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in this episode we’ll practice graphing periodic functions that involve secant

and cosecants let’s do some math [Music] all right everyone welcome back and

we’re going to begin by taking a look at

what happened previously on trigonometry

is fun so we went over these uh episodes right here so far

and so we talked about what are trigonometric angles degrees and radians

and we talked about the unit circle special angles on the unit circle we

also talked about reference angles and we talked about the trig functions

we talked about trig functions of acute angles and uh for real numbers

and in the last episode um we worked um quite diligently on graphing sines and

cosines functions that involve sines and cosines we changed the period we worked

with the amplitude we even did some shifts but today we’re going to focus on uh

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secant and cosecant and so let’s see here what the graph of secant looks like

first so let’s start here with secant and then we’ll look at cosecant in a

minute now to graph co to graph secant i’m going to be looking at the function

cosine so remember the relationship between cosine

and secant right whenever we define the cosine function and the secant function

uh secant was the reciprocal of cosine in other words if if we look at this

right here this function right here is one over cosine of x

and so uh i this you know these this right here the shape of this right here

is related to the shape of cosine right here

so um that’s why we’re looking at cosine so i’m going to look at cosine first

here uh before i look at what secant looks like so actually going to make a

good graph right here let’s put it right about here

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and the cosine is going to go up and then come back down and go back up

and come back down and over here also and so yeah we talked about the shape of

cosine in the last episode now because of this relationship right

here this is 1 over cosine of x we have to be worried about where cosine

x is zero because secant is not going to

be defined where cosine is zero so where is this cosine graph right here is zero

so it’s zero right here and right here in here and so on and then it’s also 0

right here and here and so on and so let’s actually put some labels down here

so remember cosine had a period of 2 pi so this is 2 pi right here

and this is pi right here and this is pi over 2 and this is 3 pi over 2.

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so those are those tick marks there so now we’re going to try to sketch the

graph of secant and we’re getting this graph because of this

this relationship right here secant is equal to 1 over cosine x so whenever

x is 0 the cosine will be 0 and this will be 1

over zero right so that’s just not defined right so what we’re gonna have

here is isotopes right here at x equals pi over two

and where else is cosine zero right here so we have another isotope vertical

isotope and this will be three pi over two

and then we have another one right here in fact we have infinitely many of them

and we’d like to figure out how we can write them all down

so this is one pi over two three pi over two

this one will be what five pi over two and also we have a vertical isotope here

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cosine is 0 which means secant is undefined and this will be negative pi over 2

and then we get another one right here and this one will be negative

three pi over two and the pattern continues

and so now we’re ready to look at the graph of secant let’s put that in purple

maybe let’s see how this goes so this is the

highest right here which is uh which is a one right here

for cosine for secant that’ll be um a local minimum or it’ll just be um

it’ll be the minimum between minus pi over two and pi over two

but this this is going to go down here so it won’t be an absolute minimum in

any case we have the graph right here shape like that

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and shape like this right here and we can get a little part of it over

here maybe and we got this part right here so this

purple part here is the secant graph so i’ll put it in purple secant of x

there we go so knowing what the graph of cosine

looks like we can sketch the graph of secant and we can remove the cosine

graph there if we wanted to there’s the graph of secant right there and so it’s

worthwhile to write down the information about it just to make sure we’re clear

so let’s do that over here so we’re going to have the domain what is the domain

of secant and so as you can tell here there’s some

vertical asymptotes so the domain is going to be all the real numbers x

but x cannot be equal to so how can we write down infinitely many values like

there like this so we’re going to kind of need a parameter notice they’re all

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odd so we’re going to say 2n plus 1 times pi over 2 and then we’re going to say

where n is an integer where n is an integer

good there’s the domain that’s set right there

so it’s all real numbers x but x cannot be equal to

an odd multiple of pi over two it cannot

be pi over two three pi over two five pi over two and so on and when

you know n is an integer so it could be negative and so

these negative values here the range here is so what is the range so now we’re

looking along the y-axis right here if i want to label this as the y-axis right

here and so the range here let’s write that one in interval notation perhaps

so it’s going to be from 1 to infinity we’re going to include the 1 it’s going

to hit the 1 right there so 1 to positive infinity union and then minus 1 to

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and we’re including the minus 1 because it touches right there at minus one

