**SINE AND COSINE FUNCTIONS
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to
ingrid.stewart@csn.edu.**

Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!

1. Memorize the shape and characteristics of the graph of the basic sine function.

2. Memorize the shape and characteristics of the graph of the basic cosine function.

3. Find the amplitude of transformations of the basic sine and cosine functions.

4. Find the period of transformations of the basic sine and cosine functions.

5. Sketch certain transformations of the basic sine and cosine functions.

We will now create functions out of the trigonometric ratios. In this lesson, we will look at the basic sine and cosie functions and their graphs. We will also examine some transformations.

The Basic Sine Function and its GraphThe sine function is called "periodic" because its graph consists of identical pictures that repeat forever.

The domain

consists of all real numbers. The domain consists of all number replacements forthat don't makexundefined or imaginary.y = f(x)Please note that the numbers in the domain of ALL trigonometric functions are considered to be radians. NO degree measure can be used in their domain.

We will use the Desmos online graphing calculator to make a picture of .

You can find instructions on how to use Desmos at http://profstewartmath.com/Math127/A_CONTENTS/desmos.htm

YOU MUST MEMORIZE THE SHAPE AND CHARACTERISTICS OF ITS GRAPHGiven , we can create the following table using the

x-values (in radians) labeled in the picture above. They are allQuadrantal Angles!

NOTE:

etc. (use the calculator, if necessary!)We will plot these points in a coordinate system and then connect them keeping in mind the shape of the graph.

We call this portion of the graph of the sine function its

representative picture. It lies on the interval interval which is called thePERIODof the sine graph. Clearly, it is in length.The

representative picturerepeats "forever" along the positive and negativex-axis and creates the graph of the basic sine function.Please note that its peak and valley are U-shaped!

Note that the

representative pictureis divided intofour equalintervals.

- The graph starts at the origin where there is an
x-intercept.- At the end of the first interval the graph has a peak.
- At the end of the second interval the graph has an
x-intercept.- At the end of the third interval, the graph has a valley.
- At the end of the fourth interval the graph has an
x-intercept

The Basic Cosine Function and its GraphThe cosine function is also called "periodic" because its graph also consists of identical pictures that repeat forever.

The domain consists of all real numbers.

We will use the Desmos online graphing calculator to make a picture of .

You can find instructions on how to use Desmos at http://profstewartmath.com/Math127/A_CONTENTS/desmos.htm

YOU MUST MEMORIZE THE SHAPE AND CHARACTERISTICS OF THE GRAPHGiven , we can create the following table using the

x-values (in radians) labeled in the picture above. They are allQuadrantal Angles!

NOTE:

etc. (use your calculator, if necessary!)We will plot these points in a coordinate system and then connect them keeping in mind the shape of the graph.

We call this portion of the graph of the cosine function its

representative picture. It lies on the interval interval which is called thePERIODof the cosine graph. Clearly, it is in length.The

representative picturerepeats "forever" along the positive and negativex-axis and creates the graph of the basic cosine function.Please note the difference between the representative picture of the sine function and that of the cosine function!Please note that its two peaks and one valley are U-shaped!

Note that the

representative pictureis divided intofour equalintervals.

- The graph starts at (0, 1) where there is a peak.
- At the end of the first interval the graph has an
x-intercept.- At the end of the second interval the graph has a valley.
- At the end of the third interval, the graph has an
x-intercept.- At the end of the fourth interval the graph has a peak.

Transformations of the Basic Sine and Cosine Functions- see #1 through 5in the "Examples" documentNext, we will investigate

transformationsof the basic sine and cosine functions. We will let,a,b, andcbe real numbers withdandaNOT equal to 0.bThe transformations will then be described as follows:

and

NOTE:

From algebra, we should remember that

transformationsallow us to move and resize basic functions by shifting them vertically and horizontally; by reflecting them in thex- andy-axis; and by vertically and horizontally stretching and compressing them.The number

aindicates a vertical stretch or compression of the graph of the basic sine and cosine functions. The absolute value of

ora|is called thea|AMPLITUDE.EXAMPLES:

Following are the graphs of

,y = sin(x), andy = 2 sin(x).y = 2 sin(x)We used the Desmos online graphing calculator to make their pictures.

You can find instructions on how to use Desmos at http://profstewartmath.com/Math127/A_CONTENTS/desmos.htmPlease note that if the number

is negative, thearepresentative pictureis reflected in thex-axis. Please compare the green graph and the blue graph!red graph:

,y = sin(x)amplitudea= |1| = 1blue graph:

,y = 2 sin(x)amplitudea= |2|, a vertical stretch ofy = sin(x)green graph:

y = 2 sin(x), amplitudea, a vertical stretch of= |2| = 2.y = sin(x)Additionally, since

is negative, we also have a reflection in theax-axis. Notice that therepresentative picturehas a valley first and then a peak.The number

b

indicates a horizontal stretch or compression of the graph of the basic sine and cosine functions. This will affect the length of the

PERIOD.If

is betweenband0, the period of the basic sine and cosine function gets larger.1If

is greater thanb, the period of the basic sine and cosine function gets smaller.1The formula calculates the period

of the graph of the sine and cosine functions.PEXAMPLES:

Following are the graphs of

,y = cos(x), andy = cos(2x).y = cos( x)We used the Desmos online graphing calculator to make their pictures.

red graph:

then they = cos(x),b = 1periodisPblue graph:

,y = cos(2x)then theb = 2periodis . This is a horizontal compression ofPbecause the period is less than .y = cos(x)green graph:

,y = cos( x)b =then theperiodis .PThis is a horizontal stretch of

because the period is greater than .y = cos(x)The number

cindicates a horizontal shift to the right or left of the graph of the basic sine and cosine functions. This is often called a

PHASE SHIFT.EXAMPLES:

Following are the graphs of

andy = sin(x).y = sin(x + /2)We used the Desmos online graphing calculator to make their pictures.

red graph:

y = sin(x)blue graph:

,y = sin(x + /2)which indicates a shift to thec = + /2LEFTof the graph ofy = sin(x)EXAMPLES:

Following are the graphs of

andy = sin(x).y = sin(x/2)red graph:

y = sin(x)green graph:

,y = sin(x /2)which indicates a shift to thec = /2RIGHTof the graph ofy = sin(x)The number

dindicates a vertical shift up or down of the graph of the basic sine and cosine functions.

EXAMPLES:

Following are the graphs of

,y = cos(x), andy = cos(x) + 1.y = cos(x) 1red graph:

y = cos(x)blue graph:

,y = cos(x) + 1which indicates a vertical shift up ofd = +1y = cos(x)green graph:

,y = cos(x) 1which indicates a vertical shift down ofd = 1y = cos(x)

Strategy for Sketching Some Sine and Cosine Functions- see #1 through 5in the "Examples" documentWe will only sketch functions of the form

andy = a sin(bx).y = a cos(bx)

- Determine the
amplitude.- Determine the
period.- Keep in mind the graph and characteristics of the basic function.
- Mark off a distance along the
x-axis to represent theperiodof therepresentative picture.- Divide the
periodinto four equal intervals.- Create the appropriate
representative pictureby using the beginning and ending point of each interval. Keep in mind theamplitude. Be mindful ofbeing positive or negative!a- Connect the points in (6) to form the
representative picture.- Copy the
representative pictureseveral more times along the negative and positivex-axis.