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what is the amplitude of a sine graph what about the period what about cosine

in this episode graphing trig functions using transformations we’ll practice

graphing lots of functions that involve sines and cosines

you’ll learn what all this means and best of all you’ll learn how easy it is

let’s do some math [Music] hi everyone welcome back we’re going to

begin with the sketching the graphs of trig functions

um so we’re going to start talking about um the um

even and oddness of a trig function of in the following form

so let’s get on here going um so we’re going to be looking at uh graphs of

these four types here uh where a and b are real numbers

so a is going to be a positive or negative real number

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and b is going to be a positive or or negative real number um [Music] now

in the previous episode um we talked about um trig functions of these forms

here but we were assuming that b was positive so now we’re not going to

assume b is positive so b could be negative and we’re going to deal with

that case and we’re going to see how easy it is

but before we get started on this episode i wanted to mention

that this episode is part of the series trigonometry is fun step-by-step

tutorials for beginners link is below in the description

and so you can follow along in the whole series

and so like i said in the previous episode we talked about uh what happens

uh how these graphs look of these type of functions when b is positive

so now when b is negative so we’re going to use the fact that sine

and cosecant functions are odd functions

and cosine and secant are even functions and so what does that actually mean

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well if we assume that b is positive and then we have a minus sign in front

of here then we have a negative number in front of the x

and what we can do with that negative so we can pull the negative out because uh

the sine of negative x is equal to sine of x

and so this is what i mean by that sine is an odd function

just like if you put a negative x into a third power here

you know that would be minus x to the third here right

so the negative comes outside of the third or here the negative comes outside

of the sine function right here so um now cosine is an even function and so

what that kind of means is it absorbs the negative right so if if b is if

we’re assuming that b is positive then this will be a negative

times a positive number for example something like 5 cosine

minus 6x right so now the b so the a is 5 and the b is 6

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and we have negative 6 here and so how can we deal with that negative right

there so we could this will be just 5 cosine 6x because

cosine is an even function cosine of minus x is cosine of x

um and you can see all these um identities you can if you look on the

graph you’ll you can see them on the graph so if we enter say pi over six

that output will be the same output as if we entered minus pi over six so

cosine of minus pi over six is equal to cosine pi over six

of course that’s just not true just necessarily for

um the quadrant one and quadrant four angles but it extends on but in any case

the secant is also even so secant will absorb a negative sign

and that’s true because well the secant is the same thing as 1 over cosine this

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is our reciprocal identity that we talked about since the beginning since

we defined cosine and secant and so if i was to put a negative into

secant negative x a negative angle for example that would be 1 over cosine of

minus x but that is one over cosine of x and so that is secant x

yeah so secant of minus x is the same thing as secant x

and then lastly the cosecant as you might imagine because cosecant is the

reciprocal of sine so if we have a cosecant here then this will be one over sine

and so if we try to substitute in a negative into a cosecant then we’ll have

sine here and then we’ll pull the minus sign out and so that’ll be minus sine

and so that’ll be minus 1 over sine and so that’s just negative cosecant x

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okay so cosecant minus x however you want to think about it is just the same

thing as minus cosecant x so if we know how to sketch these graphs over

here we did this in a previous episode when b is positive

and so if you have negatives in here well we can use these identities right

here so this is assuming that b is positive because we know how to graph

these and so now we can graph these right here if we want to graph something

that looks like this we’ll just sketch this graph right here because they’re

equal to each other all right so um keep in mind though that the period is 2

pi over b and again we went over that in the

previous episode so here we’re just kind

of extending this using these identities here and practice graphing some more so

let’s get started let’s uh graph our first one here 2 sine 3x

so this is definitely something that we covered in a previous video because the

b here is positive and the a is just a 2. so let’s just refresh our memory how

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to sketch this real quick so the period is 2 pi over the b and the b is three

so the period is two pi over three and the amplitude is going to be two or

the amplitude is absolute value of two which is just two all right so we can

sketch the graph now now what i’m going to do is i’m going to

sketch the shape of sine and then i’ll adjust the tick marks for

the period so it’s important that we get the shape in now you don’t want the

shape to be like making made up a bunch of you know lines right there you want

to add the appropriate curviness to the graph

the shape of the graph is important so i’m going to focus on that

now when we substitute in 0 we get out 0 sine starts right here at 0

or at least the primary period of it and so i’m going to go up

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and then i’m going to come back down and that looks reasonable i guess but