so minus one to minus infinity so the union of those two intervals that

is the range right there um some people like to write this one first

but it’s a union so it doesn’t really matter but just because i’ll

write it again so minus 1 to or actually minus infinity will go first

or write like that minus 1 and then union 1 to positive infinity

all right and so either way you want to write it there’s the range

um so notice that the secant here is discontinuous for all values um

right here so let’s write that down discontinuous and discontinuous for

uh these x values right here whenever we’re equal to these x values so we’ll

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say the odd multiples of pi over 2. discontinuous for those also notice that

there’s no x-intercepts no no x-intercepts

right there’s nowhere where it crosses x-axis see it’s just up here it’s down

here and just keeps repeating that pattern right there there’s no

x-intercepts um the period is the same as cosine so period is 2 pi

so for example we can start right here at 0

and go up for this part and then we’re coming up and we’re going down and then

we’re going down and then repeats it’s going up and then

down down and then up and then it’ll repeat again

all right uh so another way you can think about it is

you know this is pi over two to minus pi over two so this is the whole pie right

here in between not including the end points

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so there’s one branch and then here’s another branch right here so you can

think about it as the period is two pi right there

also notice that there’s no max and no min no max no min

and by that i mean no absolute maximum no absolute minimum um

okay so there’s secret so i feel like there’s

you know some basic information that everyone needs to know about secant in

particular to sketch it what i’m always going to do is i’m going to

sketch first the corresponding secant graph or cosine graph and then i’ll

purple in the actual graph that i’m looking for which will be the secant

graph right there all right so now before we go on

let’s look at the cosecant graph here so let me erase this real quick

so as you might guess the cosecant graph is going to be based on

the sine function so this right here and that’s because

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this right here is 1 over sine of x and so we’re looking at the reciprocal of

a known function we know what sine looks like and we’re looking for the

reciprocal of that so let’s draw a sketch of the sine graph first

and so let’s see here we go now sign starts here so i’m going to go up

and then down and then up and then down and i’ll try to do the same thing over

here and so on all right that’s good enough so this is the sine graph

just y equals sine x now of course this continues and that

and it continues over there um but to make the

cosecant graph well we need to worry about where sine is zero now because

that’ll be one over zero that’ll be undefined so where is sine zero it’s

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zero here and here and here and here and so on and also here and here

and so wherever sine is zero we’re going to have vertical isotopes

so let’s do do these vertical isotopes right here so the y-axis is going to be

a vertical isotope right there for the cosecant graph and we’re going

to have a another vertical isotope right here and actually let’s go ahead and

make a period so this is 2 pi and then this is pi and this is pi over 2

where we hit our height right here of one and this will be three pi over two

where we hit our minimum right there at minus one

and so we have another vertical isotope where sine is zero

another vertical isotope goes sine of zero again and again

and so this is x equals pi x equals two pi and then this will be three pi

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and this would be 4 pi and this is the period twice so go

through it once we go through it again and now we can go through it this way

and so this will be minus pi minus pi and then this right here will be -2 pi

and so on and so there we have the vertical

isotopes and that will help us draw the sketch of the secant graph

all right so here we go these are vertical isotopes so let’s

draw them like we’re getting close to them

some people like to put arrows on them so people don’t

and then let’s put them right here like this and like this and

hopefully you can tell that purple from the black

and then just to make sure that they’re all touching

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let’s put some points on them right there

there’s there we go there’s a height of one and they’re touching and they’re

touching and here’s a height of -1 oh and i forgot this little branch over here

something like that any case there’s a couple of branches

of the sine graph right there so i mean the cosecant so the label down

here so this is y equals cosecant x is the one in purple the one in black was

just to help us define it and the red lines the red dashed lines are the

vertical isotopes they’re not part of the graph either but just to make clear

let’s write down over here what the domain and the range is

so the domain let’s let’s put that up here let’s put it over here i guess so

the don’t let’s put it over here the domain

is the all the real numbers such that so um you know when we’re looking at

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cosecant x uh it’s almost defined everywhere except

