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About this Lesson
 Type: Video Tutorial
 Length: 10:13
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 110 MB
 Posted: 01/23/2009
This lesson is part of the following series:
Trigonometry: Full Course (152 lessons, $148.50)
PreCalculus Review (31 lessons, $61.38)
Trigonometry: Trigonometric Functions (28 lessons, $26.73)
Trigonometry: Graphing Sine and Cosine Functions (4 lessons, $6.93)
In this lesson, you will examine the graphs of both the following trigonometric functions: sine and cosine. Professor Burger will show you how to graph sin and cos and teach you the acronym ASTC (All Students Take Calculus). Prof Burger also defines and shows you where to look for to evaluate the amplitude, period, and zeros of the sine and cosine graphs and shows you how to find and determine the maximums and mininimums for both sine and cosine functions. Finally, he will compare the graphs of the two functions, demonstrating that they have an identical shape with merely a shift between them to differentiate the two functions from each other. You'll also learn the importance of the π/2 interval in plotting and remembering the trigonometric function graphs of cosine and sine.
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 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
 2174 lessons
 Joined:
11/14/2008
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/.
Thinkwell lessons feature a starstudded cast of outstanding university professors: Edward Burger (PreAlgebra through...
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Introducton to the Graphs of Sine and Cosine Functions
Now I want to start to actually look at the graphs of some of these trig functions and I want to start with just the sine. So, let’s take a look at the graph of y =sin x, and so here the independent variable here is x and if I knew x, I’d just compute the sine, and that’s going to be the y. So, I want to look at the graph of this like we graph all sorts of other functions. But what do I do here? Well, we might want to start by just sort of looking at a table of values, and plotting those points. So, you can put a table of values up here. You can see them here. And this is the table we just made. So, it’s got zero, 6?, 4?, 3?, 2?, and so forth. And let’s start graphing and see what this thing looks like.
Now, what I’m going to do since we’re graphing this, by the way, in radians, so let me put this value right here, I’ll make this 2?. That’s the angle that goes once around, it’s the same thing as zero. This would be ?. This is 2?, this would be 32? and let’s see how we make out. So we plot some points. What is the sine of zero? If I plug in zero, how high do I go? I plug in zero, I see I just get zero. So, in fact, this is just zero. When I plug in 2?, you can see from the chart, we just have 1. So we go up a height of 1. In fact, I should probably label 1 up here. Here’s 1, here’s –1, so we go all the way up to 1. Now what about at ?? Well, if you think about what’s happening at ?, that’s ?, 180, so what’s the reference angle? The reference angle is actually zero, right? It’s the angle I sort of flipped over. So, zero. And, so, what is the sine of zero? We know it’s zero. So, in fact at ?, I’m back to zero again.
What about 32?? Well, 32? comes sort of way down here, so I get something down here and so, well, it’s sort of hard to say what the reference angle would be. I guess the reference angle might be like, you know, 2?, and if it’s 2?, and what’s the sign of the sine, is it positive or negative? “All students take calculus.” So, I see that the sign is positive here but negative down here, so in fact, this is going to be a negative value, but what is it at 2?? It’s 1. So, it’s going to be negative that, so it’s going to be –1. And what about 2?? Well, 2? is the same thing as zero. And we know that the sine there is zero. And what if I keep going? Well, see if I keep going, now I’m just going to repeat what I’ve already seen. Because, for example, when I get up to here, that’s the same thing as 2?. So, now, once I’ve started going around again, I’m going to start spinning. I’m going to see the exact same thing. So, in fact, this is going to repeat this picture. So, let’s connect the points, and what I see is a function that looks like that. And if I were to keep going, I would see the exact same picture, because now I’m spun around once, the sine function has no memory so it just thinks that I’m seeing this, and then it goes to zero, and so on. If I were to go negative angles, this graph would continue. And so I get a sine function. And a sine function has this really pretty graph. Mine’s not that great, but this is much better, you can really see it. It has this beautiful wavy kind of thing. And so this is what the sine function looks like, so a standard sine function. There’s some basic, a little bit of language here. Let me tell you about the language. The maximum/minimum values, by the way, you’ll notice that they actually occur at 1 and –1. That height is called the amplitude. So the amplitude is just the absolute value of sort of the height, which in this case is just 1. This repeats. This picture just repeats. And you can see it even here. I just take this picture and this repeats. Take that picture right here; that’s a complete cycle. And then it just repeats again. And it repeats again. This is called the period. So, the period is the length of time it takes to sort of repeat. In this case, this would be 2?. So the period of this is 2?. And you also ask, “Well, where is the location of the high points?” Well, the high points, those sort of top points, see those “maxes”, occur where? Well, it occurs at 2? and then here. And where’s this point? Well, if you think about it, this is 32?, this is 42?, this is 52?. What about the mins? Where are the mins?
