Question

Describe the graph of the cosine function.

Answer 1

@ganeshie8

Answer 2

@nincompoop @Preetha

Answer 3

@campbell_st

Answer 4

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

Answer 5

Hehe, not really

Answer 6

Let me check online

Answer 7

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

Answer 8

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

Answer 9

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??

Answer 10

@campbell_st

Answer 11

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

Answer 12

Degrees

Answer 13

@campbell_st

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

you could...

Answer 18

would it be right?

Answer 19

@ganeshie8 could u help me out pleease??

Answer 20

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

\(\large\color{black}{ \displaystyle x=90^\circ -(180^\circ\times {\rm k});~~~\color{blue}{\rm \forall~~k\in{\bf Z}}}\)

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

Its alright....

Answer 25

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

Answer 26

@SolomonZelman

Answer 27

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.

Answer 28

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)

Answer 29

Is 180 an x-intercept? No right?

Answer 30

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

Answer 31

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

Answer 32

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.
`~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~`

Answer 33

so they are
....... -450º, -270º, -90º, 90º, 270º, 450º, ....

(adding or subtracting 180 each time)

Answer 34

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

Answer 35

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

Answer 36

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?)

Answer 37

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

Answer 38

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

Answer 39

right?

Answer 40

the rest of the cosine graphs?

Answer 41

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

Answer 42

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

Answer 43

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

Answer 44

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

Answer 45

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

Answer 46

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

Answer 47

this way
y=cos(x+a)+b

has a range of [-1+b,1+b]
y-intercept 1-a

Answer 48

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?

Answer 49

Yes, negative or positive multiples of 90

Answer 50

I mean negative or positive, odd multiples of 90

Answer 51

Thank you soooo much!!!

Answer 52

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).

Answer 53

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

Answer 54

Thanks!