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Pre-Calculus: Fundamental Trigonometric Identities


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About this Lesson

  • Type: Video Tutorial
  • Length: 7:24
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 80 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: Basic Trigonometric Identities (2 lessons, $2.97)

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 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

2174 lessons

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 or visit Thinkwell's Video Lesson Store at

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


Recent Reviews

~ hmt79

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!

~ hmt79

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!

Trigonometric Identities
Basic Trigonometric Identities
Fundamental Trig Identities Page [1 of 2]
Now what I?d like to look at are some basic identities. And what we?re going to start doing is thinking about how can
we verify some really complicated identities with trigonometric functions actually hold true. And the way we?re going to
see that is by starting off by just finding some really basic facts about how the trig functions sort of interact with
themselves. So, it?s going to be like a little trig-fest. So, let?s start off just very simply by first of all just recapping what
trig is all about. So here?s a real fast recap of trig.
We?ll start off with this little right triangle. I?ll call this angle ?. So, this is going to be the adjacent side. This is the
opposite side. This is the hypotenuse side. And so, what do I know? Well, I know that, let?s see, I can make a list of
things over here that I know. So I know that sine of ? equals opposite over hypotenuse. I know that cosine of ?
equals adjacent over hypotenuse. And I know that tangent of ? equals opposite over adjacent. So one thing that we
can see is -- well, what would happen if I just took sine and put it over cosine? So, if I take
, what would this
equal? Well let?s see. Well, sine is opposite over hypotenuse, so that?s opposite over hypotenuse. And then what?s
cosine? Well, cosine is adjacent over hypotenuse. And when I invert and multiply, I see that, in fact that the hypoteni
cancel, because I have a hypotenuse on the bottom and when I invert this, watch what happens. I have the
hypotenuse on top and so what I see here is opposite over adjacent. And what?s the name of that ratio? Well, that?s
exactly tangent. So, we just proved, in fact, our first basic identity. That tangent =
. So that?s a really important
one to remember, and not hard to verify for yourself. It?s just a matter of thinking about definitions and you?re sort of
on your way.
Now, how else can we use that triangle? Well, we can use this triangle because since it?s a right triangle, I can
actually _________ the Pythagorean Theorem. So what?s the Pythagorean Theorem say? Well, the Pythagorean
Theorem says that ?one leg squared plus the other leg squared is the third leg squared.? So, we have adjacent
squared plus opposite squared equals the hypotenuse squared. Okay, now what can I do here with this? Well, let me
divide everything through by the hypotenuse squared. So, I?m going to divide everything through by this term, just to
see what I get. If I do that, I would see adjacent over hypotenuse, all squared, plus opposite over hypotenuse, all
squared, equals one. I just divided everything through by the hypotenuse squared. But look, what do you see? Well,
adjacent over hypotenuse is just cosine. So this is just cos2.. And opposite over hypotenuse, well that?s just sin2. So,
on this side, I see cos2 + sin2 = 1. So, in fact, that is another identity. Which just follows from the Pythagorean
Theorem. So that one says that -- I?ll write it this way -- sine2? + cosine2? will always equal 1. And that?s just the
exact same statement as the Pythagorean Theorem, but in a trig kind of form, and really easy to prove. It?s literally
just this proof right here. Just take the Pythagorean Theorem, divide through by hypotenuse, and you get this.
What if we divide through by something else? Suppose, for example, we divide through by adjacent. If I divide
through by the adjacent here, what would I see? Well, if I divide through by adjacent, everything through by adjacent
squared. So this becomes a 1. This becomes an opposite over adjacent, all squared. And this becomes a
hypotenuse over adjacent, all squared. So what do I see now? I see 1 + -- and what?s opposite over adjacent? Well,
opposite over adjacent is just a fancy way of saying tangent. So, in fact this is tan2. And what?s hypotenuse over
adjacent? So what?s that? Well, hypotenuse over adjacent is sort of the flip of adjacent over hypotenuse, which is
cosine, so what?s the flip of cosine? Well, that?s secant. So this would be secant2. So, I just discovered another
identity by using the Pythagorean Theorem. I see that 1 + tan2 = sec2. So there?s another identity. So that?s pretty
cool. I just got that by taking this identity basically and dividing everything through by cosine squared, right? Because
here I see
, that?s tangent. Here, I see a 1. And here I?d see 2
which is sec2. What if took this thing and
divided everything through by sin2? If I divide everything through by sin2, here I would see a 1. I?m taking this
equation and dividing everything through by the sin2. So, here I?d see a 1, and here I?d see a
2 cos
? ?
? ?
? ?
. What?s
? Well,
is tangent,
must be cotangent. The flip. So, in fact, this must be cotangent2? = 2
. Well
Trigonometric Identities
Basic Trigonometric Identities
Fundamental Trig Identities Page [2 of 2]
is the same thing as cosecant2. And so, I see now a third identity. All these last three identities just follow from
the Pythagorean Theorem and dividing everything through by one of the terms.
So, in fact, these are three really fundamental identities, which are really important to remember. And I?ll tell you the
truth. The truth is I only remember basically these two. This one sort of makes sense to me ? tan=
, and sin2 +
cos2 is always 1, if it?s of the same angle. This is of the same angle. And these other ones just follow from this one.
If I divide everything through by, you know, either sine or cosine, and so forth. So, in fact these you don?t have to
memorize. In fact, to be honest with you, I don?t memorize these. I just know this one. I can always find these by
dividing everything through by the cosine, which gives me this one, or dividing everything through to this by sine,
which gives me that one.
Now, how can you use these things? Well, suppose you knew just one trig value of an angle and you want to find all
the rest. Well, it turns out you can actually use these inequalities to actually find those other values. So, up next,
what I?m going to do, is take a look at how, just using one information about one trig function in an angle, we can
actually resolve all the trig values, just by using these identities. Take a look at that, it?s coming up next. I?ll see you

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