anyways this is two pi over three that’s one period we go up and down

and you know if you don’t like it i recommend you know just keep practicing

go up come back down and so this i’ll put this mark here as two pi over three

because that’s one period and we repeat start repeating after that and so what’s

halfway halfway will be two pi over six but two pi over 6 reduces to just

2 pi over 3. and then we’ll cut halfway again to get

the height here which is happening at 2 and so cut halfway again is pi over 6

and now we can add up our pi over six one pi over six two pi over six

three pi over six what’s three pi over six that’s just pi over two

and the minimum right here is minus two and so there’s one period and we can

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extend this to to other period we could extend this graph as far as we’d like

we just go down and then come back up and so yeah we can extend this again

this is 1 pi over 6 2 pi over 6 3 pi over 6. this is 4 pi over 6

right it just reduces to 2 pi over 3 but we can count keep counting our pi over

6’s so that’s four pi over six so the next one will be five pi over six

and then six pi over six right and so we can keep counting our pi over sixes

and we’ll get all the uh tick marks that we see here and we can just keep

repeating that and so on and we can you know we can

subtract our pi over sixes and we can get the graph this way as much as we

want now typically i just sketch though one period of graph there all right so

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there’s a graph right there for number one here

all right and so let’s graph if we have a minus three in there

so now we have to work out our identity here

so let’s say this is going to be minus 2

sine 3x remember sine is an odd function so

this is the same graph that we just had but now has a minus so it’s been

reflected so oh we’ll just sketch this one right here let’s just uh

remark on the period the period is 2 pi over the b which is positive which is 3

and the amplitude is absolute value of minus two so the amplitude is still two

and we can sketch the graph let’s sketch it say oh right here

now instead of going up because it’s being reflected now i’m

going to sketch it going like this and then like this

and then there’s going to be one period this is 2 pi over 3

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and then this is halfway which is you know two pi over six or pi over three

pi over six one pi over six two pi over six

three pi over six which is pi over two and the height here is two

and the minimum right here is minus two and so there’s the graph of number two

there um and so you know the minus there gave

us no problem at all because we know that sine is an odd function

all right so now what happens here if we have this one right here

so now let’s write it this as this minus sign can come out which case

we already have a minus sign so it’ll be a minus times a minus so this will be 2

sine pi x and so now the period on this graph is 2 pi over the b which is pi

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pi’s cancel the 2 period is and the amplitude is absolute value of

well it’s just 2. it’s always positive just just remember

the amplitude is positive there all right and so let’s go ahead and

sketch the graph uh i might be we’re making it a little bit sideways all right

um and so yeah we have a we’re going to go up

and come back down and we have a nice sine shape to it right there

and this is period is two we’ll chop it in half we get one

we chop it in half we get half and now we count up our halves one half

two halves three halves so three over two and then two

and our height here is a two and our minimum right here is minus two

all right so there’s a graph for number three there

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and let’s look at something like this now so this is

a minus sign again sine is odd function so the minus is going to come out

so this right here is going to be minus 1 4 sine of pi over 3x okay

and so the period is 2 pi over pi over three

or so differently two pi times three over pi pi’s cancel we get six

and the amplitude is uh the absolute value of minus 4 which is just 1 4.

all right so let’s sketch the graph uh let’s put it down here

um so our amplitude is 1 4 the period is six and it has a minus on it

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and so i’m going to start going down so it’s been reflected

and then come back up and so on and so here’s one period so the tick mark is six

i’ll chop this in half is three i’ll chop it in half again three over

two and now i’m gonna count up my three over twos one three over two two three

over two three 3 3 over 2’s which is just 9 halves

and then another 3 over 2 which would be 12 halves which would be 6. and so

here the height here is 1 4. and the minimum right here is minus one

fourth there right there so there’s a graph of number four right there

all right so now let’s do um let’s do the cosecant graphs now now

we’ve done cosecants before in a previous episode also but we focused

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when the b was positive so let’s do some here where the b is negative

so let’s look at this graph right here now i’m going to be

thinking of this as 3 over sine of 2x so the point is is that in order to

graph the sketch the graph of cosecant i’m going to look at the sketch of of

the graph of sine so of sine 2x so i’m going to be sketching this graph

but before i do so i’m going to first sketch the graph of sine 2x

so let’s do that right here so i’m looking at the period the period is um

2 pi divided by the b which is 2 and so the period is pi

and this is a sine graph and there’s a positive sign in front of it so it’s

going to be going up and this will be coming down

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something like that the period is pi so that’s pi pi over two pi over four

and this is one pi over four three two pi over fours three pi over fours

and then of course four pi over four and the height here is a 3 and -3

and that’s the graph of let’s see here so yeah so we have the amplitude here of

minus three three and minus three so for example

what happens when we use pi over four here so sine of two times pi over four

and then so sine of two times pi over four

that’s the same thing as sine of pi over two

which is 1 which is 3. so pi over 4 we hit a height of 3.