for these isotopes here so we’re going to say all real numbers and then we’re

going to say the exceptions there are infinitely many exceptions

so x is not going to be equal to and so what are the values we’re going

to roll out here so in this case right here we’re going to say n times pi

and where n is an integer this will allow us to make infinitely

many values here where n is any integer all right so the domain is that set

right there and the range here we can write it out

in interval notation because it’s just got two pieces to it it’s going to be

one to infinity right here it’s going to be minus infinity to minus one so let’s

say minus infinity to minus one and then oh it includes minus one here so union

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and then one to positive infinity right there all right so there’s the range

so let’s notice where it’s discontinuous so where do you have to pick up your

pencil right so i’m drawing i’m drawing cosecant i’m drawing i’m drawing i’m

drawing i’m drawing and now we got to pick up our pencil to get over here

right so discontinuous so discontinuous mean

as long as you’re defined right so it’s discontinuous where x is equal to the n

pi where n is an integer that’s where it’s discontinuous

so the domain is where you can plug in numbers and you can plug in anything

where it’s not equal because where it is equal well it’s not defined it’s

actually discontinuous there the period is also 2 pi

we can see that by looking at the graph here’s one branch and here’s pi and

here’s another branch and here’s also pi and then it starts to repeat itself so

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the period is 2 pi and again there’s um no maximums no minimums no max

why is there’s no max well this part right here for example

just keeps going up and up there’s no maximum and this branch right here just

keeps going down there’s no minimum um yeah and so

now we’re going to start looking at the um you know

functions that trig functions or periodic functions that involve

secant and cosecant like for example you know what if we change the period

and things like that so let’s look at some examples

all right so here we go let’s get rid of this real quick

all right so let me outline the strategy for you that i’m going to use here so

let’s look at the y equals a and then we’re going to have cosecant bx

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and y equals a secant bx and b is going to be greater than zero for right now

all right and so how do we sketch these graphs here

right so we have a b and if you remember the last episode the b was changing the

period and the a was the the absolute value of the a

was the amplitude right and so what we’re going to do is we’re first going

to sketch the graph of a and then this one will correspond to sine and we’ll

first graph this right here and this will correspond to cosine

and so we’ll first graph these and we graph these right here in the previous

episode so again the previous episode here’s the trailer for it

trigonometry is fun step-by-step tutorials for beginners if you haven’t

checked out the previous episode the link is below in the description but

we’re going to graph these and we’re going to graph them pretty quickly too

because we already know how to do them so we want to sketch these right here

first and then we also want to sketch the

vertical isotopes the vertical isotopes are important because they show the

behavior of the graph and that’s important right here and then step three

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we’ll actually try to you know sketch the graph we’ll sketch

the graph that we’re looking for so those are more or less the three steps um

all right so yeah let’s look at a couple of examples so

our first example is going to be um i’m going to give this the

sine cosine version and we’re going to graph the secant cosecant version of

this right here so sketch the graph so we did this one in our previous episode

and so now in this one we’re going to do y equals

2 cosecant x so how do we graph this right out here 2 cosecant x well first

we’re going to graph this one so this is just a basic sine graph i’m going to

come up and then down and i’m only going to be gra i’m i’m only going to be

graphing one period and so this will be pi pi over 2

and then this will be 3 pi over 2 and then this will be a 1 oh sorry no

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this will be a 2 and this will be a minus 2 right here

and so this is y equals sine 2x which we could put right there if we wanted to

so now i’m going to sketch the graph of 2 cosecant x of course this repeats

and it goes on but now i want to draw the vertical

isotopes so this will tell me right here where sine is 0 that will be a vertical

isotope for this graph we’ll have a vertical isotope right

through there at x equals zero and we’re going to have a vertical

isotope right through here at x equals pi so we’re labeling our

vertical isotopes and then we have another vertical isotope right here

at x equals 2 pi and so now in purple i’m going to sketch

the graph i’m looking for which is y equals 2 cosecant x that’s the graph

that’s new for this episode right here and we’re just going to sketch it in

right here and you want to get close to the isotopes you don’t want to do

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something sloppy like that you know you want to have a

