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About this Lesson
 Type: Video Tutorial
 Length: 6:50
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 74 MB
 Posted: 07/02/2009
This lesson is part of the following series:
Trigonometry: Full Course (152 lessons, $148.50)
Trigonometry: Trigonometric Functions (28 lessons, $26.73)
Trigonometry: Graphing Sine and Cosine Functions (4 lessons, $6.93)
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 UTAustin 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, padic 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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11/13/2008
Founded in 1997, Thinkwell has succeeded in creating "nextgeneration" 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 technologybased textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.
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Finding Maximum and Minimum Values and Zeros of Sine and Cosine
Now, it turns out you can actually use the graph of the sine and cosine functions in looking at these amplitudes and periods. To actually find sort of maxima and minima values for a particular function and even where the zeros are for a particular function, by just sketching a rough graph. Let me try to illustrate this with an example. Let's consider y = 2cosx. And suppose we want to find out where the maximum values of this function actually are taken on and what those values are. Same thing with the minimum. Where do the minimum values occur and what are those values? And what are the zeros of this? That is, if I set this equal to zero, where are the x's that make that equal to zero? Well, it turns out that even just a rough sketch will give us the answer. So, let's try to sketch this. This looks very much like a cosine function except notice the 2 out here changes the amplitude. The amplitude will actually change and actually be 2 now, so I'm going to sort of stretch the cosine function, but there's no change in the period, so I'll have one complete cycle in 2. So, if I sketch this, I see that in 2 I'll have a complete period. So this is going to be , and this is going to be and 3, and instead of having an amplitude of 1 like you would normally have, we're going to double that. This is going to be sort of a stretchy looking cosine function. Now, it's a good oldfashioned cosine function so I start way on the top here, but now at 2, rather than at 1, and I'm going to zoom down, come up and go up, it's standard shape. It's a little more dramatic because of that, and it keeps going. I just want to show you one, sort of, period. Okay, well now we can sort of answer these questions. For example, where do the maximum values occur for this? Well, they occur at zero and then again at 2 and in fact, they're going to be at zero plus any multiple of 2. So, basically any multiple of 2. So, zero, 2, 4, 6, and similarly, even negative 2. It's going to come down and go up. So, in fact, the maximums occur when x is an even multiple of or just any multiple of 2.
And what is the maximum value? The maximum value is 2. That's the amplitude. Where do the minimum values occur? Well, the minimum values occur at and then again at 3, and then again at , so in fact, all the odd multiples of . So, , and then 3, and so forth. So, basically take and then add any multiply of 2 you want to it. So, you get all the odd multiples of . That's where the minimums occur, and what are those minimum values? In this case, negative two. It's the negative of the amplitude.
Last question up. What about the zeros of this? Where does this cross the x axis? Well, here and here. And you can see that's at and then at 3. And the next time it will cross will be at 5 and so on. It crosses here at , so in fact, any odd multiple of , or any odd number times 2. Those are precisely all the zeros of this particular function. So you can find all the zeros and the max and the mins and where they're located just by sketching a graph.
Let's try one more example. Let's take a look at y = sin2x. Okay, well the amplitude is going to be 1, but notice the negative sign means I'm going to have a flip. So, it's not going to be the usual sine curve that starts up and goes like this. This one is going to sort of drop first and then come back up. It's going to be the negative of the traditional one. But the period is going to actually be changed here. Let's see how. So the period is going to be 2 divided by the absolute value of this coefficient, which is just 2, and so I see . So, in fact, what I see is, instead of waiting until 2 to get a complete period, I'm going to see one just once I get to . So, I'm going to just sort of "compactify" the sign curve. There'll be more ups and downs. Sort of like life, you know, it has it's ups and downs. Okay, so let's put 2 right here. Here's . Here's Here's 3. But I know that I'm going to have a complete period just up to here. And so let's see, this is going to be , and this is . Now the amplitude will remain at 1, but I'm going to squeeze in a whole period right here. And the negative sign, remember, just means we start and go down first. So, we're going to go down, and do the whole thing in 1 unit. So, see how I sort of made a whole sine curve where usually I make one out to here? So I get to put another one in. And notice it's sort of negative of the usual sine curve. So there's the graph. \
Well, now we can answer all these questions. For example, where do the maxima occur? Well, the maxima occur at, let's see, and then when's the next one? Well, you notice that this thing has period . So, all you have to do is add to this and that brings me right to here. And that's the next one. So, in fact, all the maxima occur at plus or minus any multiple of , because I just take that thing and then add and I get this, add another , I get another maxima. If I add another way, I get another max. So, it's just plus or minus any multiple of . And why? Because the period is . What about here at the mins? Oh, and what is the maximum value? It's just 1, right? At the height it's 1. And the minimus, where does that occur? The mins occur at , and then + , + another . So, plus or minus any multiple of will give me all the minimum values. And what are those values? They turn out to be 1. And where are the zeros? Well the zeros are at zero, , , 3, 2. In fact, they are all the multiples of . This is zero , 1, this is 2, 3, 4, and so on. So, in fact, any multiple  and negative  so any multiple of will, in fact, give you all the zeros of this thing. So you can actually find the zeros and find the maximum values and where they occur and the minimum values and where they occur, just by sketching a rough picture of the graph. And then you can sort of read off the answer. It's just solving equations.
Okay, up next, we'll take a look at even more exotic graphs and figure out what is going on with all these trig function things? I'll see you there.
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