# 6.1 Graphs of the sine and cosine functions

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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. 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.

 $x$ $0$ $\frac{\pi }{6}$ $\frac{\pi }{4}$ $\frac{\pi }{3}$ $\frac{\pi }{2}$ $\frac{2\pi }{3}$ $\frac{3\pi }{4}$ $\frac{5\pi }{6}$ $\pi$ $\mathrm{sin}\left(x\right)$ $0$ $\frac{1}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{3}}{2}$ $1$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{1}{2}$ $0$

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

Notice how the sine values are positive between 0 and $\text{\hspace{0.17em}}\pi ,\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\pi \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}2\pi ,\text{\hspace{0.17em}}$ which correspond to the values of the sine function in quadrants III and IV on the unit circle. See [link] .

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.

 $\mathbf{x}$ $0$ $\frac{\pi }{6}$ $\frac{\pi }{4}$ $\frac{\pi }{3}$ $\frac{\pi }{2}$ $\frac{2\pi }{3}$ $\frac{3\pi }{4}$ $\frac{5\pi }{6}$ $\pi$ $\mathbf{cos}\left(\mathbf{x}\right)$ $1$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{1}{2}$ $0$ $-\frac{1}{2}$ $-\frac{\sqrt{2}}{2}$ $-\frac{\sqrt{3}}{2}$ $-1$

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

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 $\text{\hspace{0.17em}}\left[-1,1\right].$

In both graphs, the shape of the graph repeats after $\text{\hspace{0.17em}}2\pi ,\text{\hspace{0.17em}}$ which means the functions are periodic with a period of $\text{\hspace{0.17em}}2\pi .\text{\hspace{0.17em}}$ A periodic function    is a function for which a specific horizontal shift    , P , results in a function equal to the original function: $\text{\hspace{0.17em}}f\left(x+P\right)=f\left(x\right)\text{\hspace{0.17em}}$ for all values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the domain of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ When this occurs, we call the smallest such horizontal shift with $\text{\hspace{0.17em}}P>0\text{\hspace{0.17em}}$ the period    of the function. [link] shows several periods of the sine and cosine functions.

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 $\text{\hspace{0.17em}}\mathrm{sin}\left(-x\right)=-\mathrm{sin}\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ Now we can clearly see this property from the graph.

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich

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