reasonable graph that shows that you understand what a vertical isotope is

that’s good enough something like that you can put arrows or not

all right there we go you definitely don’t want to show

crossing but you want to show it getting closer that’s good so there’s our two

cosecant graph right there this is a point on it right here and this is a

point on it and we have the isotope sketched and labeled so that that shows

behavior right there and if you want we could get rid of the the the sign the

sine graph we don’t need any more if you wanted to get rid of that

there it is right there and now we can put in the axes a little better

and perhaps re-dash the isotopes there so there’s a reasonable graph of

2 cosecant 2 2 cosecant x right there all right so there’s our first example

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right there let’s go do another one right here what happens if we do four

sine x what happens if we change that to a four

right all right so we did that one in our previous episode so now what happens

if we graph this one right here four cosecant x so what does the four

change we change it to a two to a four well over here we’re going to go and

change this to a four right here so this will be four

and this will be minus four and so this would be

a graph right here that you could use right here to to sketch to have the graph

now however it it it look it looks nice if you could compare the two to make

sure that you understand what’s going on what does that four do right here’s a

graph of it but you know how did how do these actually compare

so like if you had a sine graph that just went up to a height of one

and then you had a sine graph that went up to a height of

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two it still hits 0 at the same place now if we have 4 right so that will come

up to a height of 4 and then it’ll come down to

a minimum of 4 but they still go through the same zeros right there so there’s a

4 and there’s a minus four right there and so that’s that would be the fill in

part right there and so you can see what’s happening with the secant right

that’s what’s happening with the sign we talked about that last episode but now

it’s happening with the secant so the secant is coming down here to a one

and then the two is going to get you know it’s it’s going to it’s only going

to come down it’s going to go like this it’s going to come down to a 2

and then it’ll come down to a 4 and so you can kind of see

you know what that’s kind of doing there

but in terms of having to sketch a graph there it is right there

all right so let’s do another one here um let’s do a minus now

so the problem that we’re trying to solve we did this one in our previous

episode just want to keep saying that and then minus and then we have a three

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and then we have a cosecant so now let’s grab this one right here so the first

step is to graph this one right here so we have the graph of sine sine comes up

like that but now this is a minus three so now we’re going to go down like here

like this then we’re going to come up here like this and this is a three and a

minus three and we still have a two pi and a pi

and a pi over two and one pi over two 2 pi over 2 3 pi over 2

and so there’s a quick sketch of y equals minus 3 sine x

and so now let’s graph this one right here minus 3 cosine x

so next step is to put in the vertical isotopes we have vertical isotope right

there at zero we have a vertical isotope right there at pi

and we have a vertical isotope right here at two pi

and we can do more but that’s one period that’s enough and then

we’ll come in here now with our cosecant graph it’s going to come in

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here it’s going to be an isotope so let’s just draw it draw the behavior

like that and draw the behavior like this right there

um and so that would be the graph of minus 3 cosecant x and then we can erase

the sine graph there if we want all right let’s do another one minus cosine

and so now let’s do a secant so now we’re going to be trying

to sketch the graph of y equals minus secant x

that’s that’s the new problem that’s the previous one we did in the last episode

that’s the new problem right there all right so let’s sketch it

so first i’m going to sketch the graph of cosine right here here we go

uh minus cosine and so cosine normally starts up here now it’s going to start

here i’m going to go like that and come in through like that and then

go back down and then there’s one period right there 2 pi halfway is pi

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halfway again is pi over 2 and so that would be three pi over two

all right so that’s the graph of minus cosine x right there in black

and now let’s put in the isotopes so where is cosine zero

so cosine 0 is right here so i’m going to have an isotope right there

cosine 0 right here so this will be the isotope x equals pi over 2

x equals 3 pi over 2 and then we could draw another one but

we don’t need to we could draw another one we got one period right there

so now i’m going to draw the um the secant now to do that i’m going to extend my

vertical isotope up here and here to show behavior

and so we’re going to get it like this in purple and so i’m going to come in

here like this and the amplitude here is 1

so this highest point right here is 1. put a one right there

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and so let’s see here we have a another branch right coming right through here

and another branch coming right through here excuse me all right so there we go

the purple one is the secret the minus secant graph right there

um i wonder if orange would show any better than the purple here

and then it’s hard to tell between orange and red any case

let’s go on to another one um what about one-fifth cosine

alright so that was our previous one now this one is going to be y equals

1 5 secant x so how do you sketch the graph of one fifth secant x

first i draw my cosine now it’s just a regular cosine so just

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coming through like this this is a 2 pi pi pi over

