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If you have not watched the PowerPoint on the unit circle you should watch it first. After you’ve watched that PowerPoint you are ready for this one. If you watched it, just click to begin this part of the section.

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**Let’s think about the function y = sin x**

What is the domain? (remember domain means the “legal” things you can put in for x ). You can put in anything you want so the domain is all real numbers. What is the range? (remember range means what you get out of the function). The range is: -1 sin x 1 (0, 1) Let’s look at the unit circle to answer that. What is the lowest and highest value you’d ever get for sine? (sine is the y value so what is the lowest and highest y value?) (1, 0) (-1, 0) (0, -1)

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**Let’s think about the function y = cos x**

What is the domain? (remember domain means the “legal” things you can put in for x). You can put in anything you want so the domain is all real numbers. What is the range? (remember range means what you get out of the function). The range is: -1 cos x 1 (0, 1) Let’s look at the unit circle to answer that. What is the lowest and highest value you’d ever get for cosine? (cosine is the x value so what is the lowest and highest x value?) (-1, 0) (1, 0) (0, -1)

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**Look at the unit circle and determine sin 420°.**

In fact sin 780° = sin 60° since that is just another 360° beyond 420°. Because the sine values are equal for coterminal angles that are multiples of 360° added to an angle, we say that the sine is periodic with a period of 360° or 2. All the way around is 360° so we’ll need more than that. We see that it will be the same as sin 60° since they are coterminal angles. So sin 420° = sin 60°.

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**The cosine is also periodic with a period of 360° or 2.**

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**SINE AND COSINE FUNCTIONS**

GRAPHS OF SINE AND COSINE FUNCTIONS

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**We are interested in the graph of y = f(x) = sin x**

Start with a "t" chart and let's choose values from our unit circle and find the sine values. plot these points x y 1 - 1 x y = sin x We are dealing with x's and y's on the unit circle to find values. These are completely different from the x's and y's used here for our function.

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**y = f(x) = sin x choose more values x y = sin x plot these points**

join the points x y 1 - 1 If we continue picking values for x we will start to repeat since this is periodic.

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**Here is the graph y = f(x) = sin x showing from -2 to 6**

Here is the graph y = f(x) = sin x showing from -2 to 6. Notice it repeats with a period of 2. 2 2 2 2 It has a maximum of 1 and a minimum of -1 (remember that is the range of the sine function)

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From College Algebra recall that an odd function (which the sine is) is symmetric with respect to the origin as can be seen here

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**What are the x intercepts?**

Where does sin x = 0? …-3, -2, -, 0, , 2, 3, 4, . . . Where is the function maximum? Where does sin x = 1?

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**Where is the function minimum?**

Where does sin x = -1?

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Thinking about transformations that you learned in College Algebra and knowing what y = sin x looks like, what do you suppose y = sin x + 2 looks like? This is often written with terms traded places so as not to confuse the 2 with part of sine function y = 2 + sin x The function value (or y value) is just moved up 2. y = sin x

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Thinking about transformations that you've learned and knowing what y = sin x looks like, what do you suppose y = sin x - 1 looks like? y = sin x The function value (or y value) is just moved down 1. y = sin x

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Thinking about transformations that you learned and knowing what y = sin x looks like, what do you suppose y = sin (x + /2) looks like? y = sin x This is a horizontal shift by - /2 y = sin (x + /2)

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Thinking about transformations that you learned and knowing what y = sin x looks like, what do you suppose y = - sin (x )+1 looks like? y = 1 - sin (x ) This is a reflection about the x axis (shown in green) and then a vertical shift up one. y = - sin x y = sin x

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**What would the graph of y = f(x) = cos x look like?**

We could do a "t" chart and let's choose values from our unit circle and find the cosine values. plot these points x y 1 - 1 x y = cos x We could have used the same values as we did for sine but picked ones that gave us easy values to plot.

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**y = f(x) = cos x Choose more values. x y = cos x plot these points y 1**

- 1 cosine will then repeat as you go another loop around the unit circle

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**Here is the graph y = f(x) = cos x showing from -2 to 6**

Here is the graph y = f(x) = cos x showing from -2 to 6. Notice it repeats with a period of 2. 2 2 2 2 It has a maximum of 1 and a minimum of -1 (remember that is the range of the cosine function)

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Recall that an even function (which the cosine is) is symmetric with respect to the y axis as can be seen here

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**What are the x intercepts?**

Where does cos x = 0? Where is the function maximum? Where does cos x = 1? …-4, -2, , 0, 2, 4, . . .

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**Where is the function minimum?**

Where does cos x = -1? …-3, -, , 3, . . .

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**Let's try y = 3 - cos (x - /4) y = - cos x y = cos x**

You could graph transformations of the cosine function the same way you've learned for other functions. moves right /4 moves up 3 Let's try y = 3 - cos (x - /4) y = - cos x y = cos x reflects over x axis y = 3 - cos x y = 3 - cos (x - /4)

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**unit circle y = sin t x = cos t value of angle t**

These graphs illustrate how as you go around the unit circle and plot the y value (upper left) or the x value (lower right) you generate the sine and cosine graphs. The lower left shows the value of the angle t at any given time. Notice the axis for cosine are reversed here so you can see how the x value moves but you can rotate this graph to have t horizontal and x vertical and see the cosine graph like it is traditionally graphed.

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**amplitude of this graph is 2**

What would happen if we multiply the function by a constant? All function values would be twice as high y = 2 sin x amplitude is here amplitude of this graph is 2 y = 2 sin x y = sin x The highest the graph goes (without a vertical shift) is called the amplitude.

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**For y = A cos x and y = A sin x, A is the amplitude.**

What is the amplitude for the following? y = 4 cos x y = -3 sin x amplitude is 3 amplitude is 4

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**absolute value of this is the amplitude**

This is the phase shift (horizontal translation) remember it is opposite in sign This is the vertical translation

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**Given this graph, let’s see if we can find it’s equation in the form y = A sin (x – C)+ D**

y = 2 sin (x) + 3 So what is A? A = 2 A D So what is D? There is no horizontal shift so C = 0 and we have our equation. Now let’s determine the vertical shift D. D = 3 First let’s find the vertical center of the graph and then we can determine the amplitude.

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4.5 Graphs of Sine and Cosine Functions

4.5 Graphs of Sine and Cosine Functions

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