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so that’s a 1 sine of pi over 2 is a 1 so we get a height of 3 out there

all right so in any case now we’re ready to sketch cosecant all

right so what we’ve sketched so far is y equals 3 sine 2x

and now to get the reciprocal so now i’m going to sketch that in red

and so i’m looking where the sine is zero so i’m going to put the vertical

isotopes in here x equals zero and another one right here at x equals

pi over two and another one right here at x equals pi

and i’m going to sketch the graph here in purple for cosecant

and so the graph is going to look like this so these are isotopes so we got to

get close but not too close and then down here

right there we got that point and we got this point right here the height is

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three and this is minus three right here at three pi over four

and so there’s the graph in purple and so

you know we may erase the graph in black there that was just there to guide us

the graph of cosecant is the one in purple there

all right so let’s do another one let’s do this with um a minus 5x in there

all right so let’s sketch that one now so again i’m going to first sketch the

graph of y equals minus 3 sine of minus 5x

and so let’s sketch that one right here now this is the graph of minus this is

the function minus comes out remember sine is odd so this is just

going to be 3 sine of 5x in other words this one right here is just

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the 5 comes out the 3 com sorry the minus sign comes out so that gives us a

three cosecant five x so this is what i’m really sketching right here

so i first want to sketch this one right here three sine five x

and what’s the period on this one it’s two pi over five two pi over five is the

period so i’m going to sketch a regular sine graph as a guide

and so i don’t want to make it too tall because i want the cosecant graph to

stand out more all right and so there’s one period

and the period marks in at uh this is period period and so that’s 2 pi over 5.

so halfway would be just pi over 5 and then halfway there would be power ten

and so what we’re really doing is counting up pi over tens one pi over ten

two pi over ten and so this would be three pi over ten

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and now i’m looking for the isotopes to get the cosecant graph so where is sine

0 right because this is this is cosecant 1 over sine we cannot divide by zero so

we’re looking at isotopes right here x equals zero x equals pi over five

x equals two pi over five and you can see the pattern you know

zero pi over fives one pi over five two pi over fives three pi over fives four

pi over fives and so on those would be the isotopes

so now we can see the different branches

and maybe i’ll try to put this in orange maybe it’ll stand out a little bit more

right here there’s that branch and now we’re graphing the this one

right here in orange and then we got this branch right here

and there’s the graph of cosecant y equals 3 cosecant 5x

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and of course we have the x and y axis all right so there’s a reasonable sketch

there of the cosecant all right so you see that this episode

is very similar to the last one we’re just making the small modification

and the usefulness of because knowing the cosecant function is odd

all right and so let’s do this right here and so i’m going to first sketch this

this is odd so this is going to be pull out this minus sign it’s going to cancel

with this one or it’s going to multiply to a positive cosecant of 2 pi over 3 x

so this is what we’re going to sketch the graph of

and so to do that we’re going to sketch the graph of y equals

sine of two pi over three x and so what’s the period is two pi over

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two pi over three which is the same as two pi times three over two pi

in other words just three i can’t see that just three

all right so the two pi’s cancel you just get a three all right and so what’s

the sketch look like all right so um let’s go here with this is

so this is a negative times a negative this is positive here so i’m looking at

a positive so i’m going to go up and come down

and this is a 3. i chop three and a half i chop that in half

now i add up my three over fours one three over four two three over four

three three over fours which is nine over four

and then twelve three over 4s which is of course three

and then the height here is a one and minus one and now i look where the

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isotopes are coming in here where sine is zero so x equals zero [Music]

x equals three over two x equals three and we can see the pattern

zero three over twos one three over twos two three over twos

four two three over twos and so on and now we can sketch the graph now that

we have the isotopes we can shape it nicely without the isotopes it’s not quite

clear what the graph is doing so i like the isotopes on the graph there

and then if you want we can just erase the

sine graph it’s not really part of the graph

and occasionally i like to do that just so i can see the graph all on its own

i’ll put this back in here x and then y and then in red you have the isotopes

there and so that’s the cosecant graph and so

this would just repeat you got the upper branch the lower branch an upper branch

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a lower branch and so on all right let’s do one more