one pi over two two pi over two three pi over two

now i’m going to draw my isotopes right here got one coming right through here

at zero at sorry pi over two and another one coming right through here

and so now in the purple we can sketch the secant graph

right there and right there and right there

i’ll extend this up here a little bit and what are these right here these tick

marks right here that’s a one-fifth and this one right here is a minus one-fifth

and so there’s the sketch of one-fifth x it’s in purple and let’s just go ahead

and erase this one right here just to erase one of them at least and then

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this would be what we’re left with is the actual graph that we were looking for

this graph right here so there it is there it is right here

it’s going to bounce right there at the one-fifth

and bounce right here at minus one-fifth um

at pi minus one-fifth right there all right so there’s the graph of one fifth

secant and let’s do [Music] uh sine two x now now we’re gonna be

changing the period a little bit so now we’re going to sketch the graph

of instead of the sine one that was our previous one

so now let’s graph the sketch the graph of cosecant cosecant of 2x

so how does this change right so let’s see here um remember the period

is going to be 2 pi over the b the b is positive so it’s 2 pi over b

so the period is pi and so i’m sketching this one first just

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like we did last time so it’s just a sine graph so i’m gonna come up

and then down and the period is pi so halfway is right here pi over two

halfway again is pi over four and then now we count up our pi over

fours one pi over four two pi over four three pi over four four pi over four

and so there’s one period of the graph of y equals sine two x

and now time for the secant graph time for the vertical isotopes first so where

is this one right here zero so it’s a zero right here zero right here

and right here so keep in mind how these are related to each other right um

this right here cosecant 2x is the same thing as 1 over sine 2x so

we have to sketch the graph of y equals sine 2x first nothing not just the graph

of sine x sketch the graph of y equals two sine x

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where is the graph of y uh of sine two x where is that zero because that’s where

this one right here will be undefined okay so i just wanna mention that here

at least once but in any case time for the

sketch of this one right here let me see if i put it in orange if it will look

nice here let’s put it in orange it will go right here

and then we’ll come right here and let’s figure out the height now

the amplitude is for the sine one right here the amplitude is one so

this one right here is one right here and this one here is minus one right here

all right so there’s the graph right there in orange

is the graph we’re looking for right there i think it looks so much nicer if

you put the isotopes down but the sine graph right here which i always put

first but you know that’s optional you don’t necessarily need to do that

especially if you can visualize this if you do enough of these and you can just

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visualize it but you know you got to get to tick marks

all right so the next one is cosine 4x so we’re going to be graphing y equals

secant 4x so how do we what is the graph of secant 4x look like

so let’s get a cosine graph going so cosine we’re going to be looking at

the period first is 2 pi over our positive b which is 4.

and so that’s pi over two and so our period is pi over two now

we’re going to be looking at cosine cosine is going to start right here and

so we’re gonna we’re gonna start here and then come down and come back up

uh don’t make it look too straight it’s got to be nice and curvy

all right and so here’s where it starts repeating at pi over two

and then i chop that in half pi over four and i chop that in half again pi over

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eight so we get one pi over eight two pi over

eight and this will be three pi over eight and so now we’re going to look for

vertical isotopes and so where is this graph zero right here at pi over eight

so that’s the vertical i um let’s make it a little longer so this is the

vertical line x equals pi over eight and we have another one right here this

is where cosine four x is zero at three pi over 8 right so we’re going

to go right through here and say this is 3 pi over 8

and now we’re going to have here the graph

we’re going to come in through here like this and like this and

maybe we should extend this one up a little bit more

and then we’re going to come in here like this one right here there we go

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all right so let’s see what’s missing now is this label right here which is

