly what the combination of them is. So that's sort of interesting. This really complicated expression turns out to equal 1, and this really complicated expression turns out to be 1, so therefore if anyone ever gives you a really complicated expression with trig, the answer must be 1! Well, not quite. But anyway, try these and see if you can figure out which one is 1 and which one isn't 1. See you soon.

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