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Pre-Calculus: Graph Sine, Cosine with Phase Shifts


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About this Lesson

  • Type: Video Tutorial
  • Length: 7:21
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 79 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 Functions (28 lessons, $26.73)
Trigonometry: Graphing Sine and Cosine with Shifts (2 lessons, $2.97)
Pre Cal: Graphing Sine and Cosine With Shifts (2 lessons, $2.97)

Now that you have learned how to graph the sine and cosine functions, Professor Burger asks the question ""How does changing the x-value affect the graph?"" He shows you how adding or subtracting to the x-value can actually change graphs of the sine and cosine functions, a process called translation. Professor Burger also warns you about classic mistake #8, reminding you that adding and subtracting to the x-value actually creates the opposite effect when graphed (adding to X moves the graph in the negative direction). Finally, Professor Burger shows you how to simplify the equation y = 3sin(x + Pi/2) using translation. The key lies in the fact that adding or subtracting pi/2 or 2*pi to a sine or cosine function means there are some shortcuts that you can take to determine what the graph of the function looks like (e.g. the graph of sine of (x+pi/2) is the same as the graph of cosine and the same as the graph of sine of (x+2*pi)).

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

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The Trigonomic Functions
Graphing Sine and Cosine Functions with Vertical and Horizontal Shifts
Graphing Sine and Cosine Functions with Phase Shifts Page [1 of 2]
Now I want to take a look at what happens if we take a look at these sine and cosine things and look at sort of a
horizontal translation, so we sort of move things to th right or to the left. Now, when we see this on other graphs, we
know exactly what happens. We’re changing the x values by taking x and replacing it by x plus something. But now
the question is, sort of, how does that have an effect on the graph itself?
Well, first of all, let me just sort of pose this as sort of a basic observation. If I’m finding any trig function, for this angle
let’s say, this value for x is this angle here. What happens if I add 2? to it? Well if I add 2? , remember, that’s just
360 so it’s just a take this and just loop it around but then I get right back to here. It has no effect. It has no effect on
anything. So, in fact, if I take one of these trig functions and add 2? to the angle, it shouldn’t change the value, right?
This thing sort of spins around. So, in fact, what I see here is the following. In fact, you can even see that visually.
Let’s take a look at it visually. Suppose I take a look at this. This is just a sine function graphed. And suppose, I say,
“Okay, I want you to take a look now at sine of x + 2? .” Now what does that mean? What does it mean to look at
sine of x + 2? ? Well that’s going to be a shift in the x direction, but now which way? Well, the plus 2? , you may
say, “Oh, wait a minute, I should be going to the right because it’s a plus.” But, no, that’s a classic mistake. In fact,
that’s a classic mistake on my top ten list of classic mistake, it’s Classic Mistake No. 8. The shifting mistake;
remember, add to y, go high, add to x, go west. So, in fact, I should be going in this direction, to the left, west, by 2? .
So, what does that mean? Well, let me actually sort of start me off at the right foot here. So, here’s the sine function,
right here. And now I’m going to shift this way by 2? , that means that this point here at 2? will now be at zero. Now
look at the sine function. Look what happens when I shift this over. The 2? , it’s the exact same function. It doesn’t
change. Because when you loop around, you’re back at the same angle you were before. So, in fact this equals just
sine, of x. It’s the same function. Because, if I translate it by one of its periods, it just lines up perfectly, right?
Another way of seeing this, maybe, is to take a look at the sine function and now see what happens when I shift it 2? .
Now watch what happens. I shift it 2? , they line up perfectly again. So, in fact, any multiple of 2? that I shift, will
not change this. So, for example, if I say, sine of x – 4? , it doesn’t change anything. I just take this picture and I shift
it now to the right 4? . So, now I just don’t go 2? , but I go another 2? . And it doesn’t change the picture.
Same thing with cosine, of course. So, if I say cosine of x, that’s the same thing as cosine of x + 8? . If I add or
subtract any multiple of 2? , it’s not going to change the function because a period, a perfect period, is 2? . So, if I
_____ translate it over by 2? , everything will match up perfectly. Just like you see in this example here. Everything
matches up perfectly every 2? I shift.
But what if I don’t shift 2? ? For example, what if I just shift
? So what’s sine of x +
? So now I’m going to shift
to the left, again left, by
. Let’s try that right here. So here you can see – this is the sine function – I’m going to
shift this now by
, which means that this
is going to be shifted over and become zero. Look what function we
get. Now, what function is that right there? Well, that actually looks like it’s a cosine function. And, in fact, it is. If you
take the sine function and just shift it over
, you get an exact copy of the cosine function. And you can see that
visually. It looks like that. So, in fact, this equals cosine of x. What happens if I take the cosine function and look at x
? So I start with the cosine function; so here’s the cosine function here and what is x -
? Remember, minus
means I’m going to go the east,
units, and look what happens. I’ll end up perfectly on sine. So, in fact cosine of x
= sine of x. These are two really neat facts about translation, which helps us in doing some of these simple
translation problems. For example, suppose I want to graph the following: someone said, “Graph y = 3 sine of x +
The Trigonomic Functions
Graphing Sine and Cosine Functions with Vertical and Horizontal Shifts
Graphing Sine and Cosine Functions with Phase Shifts Page [2 of 2]
. We might say, “Gee, first I’ll have to graph sine and then shift it and then multiply by 3 and so forth.” Or, you
could just notice that sine of x +
, that’s actually going to be just cosine. So this is just 3 cosine x and graphing that
is a breeze because its period is just going to be 2? , which doesn’t change, sort of the same as standard period. Of
course, the amplitude is going to change, because the amplitude is going to be up to 3. So, instead of being at 1, we
have to be way up here at 3. So, there’s 3 and here’s -3, so what we get is a really high, and we get cosine, so we get
a really high cosine thing going. And, it keeps going. And that’s actually a graph of 3 sine x +
. And why?
Because it’s just the same thing as 3 times cosine x. So, in fact, translating and shifting is actually not that big of a
deal as long as you remember in some cases, that in fact there are these nice formulas that show one is the same as
the other and you could just think about those again just by visualizing the shifting. When I go this way, I get cosine;
when I take the cosine, and go this way, I get sine. So the first thing says sine of x +
, and I get cosine. And this
says we start with cosine, but now go x -
, that shifts back and I get back to sine. These two things are actually
handy and as long as you remember that when you add to x, go west, that will always make sure you get the right
That’s all there is to it and so translating by ? ’s over 2, no problem. Translating by multiples of 2? , it’s trivial.
Nothing changes because, in fact, these things have period 2? . We’ll take a look at some more graphs of more
exotic trig functions up next.

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