um the amplitude is one for this so the for the secant 4x it’s going to

bounce here at 1 label about right there and this will be -1 right there all

right there we go so the period of secant 4x is pi over 2 same period

and we have a branch going right here a branch here and a branch here so we’ve

covered enough for one for one period of this for one period of this graph right

here all right so next one is sine of 1 3 x

so what’s the new function that we’re graphing this time will be y equals

00:34

cosecant of one-third x and so now what is the period so the period is

2 pi over 1 3 which is 6 pi so i’m going to graph the sketch to

graph a sine real quick but i’m going to change the period to 6 pi

and that will guide us towards graphing this graph right here so here we go so 6

pi this is sine it’s it’s a positive sine function right

here so it’s just going to come up and then back down

and then i’ve already gone through 6 pi right here so halfway right here is 3 pi

and then halfway of that is 3 pi over 2 so i need to count up my 3 pi over 2’s 1

3 pi over 2 2 2 3 pi over 2’s and this would be 4 3 pi over 2’s which is just

you know wait one two three i forgot three pi over

twos so three three pi over twos which is just nine pi over twos all

right so nine pi over twos all right so where are the vertical

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isotopes right here we have a vertical asymptote right here

and a vertical asymptote right here and right here

and so now we can sketch the graph for the cosecant we’re going to get this

graph right here this branch right here and

that’s already one period right there so all we’re missing is the

what labeling on the y-axis right here one and minus one there

all right and so these are the isotopes right here right x equals 0

x equals 3 pi and then x equals 6 pi there we go all right so next one is

cosine of one fifth x this won’t take us long cosine one fifth so

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we’ll sketch the graph of secant of one fifth using this as a guide

um that looks strange so one-fifth x all right and so now let’s see

so what would be the period so let’s put the period right here period is

2 pi divided by 1 5 1 5 or in other words 10 pi

all right so that’s the period so here we go um

we’re going to sketch the graph of cosine first so cosine starts up here

here we go cosine and then right there and right about there

and so we’re going to get one period here which is 10 pi

so it grew really slowly right just go on really slow it took 10 pi to get

through one cycle one period all right so halfway is five pi

and then halfway again is five pi over two and then halfway again

or this one right here is 1 pi over 2 5 pi over 2 two of them and then three of

00:37

those this is 15 pi over two and that’s 20 pi over two

all right and so now we have our vertical isotopes right here and right here

and let’s label them 5 pi over 2 and 15 pi over 2

and so now we’re going to have the sketch of the graph which i’ll put in

purple right here so let’s go right here bounce it and go up

we get this lower branch right here and this branch right here

and so if we want to we can take off the cosine graph

here and then every now and then i just like to take one off and so

there’s roughly the sketch right there where this is a amplitude is one again

so this will be minus one and this one right here will be one right there

00:38

all right so all right so now um let’s um i think that’s good um

we haven’t done any where where are mixed right here

so these are all amplitudes and these are all periods and let’s do a couple of

them that are mixed um let’s skip ahead a little bit to how about

this one right here let’s go to this one right here

if you’d like to see these other three right here put a comment down below and

let me know and i’ll work those out otherwise let’s just go on and kind of

mix them let’s do this right here so now the graph we’re going to sketch is

00:39

the sketch of minus 2 cosecant of 2x how do we sketch this

minus 2 cosecant of 2x right there all right so the period is going to be 2 pi

over our positive b and that’s a 2 that’s positive so the period is just

going to be pi all right so let’s sketch the graph with the sign and

this is a minus 2 so instead of starting up like that i’m going to go down here

like this so i’m gonna go like this and then like this

and so here one period is pi and there’s pi and so this would be pi over two

and so this would be pi over four and so one pi over four two pi over

fours three pi over fours and now we’re going to be looking for

the isotopes to sketch the graph for cosecant

00:40

so we hit 0 right here so we’re going to have x equals 0’s and isotope

x equals pi over 2 and right here at pi so those are vertical isotopes and now

we can sketch the graph um let’s do it in purple now here we go

let’s sketch the graph right there it’s touching right there

and we got it right here and there we go so what is the uh

labels on the y-axis right here so y-axis x-axis

and it’s a 2 right the amplitude for this function right for this right here

is a two and so that’s where we’re doing our bouncing right here two

and minus two so it’s going to go up and then it’s gonna do hit that minus two

height right there and then it’s going to start falling down again start

decreasing and here it’s decreasing and then it hits that height of two right

00:41

there and then it starts to increase again right there so we got our isotopes

we got our sketch um so yeah looks good all right so um

if you’re wondering what happens when the b is greater than zero

well we worked that out today now if you’re wondering what happens if b is

less than zero then i recommend that you watch the next

video which is starting right now if you enjoyed this video please like and

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