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Pre-Calculus: Simplifying Using Trig Identities

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About this Lesson

  • Type: Video Tutorial
  • Length: 5:28
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 59 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: Simplifying Trig Expressions (5 lessons, $7.92)

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.

About this Author

Thinkwell
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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...

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Recent Reviews

Nopic_orng
Very Well done
05/30/2009
~ isioma

Still a little confused but not because of tutorial.

Nopic_orng
Very Well done
05/30/2009
~ isioma

Still a little confused but not because of tutorial.

Trigonometric Identities
Simplifying Trig Expressions
Simplifying a Trigonometric Expression Using Trigonometric Identities Page [1 of 2]
Let?s use the trig identities that we just looked at to actually simplify very complicated-looking trig expressions.
Because it turns out that some of these real trig expressions that look awful, really can be reduced down to something
really, really simple. And these are just great fun because they?re almost like puzzles, you know, you sort of having
his awful looking thing and you have to massage and massage and massage until at the end of the day, you feel
completely relaxed after a massage and you have a nice small trig expression. Let me show you what I mean.
Let?s start of with this one. Now, this may not look that threatening to you: tan(x) x cos(x). Let?s see if we can
actually make that thing look even better. So, I want to just see if I can simplify this trig expression. Well, I can
remember what tangent is. Tangent is just
sin
cos
x
x
. So, in fact, I could write this as sin(x) ÷ cos(x) and then if I
multiply that by cos(x) and I see that these cosines cancel. Now, of course, there?s a little thing we shouldn?t be too
sloppy about. I can only cancel as long as cos(x) isn?t equal to zero. Now there are a lot of places where cosine is
zero. But as long as cosine is not zero, I can cancel, and I see this thing actually is the same thing as sine of x. So,
that?s pretty cool. So, tan(x)cos(x) is actually equal to sin(x) as long as cos(x) isn?t zero. So, you can see how I can
simplify this thing in just one little thing like that.
Let?s try some more. I just love these things. I can do this for hours, by the way. 1 + tan2? ÷ csc2? Okay, well, what
can you do here? Well, there?s two ways to proceed here. If you?re really, really sort of sharp and clever and with it,
which, by the way, I am not, you could use the Pythagorean Identity which says 1 + tan2? = secant2?. So you could
just replace this whole thing by secant2? and start simplifying. I?ll do that this one time. So the top, using one of those
Pythagorean Identities we talked about, is just secant2? and I divide that by csc2? . That looks like maybe that?s the
best we can do, but let?s convert everything to sines and cosines. Because, in fact, that?s always the strategy I use.
Always. Because it?s easier for me to know -- I know everything about sines and cosines. There?s nothing in the
world I don?t know about sines and cosines. Well, there probably is. But, I?m not going to tell you about them.
Anyway, the point is when I see tangent and cosecants and stuff like that, I usually convert them to sines and cosines.
I?d write this as
sin
cos
x
x
and I?d write this as 2
1
sin
. So, if I were to translate all these things into sines and cosines,
what would I see?
Well, secant2 is just the same thing as 2
1
cos ?
and csc2 is just 2
1
sin ?
. Now, you have to be careful here. What
happens when I invert and multiply? The 2
1
sin
becomes
sin2
1
. So I see the sin2 on top and the cos2 on the bottom.
This is just
2
2
sin
cos
?
?
. But that can be reduced even further, too. Because, what is
sin
cos
? That?s tangent. So, this is
just tangent2. So, this awful looking thing which has a tangent2 in there actually just equals tangent2 . It?s almost like
amazing magic, isn?t it? But just doing some very careful simplification, using all of the identities that we know, we can
actually write this complicated expression as a much easier expression to deal with. Neat.
One last one just for the road. 1 - cos2(t) ÷ sin(t). Let?s see if we can make this thing an easier expression. Well,
when I see a 1 - cos2(t), the thing I think about is this identity right here. Let me show it to you. It?s sin 2 ? + cos 2 ? =
1. In fact, notice that if I take this cos 2 ? and bring it to the other side, I?ll have a 1- cos 2 ?. So I see that 1 - cos 2 ? is
the same thing as sin 2 ?. It?s just this formula, this part, brought to the other side. So this thing is just sin 2 ?, or in our
case, sin 2 (t). So, again, using the Pythagorean Identity, I see the top is just
sin2
sin
t
t
. And I can cancel. Here I have
sine -- remember what this means, by the way -- this is sin(t) x sin(t). That?s what sin 2 (t) means, over sin(t). I can
cancel one of these away. Again, there?s a little proviso here. I have to make sure that the sine of t isn?t zero, but as
long as it is not zero, I can cancel and what do I get? What I get is sin(t). So, look, this really complicated looking
expression turns out to be the same thing as sin(t). So, it?s really pretty cool. You can take some of these really awful
Trigonometric Identities
Simplifying Trig Expressions
Simplifying a Trigonometric Expression Using Trigonometric Identities Page [2 of 2]
looking things and if they just sort of fit together just right, they can actually be simplified as a tremendous conspiracy,
and you?re just down to something simple.
It turns out these things will be very valuable and useful in sort of verifying certain identities and solving certain trig
equations. These techniques, even though they?re not that difficult, per se, I think they?re sort of fun and also really
important. Okay, up next, we?ll take a look at some more identity type stuff.

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