Describe the graph of the cosine function.

- anonymous

Describe the graph of the cosine function.

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- anonymous

@ganeshie8

- anonymous

@nincompoop @Preetha

- anonymous

@campbell_st

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## More answers

- campbell_st

so do you know what the graph looks like...?

- anonymous

Hehe, not really

- anonymous

Let me check online

- anonymous

Pretty similar to the sine graph if you ask me...

- campbell_st

here is a site that will graph it for you
on the left side just enter
y = cos(x)
and you'll see the graph
https://www.desmos.com/calculator

- anonymous

ok, so now, just to make sure, the domain is all real numbers, the range is -1≤y≤1, the y intercept is 1, but what would the x- intercepts be??

- anonymous

@campbell_st

- campbell_st

so where does it cut the y-axis, are you working in radians or degrees..?

- anonymous

Degrees

- anonymous

@campbell_st

- anonymous

Sorry to bother u, but I'm pretty lost...

- campbell_st

so so cos(90) =0 and cos(270) = 0
so the x- intercepts are at x = 90 and x = 270 then repeat every 180 degrees

- anonymous

Oooh, ok, so, as an answer, I put "starting at 90°, every 180°"?

- campbell_st

you could...

- anonymous

would it be right?

- anonymous

@ganeshie8 could u help me out pleease??

- SolomonZelman

There is a pattern to x-intercepts.
For what values is cos(x)=0, the solutions to this are the x-intercepts.
For example at x=90 the cos(x) is 0
and so it is for x=270, for x=450
and on adding 180º each time....
\(\large\color{black}{ \displaystyle x=90^\circ }\) is your first y-intercept
then, \(\large\color{black}{ \displaystyle x=90^\circ+180^\circ=270^\circ }\) is another x-intercept
then, \(\large\color{black}{ \displaystyle x=90^\circ +180^\circ +180^\circ+.... }\)
Each time you add 180º you get an x-intercept.
So, you can generate the pattern:
\(\large\color{black}{ \displaystyle x=90^\circ +(180^\circ\times {\rm k})}\)
(for all positive integer values of k, and for 0 as well)
You can also go -180º , subtract 180º, to get the x-intercept.
\(\large\color{black}{ \displaystyle x=90^\circ-180^\circ=-90^\circ }\) is an x-intercept
\(\large\color{black}{ \displaystyle x=90^\circ-180^\circ-180^\circ=-270^\circ }\)
\(\large\color{black}{ \displaystyle x=90^\circ -180^\circ -180^\circ-180^\circ-.... }\)
So, you can generate the pattern:
\(\large\color{black}{ \displaystyle x=90^\circ -(180^\circ\times {\rm k})}\)
(for all negative integer values of k)
--------------------------------
it follows that x-intercepts all go by the pattern
\(\large\color{black}{ \displaystyle x=90^\circ -(180^\circ\times {\rm k});~~~\color{blue}{\rm \forall~~k\in{\bf Z}}}\)

- SolomonZelman

The blue notation here means
"for all integer values of k"

- SolomonZelman

saying that k can be equal to
..... \(-5\), \(-4\), \(-3\), \(-2\), \(-1\), \(0\), \(1\), \(2\), \(3\), \(4\), \(5\), .....

- anonymous

Hehe, sorry @SolomonZelman , I'm not quite following... XD

- SolomonZelman

Its alright....

- anonymous

So what you're saying is that the x-intercepts are all multiples of 90?

- anonymous

@SolomonZelman

- SolomonZelman

the x-intercepts start from 90º.
and they are all the following:
x=90-180
x=90-180-180
x=90-180-180-180
x=90-180-180-180-180
x=90-180-180-180-180-180
x=90-180-180-180-180-180.....
and so on....
x=90+(k•180)
where K IS NEGATIVE integer or 0.
And ALSO, x-intercepts are from 90 and +180
x=90+180
x=90+180+180
x=90+180+180+180
x=90+180+180+180+180
x=90+180+180+180+180+180
x=90+180+180+180+180+180......
and so on....
x=90+(k•180)
where K IS POSITIVE integer or 0.

- SolomonZelman

this why, you can generate a pattern with k (where k is both positive integers and negative integers and 0)
and this pattern is:
x=90+(k•180)

- anonymous

Is 180 an x-intercept? No right?

- anonymous

Aren't the x-intercepts the same as the sine and tangent graphs?

- SolomonZelman

no 180 is not, but 90 plus 180 (add or subtract 180 from 90 any number of times)

- SolomonZelman

lets look at it simple.
(we are talking about the cos(x)=y graph)
the x-intercepts are the values of x that make the y=0.
these are as I posted:
`~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~`
the x-intercepts start from 90º.
and they are all the following:
x=90-180
x=90-180-180
x=90-180-180-180
x=90-180-180-180-180
x=90-180-180-180-180-180
x=90-180-180-180-180-180.....
and so on....
x=90+(k•180)
where K IS NEGATIVE integer or 0.
And ALSO, x-intercepts are from 90 and +180
x=90+180
x=90+180+180
x=90+180+180+180
x=90+180+180+180+180
x=90+180+180+180+180+180
x=90+180+180+180+180+180......
and so on....
x=90+(k•180)
where K IS POSITIVE integer or 0.
`~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~`

- SolomonZelman

so they are
....... -450º, -270º, -90º, 90º, 270º, 450º, ....
(adding or subtracting 180 each time)

- anonymous

Yes, I understand now! So examples of intercepts would be 90, 270, 450, 630, 810, etc?

- SolomonZelman

yes, or you can subtract 180 from 90 many times
90
90-180=-90
-90-180=-270
-270-180=-450
and on....

- SolomonZelman

So, for this reason they can be all given using a pattern
x = 90 + k•180
for all integers of k.
(is this still unclear?)

- SolomonZelman

(all integers =
..... −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ..... )

- anonymous

Great, and just to make sure, the rest of the cosine graph would be:
y-intercept is 1, the domain are all real numbers and the range is -1≤x≤1

- anonymous

right?

- SolomonZelman

the rest of the cosine graphs?

- anonymous

No, I mean the rest of the characteristics of the cosine graph

- SolomonZelman

yes, yes.
But provided they are not shifted sideways.
they can have any angle of cx, such that y=cos(c•x)

- SolomonZelman

and the range will change if you shift it up/down

- SolomonZelman

so y=cos(x),
has a range [-1,1]
y-intercept 1

- SolomonZelman

but,
y=cos(x+c)
has a range of [-1,1]
y-intercept 1-c

- SolomonZelman

and
y=cos(x)+c
has a range of [-1+c,1+c]
y-intercept of 1

- SolomonZelman

this way
y=cos(x+a)+b
has a range of [-1+b,1+b]
y-intercept 1-a

- anonymous

I just started learning the trig graphs, so I don't think the curves will shift. thanks anyways! One more thing, is it fair to say that the x-intercepts are all odd multiples of 90?

- SolomonZelman

Yes, negative or positive multiples of 90

- SolomonZelman

I mean negative or positive, odd multiples of 90

- anonymous

Thank you soooo much!!!

- SolomonZelman

in this case of y=cos(x), which is not the case if you shift the graph sideways such that y=cos(x+b), and which is not [neccessarily, but for most values in fact not] the case if you multiply the angle or the function times a scale factor, such that: y=cos(bx), or y=bcos(x).

- SolomonZelman

this is just graph shifts..... the rule you can get in a book or online..... my PC is about to resart because I am scanning and updating. cu

- anonymous

Thanks!

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