and if you like this video i hope that you subscribe

let us know in the comments below if you’re enjoying this video

all right so this is a cosecant which is odd just like sine so the minus sign is

going to come out and so here’s the function that we’re going to be graphing

it’s going to be minus 1 6 cosecant of pi over 4x

and so to graph this one right here we’re going to look at the sketch of

y equals minus 1 6 sine of pi over 4x and so let’s see here the period is

going to be period is 2 pi divided by pi over 3 sorry pi over four

and so yeah that’s two pi times four over pi pi’s cancel and we’re really just

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getting an eight right there all right so our period is eight

and so let’s go ahead and sketch the graph over here

let’s do it about right here and so we got an 8 and this is a negative so

it’s been reflected so i’m going to sketch the graph here

and that’s it eight right there halfway is a four halfways of two two four six

and the height here is at the uh 1 6 the amplitude is 1 6. now we don’t

say the amplitude of the cosecant is 1 6 but for the auxiliary function right

here the sine the amplitude is 1 6. and so now we can look at the isotopes here

where is sine zero right here at x equals zero and then x equals four

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and then x equals eight and so you can see the pattern here for all the

isotopes 0 4 8 12 and so on and then and then the negative

0 minus 4 minus 8 and so on and now we can sketch the graph for the cosecant

here so we’ll put the upper branch right here

and we’ll put the lower branch right here and actually just now notice that

we can’t see the isotopes labeled down there so we’ll put them up here

x equals zero x equals four x equals eight and

you know that just keeps repeating there’s just one period for this

cosecant graph it starts with a lower branch and then

an upper and then a lower and then an upper and so yeah that’s it

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so what about the secant graph now or the cosine sorry what about cosine

cosine is even so these will be even easier

so what how how do we sketch 2 cosine 3x uh this is something that we did in our

previous episode so the the period is going to be 2 pi over 3 and now

cosine right so cosine starts up here so i like to

all right and then i’ll just erase that that’s that’s part of the graph but it’s

not in the first it’s not in the first period here

so i’ll put these tick marks here this is 2 pi over 3 so halfway 2 pi over 3

so halfway is right here and this will be two pi over six or pi over three

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halfway again is pi over six and so we have one pi over six two pi over six

three pi over six two pi over pi over two and so this just keeps repeating

now the amplitude right here is two the height is two right here and then minus

two right here and so there’s the sketch of this graph

right here uh on on one uh period all right and so yeah what happens if

there’s a minus sign in there so cosine is an even function

so this is 2 cosine 3x which we just graphed so we don’t need to sketch it again

all right so what happens if we have a minus pi in here so this right here is

the minus sign gets absorbed so this is minus 2 cosine pi x

all right so this is easier to think about how to sketch the graph of

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the period is 2 pi over pi 2 pi over pi or in other words just two

and so we have the cosine graph again but it’s been reflected so now i’m going

to start down here and this is a minus two and the height here is a two

and where did it start to repeat so right about here and this is a two

halfways one and so this is a half and so this is uh three halves and so

that’s it um and now let’s do this one right here

let’s see did i forget anything right here i got the period i got the heights um

yeah okay so that’s it it just keeps repeating

okay so um let’s look at this one right here now

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um the minus sign gets absorbed so this is going to be equal to 1 4 cosine of

pi over 3 x and so the period is 2 pi over pi over three

or two pi times three over pi fives cancel we get six and

so this is a positive number in front of here so we’re gonna start up here

so we’ll just start like that go down come back up

and this height right here is uh sorry this x is a six and the height here is

one-fourth the amplitude here is one-fourth so we get one-fourth right here

and this will be minus 1 4 down here this is a 6 so let’s chop it in half we

get a 3 and then a 3 over 2 so 1 2 3 so 9 over 2. and

i’ll just put an x and a y here and we’re good to go there

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and then this just keeps repeating on and on and on

now maybe that doesn’t look like it’s the same height there

all right there we go all right so what about seek it now all right

so yeah if you um like this video please subscribe i would

really appreciate that help support the channel

and so now let’s look at uh 3 secant 2x so this one right here is going to be

equal to um i only put this down to help to help someone understand

but you know i wouldn’t necessarily do that so i need to

think about the cosine graph in order to sketch the graph of secant i like to

draw the graph of cosine 2x in fact i’m going to draw the graph of 3

cosine 2x to help me sketch the graph of this one right here

so i remember secant goes with cosecant because the reciprocal functions right

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here all right and so the period here is well we’ve already did this graph right

