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In this section, you will:
  • Graph variations of  y=sin( x )  and  y=cos( x ).
  • Use phase shifts of sine and cosine curves.
A photo of a rainbow colored beam of light stretching across the floor.
Light can be separated into colors because of its wavelike properties. (credit: "wonderferret"/ Flickr)

White light, such as the light from the sun, is not actually white at all. Instead, it is a composition of all the colors of the rainbow in the form of waves. The individual colors can be seen only when white light passes through an optical prism that separates the waves according to their wavelengths to form a rainbow.

Light waves can be represented graphically by the sine function. In the chapter on Trigonometric Functions , we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions.

Graphing sine and cosine functions

Recall that the sine and cosine functions relate real number values to the x - and y -coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let’s start with the sine function    . We can create a table of values and use them to sketch a graph. [link] lists some of the values for the sine function on a unit circle.

x 0 π 6 π 4 π 3 π 2 2 π 3 3 π 4 5 π 6 π
sin ( x ) 0 1 2 2 2 3 2 1 3 2 2 2 1 2 0

Plotting the points from the table and continuing along the x -axis gives the shape of the sine function. See [link] .

A graph of sin(x). Local maximum at (pi/2, 1). Local minimum at (3pi/2, -1). Period of 2pi.
The sine function

Notice how the sine values are positive between 0 and π , which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between π and 2 π , which correspond to the values of the sine function in quadrants III and IV on the unit circle. See [link] .

A side-by-side graph of a unit circle and a graph of sin(x). The two graphs show the equivalence of the coordinates.
Plotting values of the sine function

Now let’s take a similar look at the cosine function    . Again, we can create a table of values and use them to sketch a graph. [link] lists some of the values for the cosine function on a unit circle.

x 0 π 6 π 4 π 3 π 2 2 π 3 3 π 4 5 π 6 π
cos ( x ) 1 3 2 2 2 1 2 0 1 2 2 2 3 2 1

As with the sine function, we can plots points to create a graph of the cosine function as in [link] .

A graph of cos(x). Local maxima at (0,1) and (2pi, 1). Local minimum at (pi, -1). Period of 2pi.
The cosine function

Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval [ 1 , 1 ] .

In both graphs, the shape of the graph repeats after 2 π , which means the functions are periodic with a period of 2 π . A periodic function    is a function for which a specific horizontal shift    , P , results in a function equal to the original function: f ( x + P ) = f ( x ) for all values of x in the domain of f . When this occurs, we call the smallest such horizontal shift with P > 0 the period    of the function. [link] shows several periods of the sine and cosine functions.

Side-by-side graphs of sin(x) and cos(x). Graphs show period lengths for both functions, which is 2pi.

Looking again at the sine and cosine functions on a domain centered at the y -axis helps reveal symmetries. As we can see in [link] , the sine function    is symmetric about the origin. Recall from The Other Trigonometric Functions that we determined from the unit circle that the sine function is an odd function because sin ( x ) = sin x . Now we can clearly see this property from the graph.

Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
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don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
Practice Key Terms 5

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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