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Pre-Calculus: Using Sum and Difference Identities

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About this Lesson

  • Type: Video Tutorial
  • Length: 3:12
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 34 MB
  • Posted: 01/22/2009

This lesson is part of the following series:

Trigonometry: Full Course (152 lessons, $148.50)
Pre-Calculus Review (31 lessons, $61.38)
Trigonometry: Trigonometric Identities (23 lessons, $26.73)
Trigonometry: The Sum and Difference Identities (3 lessons, $3.96)

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.

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Trigonometric Identities
The Sum and Difference Identities
Using Sum and Difference Identities Page [1 of 1]
Okay, so let’s see if we can use these types of formulas, the sums and differences of angles formula for trig functions,
to actually now compute precisely—precisely, without even a calculator—values of trig functions that heretofore
eluded us. Let me give you an example just to illustrate how this sort of thing works.
Suppose I want to find the sine of 15°. And I’ll use degrees now for a little bit. Well, that’s not one of those standard
formula angles that we know the values of, so that’s sort of bad. But notice that 15° can be written as 45° - 30° and
each of those are standard angles that we should know, in fact precisely, those trig functions for. So in fact I see that
sine of 15° actually is the sine of a difference of two angles where I know all the information for them individually. So
through those difference and sum formulas, I should be able to find the formula exactly. Well, let’s take a look and
see.
So let’s use the difference formula for sine. How does it work for sine? Let’s see, sine is good, so it’s going to share
the limelight. So I see first of all sine of 45° but it’s going to share, so cosine of 30°. And since it’s good, since I’m
subtracting here, I’ll subtract here. And now second billing, everyone will get a shot. So I’ll see cosine of the 45 times
the sine of the 30. But each of these I should know the value of, right? Sine of 45° is
2
2
. Cosine of 30° is
3
2
.
These are the things I just read from my chart that I would make us so I could actually know these values precisely.
Cosine of 45° is
2
2
. And sine of 30° is just
1
2
. So what do I see? I see, well, 2 times 3 is actually 6
because these are both to the ½ power, so I can just combine the bases to that same ½ power. 2 x 2 = 4, multiplying,
minus
2
4
. And so what I see is the sine 15° is exactly
6 2
4
?
. Isn’t that cool? We see a new fact about the
sine. We see its exact value at 15°. It’s equal to
6 2
4
?
. I wouldn’t have known that otherwise. If you look at a
calculator you won’t get this value. You’ll get some decimal rough approximation. This is exactly what it equals.
So you can really see the power of these sum and difference formulas. The power is, they allow us to sometimes take
an angle that we don’t know and decompose it as either a sum or a difference of angles that we do know, and then
use the sum or difference formulas to sort of report the news just in terms of the angles that we know, and actually get
a precise answer. That really is the value and the importance of these sum and difference formulas. But up next we’ll
take a look at how you can actually use the sum and difference formulas in a slightly different guise. I’ll see you there.

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