here remember the period is 2 pi over 2 so just pi uh i think we did this graph

anyways um period is pi and it’s cosine and so it’s going to start up here

and that didn’t give me much room for my

secant so i’m going to put that in again and this time try to draw a straight

line so now i’m going to just try to draw it in here like this

it’s just going to keep repeating over and again the height here is a 3

and this is the minus 3 and this right here the height right here is going to be

at pi and then pi over two and then pi over four and then three pi over four

00:32

if you can see those tick marks actually you know

that doesn’t look that great let’s do it again uh i’ll try to

make it in the middle here so this will be pi halfway will be pi over two

pi over four three pi over four and then now let’s find the isotopes

where is the cosine graph zero so right here

and this would be x equals pi over four and where else is the cosine graph zero

so at three pi over four and so now we can draw the

secant graph right here i’ll draw this in orange so we have a lower branch

right here and we have a partial upper branch right here

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and another partial upper branch right here and so the

graph in orange is the seek 3 secant 2x where the height here is a 3 and this

right here is a minus 3. here we go all right so this is um

pretty straightforward now i’m hoping getting the hang of this here

so this is secant it is the same thing as minus 3 over cosine of minus 5x

remember cosine is an even function so this will be cosine 5x

so what i really want uh so what i want to sketch the graph of first is

minus 3 cosine 5x so i want to sketch this right here to

help me sketch this one right here and this this graph right here is uh so the

m the minus here gets absorbed so minus three secant of five x

00:34

um and so in order to sketch this one i’m gonna sketch this one right here first

and so i’m gonna sketch this right here here we go so let’s put it right here

minus 3 cosine 5x and this is a minus sign so i’m going to

start down here actually come up and then go back down and then

the period here is 2 pi over 5 2 pi over five

so this will be two pi over five so halfway will be um

two pi over ten or in other words pi over five

and then the halfway again will be pi over ten so 1 2 3 pi over 10

00:35

and here the height here is at a 3 and this will be a minus 3 right here

so now we’re going to look where the um where is it zero so right here

at x equals my pi over 10 and then right here at x equals 3 pi over ten

and so now we’re ready to sketch the graph i’ll do this in orange

again we have a lower branch right here split here we

have a upper branch right here and we have a branch right here split

and so there’s the sketch right there of one period of the

secant graph right there all right so what is this right here looking like so

this minus sign right here gets absorbed so this would be minus secant

00:36

of 2 pi over 3 and then for the last one right here

this minus gets absorbed so this will be 1 6 secant and this will be pi over 4 x

so this one right here has period two pi over two pi over three

or so differently two pi times three over two pi or just three

and this one right here has period is two pi over pi over four

two pi times four over pi and the pi’s cancel we get eight right

so here um we’re going to be sketching the graph of y equals minus cosine

00:37

two pi over three and here we’re going to be sketching the graph of one sixth

cosine of pi over four x so we need to sketch these graphs first

and then we’ll be able to put the isotopes in and sketch the graphs of the

secant so to sketch these two graphs right here

we really need to sketch these two graphs right here

now if if you haven’t checked out that previous episode i just want to remind

you one more time there’s a whole episode where we go

through these graphs right here but let’s just do them one more time right

here so let’s let’s sketch this graph right here first so this one’s minus

cosine 2 pi over 3x so here we go i’ll sketch it over here

and this is a minus cosine so i’m going to start down here um

let me add more room here in fact i can move the whole thing up here let’s do

00:38

that and i’m gonna start down here come up back down and this right here is a

three three over two three over four so one two three or nine nine over four

and then 12 over four right there so the height here is a one the amplitude is

a one and a minus one right here and now for the isotopes

where where’s the graph zero for cosine graph zero right here x equals

uh three over four and x equals uh so that says three over four i think

i can squeeze it in three over four and nine over four and now we can sketch the

00:39

secant in orange right here and then down here

and maybe we’ll extend these isotopes up here all right and so now

this one right here 1 6 cosine pi over four the period is eight

so let’s sketch it right here and this one is positive right here so

it’s going to start up here and looks something like that

height here is uh as at eight and it’s one sixth so minus one sixth

and i’m gonna chop it in half and then half and then add them up two four six

eight all right and then where are they 0 here right here at x equals 2

and right here at x equals 6 and i’ll draw this one in orange

00:40

so it’s right here we have a lower branch right here

for this secant graph right here lower branch and then this is split right here

and split right here so this is one period here where we’re going up right here

coming down right here and then we have this lower branch right

here and then this just repeats if we wanted to do more we could

stretch it out there and then we would have another isotope and we’d have it

again and so on yeah we can draw as many of those as we want all right so

i hope that you enjoyed the video and um yeah so

next um we’re going to be looking at the um phase shift uh in in more detail

and so that video should start right now