Section 7.6 Graphs of the Sine and Cosine Functions

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Theodora Hardy
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1 4 Section 7. Graphs of the Sine and Cosine Functions In this section, we will look at the graphs of the sine and cosine function. The input values will be the angle in radians so we will be using x is place of θ as our angle and the output values will be the functions values. From the previous section, we know that the period for the sine and cosine function is, so we will first sketch the graph for values of x between and inclusively and then use the fact that the graph is periodic to draw the rest of the graph. We will begin by making a table of values to graph the sine function: x sin(x) Now, draw a smooth curve: This is the graph of y = sin(x) on the interval [, ]. Since it is periodic, this shape repeats and thus we get the following graph:

2 47 y = sin(x) Properties of the graph of the sine function: ) Domain: (, ) Range: [, ] ) The sine function is odd so it is symmetric with respect to the origin. ) The sine function is periodic with a period of. 4) The x-intercepts are {(n, ) n is an integer}. The y-intercept is (, ). 5) The maximum value is occurs at x = (4n+) ) The minimum value is occurs at x = (4n ) where n is an integer. where n is an integer. Now, we will make a table of values to graph the cosine function: x cos(x) Now, draw a smooth curve:

3 This is the graph of y = cos(x) on the interval [, ]. Since it is periodic, this shape repeats and thus we get the following graph: Properties of the graph of the cosine function: ) Domain: (, ) Range: [, ] ) The cosine function is even so it is symmetric with respect to the y-axis. ) The cosine function is periodic with a period of. 4) The x-intercepts are {( (n+), ) n is an integer}. The y-intercept is (, ). 5) The maximum value is occurs at x = n where n is an integer. ) The minimum value is occurs at x = (n + ) where n is an integer. Objective & Graph Function in the Form of y = Asin(ωx) & y = Acos(ωx). Use Transformations to sketch the graph of the following: Ex. a y = sin(x) Ex. b y = cos(x) Solution: a) y = sin(x) is the graph of sin(x) stretched by a factor of. We will draw sin(x) and gray and sin(x) in black:

4 Now, erase the gray graph: The stretching and compression works the same way for the trigonometric fucntions as it did with the other functions we have graphed. a) y = cos(x) is the graph of cos(x) stretched by a factor of. We will draw cos(x) and gray and cos(x) in black:

5 Now, erase the gray graph: In graphing sine and cosine functions, the stretching/compressing factor A is called the amplitude. Notice that in the preceeding example, the amplitude was half the distance between the the minimum value and the maximum value of the function. We typically refer to the amplitude as being the vertical stretching/compressing factor. We can also have a horizontal stretching/compressing factor. Consider the following example:

6 Use Transformations to sketch the graph of the following: Ex. a y = sin(x) Ex. b y = cos( x) Solution: a) Since A =, we know the graph will be reflected across the x-axis. We need to examine what the in front of the x does to the graph. We know that the period of the sine function is, thus if we were to graph the sine function for one period, then the angle would be between and inclusively. Thus, x. Solving for x yields: x (divide by ) x Thus, sin(x) has a period of. Hence, the factor of three causes a horizontal compression of. Our graph starts at and then begins to repeat at : b) Since A =, there is no reflection. We need to examine what the in front of the x does to the graph. We know that the period of the cosine function is, thus if we were to graph the cosine function for one period, then the angle would be between and inclusively. Thus, x. Solving for x yields: x (multiply by ) x Thus, cos( x) has a period of. So, the factor of causes a horizontal stretch of. Our graph starts at and then begins to repeat at :

7 Objective : Finding the Ampltude and Period of a Sine and Cosine Function. Theorem If ω >, the amplitude and period of y = A sin(ωx) and y = A cos(ωx) is given by: Amplitude = A Period: T = ω Note, if ω is negative, use the even/odd properties to rewrite the function without the negative sign in front of ω. Determine the amplitude and period and then sketch the graph of the following: Ex. a y =.sin( 5 x) Ex. b y = cos( 4x) Solution: a) Since. =., the amplitude is.. The period is 5 5. Since A is negative, the graph is reflected across the x-axis: =

8 5 b) The cosine is an even function, so cos( 4x) = cos(4x) The amplitude is = and the period is T = 4 = If we graph the sine and cosine function on the same graph, we would notice that the cosine function is just the sine function that has been shifted horizontally..5.5 sin(x) -.5 cos(x) - Notice that cos(x) = sin(x + ), thus every cosine function can be rewritten in terms of the sine function. Hence, any function that has a graph of either the sine or cosine function is referred to as having a sinusoidal graph. Sketch the graph of the following: Ex. 4a y = sin( x) + Ex. 4b y = cos( 4 x) Solution: -.5 a) The amplitude is = and the period is T = = 4. The graph is also reflected across the x-axis and shifted up units. Here, it may be helpful to get some additional points to sketch the graph. Since

9 the graph starts at x = and ends at 4, then the functions value is + at these points. Similarly, halfway between and 4, at, the function value also has to + =. Halfway between and, at, the function will hit its lowest point because of the reflection, so the function value is + =. Between and 4, at, the function will hit its highest point at + = b) The amplitude is = and the period is T = 4 = 8. The graph is also shifted down by. Since the graph starts at x = and ends at 8, then the function value peaks at these points and has a value of =. Similarly, halfway between and 8, at 4, the function hit it lowest point and have a value of =. Halfway between and 4 and halfway between 4 and 8, at and at, the function value is

10 55 Given the graph below, write the equation of the function: Ex. 5a Ex. 5b Solution: a) Since the range of the function is from 5 to 5, the amplitude is 5. It is not reflected, so A = 5. It is the cosine function that starts at zero and ends at, so the period is. We can now find ω. = which implies ω = ω Plugging in A = 5 and ω = into the cosine function, we get: y = 5cos(x) b) Since the range of the function is from to 4, the amplitude is. It is reflected, so A =. It is the sine function that starts at zero and ends at, so the period is. We can now find ω. = which implies ω = ω Plugging in A = and ω = into the cosine function, we get: y = sin(x)

Section 8.4: The Equations of Sinusoidal Functions

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