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

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About this Lesson

  • Type: Video Tutorial
  • Length: 10:27
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 112 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)

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.

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_tan
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.

Nopic_tan
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

Nopic_tan
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.

Nopic_tan
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

Trigonometric Identities
Simplifying Trig Expressions
Determining Whether a Trig Function is Odd, Even, or Neither Page [1 of 2]
Now let’s start thinking about whether certain trig functions are odd or even. Let me remind you of what “odd” and
“even” even mean. A function is even if in fact it’s symmetric about the Y axis, so whatever I do on this side, on the
right side, I do the exact same thing, the symmetric thing, over on the left side. So this is an even function. And how
can we distinguish that? How can I know if a function is even or not? A function is even if f evaluated at ?x , is the
same thing as f evaluated at x . And here’s what that means. It means that if you look at the value of the function
at x and see what the height is, it’s the exact same thing as if I look at ?x . So if I look at ?x , I get the exact same
height. So x and ?x produce the same value. That’s what this means. That’s what it means for a function to be
even.
What does it mean for a function to be odd? Well, a function is odd if in fact it has some symmetry but it’s more like a
flip-of-a-mirror symmetry. Whatever I do on the right, I do the same thing on the left but sort of in reverse, so I’d see
this. There we go. So whatever I do here, I sort of do the opposite here. This is an odd function, or symmetric
around the origin because if you have a point here, it’s negative right through here. And the way to see an odd
function is if you take f(?x), how does that compare to f(x)? Well, if you have a value here that’s f(x),
f(?x)should be the negative of that. So in fact there should be a negative sign out here. So a function is odd if
f(?x)=?f(x), and a function is even if f(?x)= f(x). Okay?
Well, let’s take a look at these trig functions and see whether in fact these are odd or even. What about the sine
function? Well, we think about sine function as odd. The sine function is odd because notice that whatever I do on
the right, I sort of do the opposite on the left. In fact, if you keep going—maybe I should do a continuous picture of
this. You can see this is really an odd function, sine is odd, because whatever I do on the right-hand side, I do the
exact same thing with the negative on the left. It’s symmetric around the origin, so it’s an odd function. What that
means is sin?x=?sin x . Knowing that it’s odd means that we have this identity. If you look at sin?x=?sin x ,
odd.
What about cosine? Let’s see what cosine looks like. Well, cosine you notice is even. That is, if you keep drawing
it—you can sort of see it here—it’s symmetric right around the origin. Whatever I do on the right-hand side, I do the
exact same thing on the left-hand side. So the cosine is actually an even function, and that means that if I take a look
at cos?x=cos x . What I do on this side and on the ? f side, it’s the same value. Cosine is an even function
whereas sine is in fact odd.
Now, just knowing those two things, we can actually figure out whether other trigonometric functions are odd or even,
so let’s take a look and see. For example, what about tangent? Suppose that f(x) = tanx and I want to find out if
this is odd or even, what do I do? Well, the first thing I realize is that tangent—well, I can first of all look at the graph,
and look at the graph really fast. This is a really fast graph of tangent, just really fast. One of those speed-graphing
courses that you see advertised in the newspapers. And you can see this function actually is an odd function just by
looking at the graph. Whatever I do here and get a value, at ?x I do the opposite value, so this is actually an odd
function. It’s symmetric with respect to the origin.
Okay, so we know that now. But how can we have seen that analytically just by looking at the function? Well, you
can remember that
tan sin
cos
x
x
= . So to see if a function is odd or even or neither, what you do is you look at f(?x).
So wherever I see an x here, I’m going to plug in a ?x . So I see
sin
cos
x
x
?
?
. Now what does that produce? Well what
that produces is—well, I know that sine is an odd function. That means that sin?x=?sin x. cos? x , well cosine is
an even function so cos?x=cos x. cos? x , what’s that? Well, since cosine is an even function, I have just cos x.
Okay, now what do I see? I see
sin
cos
x
x
?
, and
sin
cos
x
x
is just tangent, so I see if I put a negative sign in front, ? tan x .
So what I see is f(?x)=?f(x). So this means the function must be odd. And of course our picture verifies that,
Trigonometric Identities
Simplifying Trig Expressions
Determining Whether a Trig Function is Odd, Even, or Neither Page [2 of 2]
confirms that as well. So you can see that to see if your function is odd or even, you just plug in ?x for x and see if
you get the function, in which case it would be even, or minus the function, in which case it would be odd. Or if you
get neither, then it’s neither even nor odd.
Let’s take a look at some more examples. How about this one, f(x) =sin2x. Even or odd? Well, to see, let’s just
say f(?x) and see what that equals. That equals sine of—and I put a ?x here, so 2 times ?x , which equals
sin? 2x . But what do we know about the sine function? We know that the sine of minus something equals minus
sine of the something, so in fact this equals ?sin 2x , which you’ll notice is just negative the original function. So I
see f(?x)=?f(x). This function is odd. And in fact it’s exactly the sine function with a change in period. But
since the sine function is odd, it’s not surprising that sin 2x is odd. I just changed the period a little bit.
How about I show you another one? How about this,
fx sin x
x
= . Let’s take a look at f(?x). That would equal
sin x
x
?
?
, because wherever I see an x I’ve replaced it by ?x . Sine is an odd function, so that means that
sin?x=?sin x . This equals
sin x
x
?
?
. But now I’ve got a -1 factor on the top and a -1 factor on the bottom, and so if
I cancel those factors, what do I see? I just see
sin x
x
, and that’s exactly equal to the original function. So I see
f(?x)= f(x). That means this function is actually even. Notice what happened, by the way. This function on top
is odd, and actually the function x is also odd. It’s just that diagonal line, so whatever you do on the upper left you’ve
got to do on the lower right. So an odd divided by an odd produces an even. It’s just like doing multiplication with
regular numbers, right? If you take a negative and divide it by a negative, you actually get a positive. It’s the same
kind of thing here. An odd divided by an odd actually produces an even function.
One last one, f(x) = xcosx. Let’s see if that’s even, odd or neither. So I replace all the x by ?x , so I see
?x cos?x . Okay, well cosine is an even function, so cos? x is the same thing as just cos x , so this equals
?x cos x . Well, that minus sign still remains, and notice that just equals minus x cos x . x cos x is the original
function. So I see that f(?x)=?f(x). This function by definition is an odd function. And so that’s pretty cool
because what it says is if you take a function that’s even and multiply it by a function that’s odd, the net result actually
is odd, which is what we see here.
So you can tell if trigonometric functions are odd or even or maybe neither. In fact, a function may be neither odd nor
even if it’s not symmetric to either the Y axis or to the origin, or you can actually see just by looking at f(?x) what
the symmetry is, if there’s any symmetry at all, even or odd. Anyway, I just wanted to point that out as another
example of looking at even function or odd functions.

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