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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

what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl 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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