Well, the mins, they occur sort of here. So what are those? Those are going to be what kind of multiples? Well, let’s be careful now. So, here we have 2? and then here’s this 52?, and so this is 32?. So the mins occur where? Well, the mins occur if you take 32? and then you just shift them over. So, how far do you shift? Well, you shift it by any multiple of 2?. So take that point and now shift it a multiple of 2?, and you get to here. Just like here, the maxes occur. You take 2? and you shift any multiple of 2?. Take this point, add 2?, you get to here. Add 2? again, you get another max. Subtract 2?, you get here. And the mins occur sort of over here. So, you can find the mins and maxes pretty easily by just taking one of the min or maxes and then just shifting it over a multiple of 2?. That’s all there is to it. That’s how you find the max. That’s how you find the mins. Where are the zeros, by the way? Where does this thing cross the zeros? Well, zero, ?, 2?, 3?, 4?, so forth. So, the zeros are going to be any multiple of ?. Zero ?, 1 ?, 2?, 3?, ?, and so forth. You can see –? is zero, right there. So the zeros are easy to see. Those are the multiples of ?. The high point are exactly 2? plus or minus any multiple of 2?, so here, here, and so forth. And the low points happen at 32? plus or minus any multiple of 2?. So, boom, boom, boom, boom, and so forth.
What about cosine? Well, cosine you can graph the exact same way. And if we make a chart, we’ll see what the cosine function looks like. And we’ll see it’s actually very similar to what we got with sine. So, again, we start here, and here’s 2?. That makes half of this ?. That’s 180°, by the way, 2?, 32?, and so on. What happens? What is cosine of zero? Well, cosine of zero is 1. You saw that before. That’s one of the standard points you should know. So, we go up to 1, but cosine of 2? is zero. So this is now zero. What about cosine of ?? Well, cosine of ? has a reference angle of zero, but it’s going to be negative there, so in fact, we’re at –1, instead of 1. We’re at –1. And at 32?, we’re at zero, and at 2?, it’s the same thing as a 2?. It’s the same thing as zero. So, it’s just 1. So, let’s graph this. What we see is a similar type of function. And so on. It sort of just goes wavy again, just like before. Wavy. If fact, it’s the exact same function except it’s just been shifted a little teeny bit. It looks like this now. See the difference with that? Here’s sine. Sine starts at zero and goes up. The cosine starts at 1 and goes down. But everything else is the same, in terms of it’s shape and it’s curvature. But now, of course, you can tell all sorts of things. You can tell, for example, where are the maxes? Well, the maxes occur where? Well, they occur at zero and at 2?, and at 4?. So, all the multiples of 2?. So, 2?, zero, 4?, 2?, and so forth. Where are the minimums? Where are the low points? Well, those low points are where you get negative 1, are actually going to happen where? Well, that’s going to happen at ? and then at 3?, and then at 5?, and then at –?. So, it’s going to be all the odd multiples of ?, plus or minus, right; ?, 3?, and so forth. The amplitude in this case is still 1, that was the maximum height here above is 1. And the period, notice that after you get to 2?, the thing just repeats again. So, in fact, the period here is going to be 2?. Where are the zeros? Well, the zeros are now at the half marks, right? 2?, 32?, so it’s any odd number, 2?. So, any odd multiple of 2?. So, this is the general shape of the cosine function. This is the general shape of the sine function. You can see they are very similar. In fact, one is just a shift of the other. I can give you the sine function, cosine function. So you can see, they’re related in a very strong way. Anyway, what we’ll start to do is start to look at more exotic graphs of these things. What happens if I put a coefficient in front of here? What happens if I put a coefficient in front of here? How does that change the general picture of these graphs. We’ll take a look at those pictures, start seeing how the period can change, and the amplitude can change. We’ll see that next.
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Sin and cos are definitely the easiest to understand graphically versus tangent and other trig signs. After watching this video I feel confident about my understanding of the basic trig functions. So happy!