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

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

begin by talking about what happened previously on trigonometry is fun

we covered these four videos so far we talked about trigonometric angles

what they are and the unit circle and we talked about all the special

angles that go on the unit circle we also talked about reference angles

and in the last episode we talked about the six trigonometric functions

and so let’s go on today to talk about what does the graph of the sine function

look like so to do that we’re going to start with an equation y equals sine x

and in the last episode we talked about if you input x as a real number

it’s going to represent an angle in radians

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and we talked about how to define that and we worked out lots of examples so

before we get that going um i just want to mention that this episode is part of

the series trigonometry is fun step-by-step tutorials for beginners uh

check out that link below in the description

and yeah so we’re going to start off by talking about the sine function

and the sine function is going to be periodic if you look at the way that

sine was defined last time we use the unit circle we used to point on the unit

circle so it’s probably not too surprising to you that if you

start going around and around and around the circle you end up with the same

output values for sine over and over again um

so for example if we have sine of 30 degrees

um and then that’s going to be equal to sine of you know 30 degrees

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and then if we go around one more time how would we find sine of 390 degrees

right so last episode we said well let’s find the reference angle and then we’ll

look at the sign in front and the quadrant to determine if it should be

positive or negative so these two values have the same so so these two numbers

are the same sine 30 degrees equals sine

390 degrees and we can keep going around and around the round circle right so

what we’re going to say is that a function is periodic

so let’s put that right here periodic function what does that mean

well it means that the function let’s call it periodic function f so f of t

plus p that says t plus p equals f of t and this is for all t in the domain for

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all t in the domain so and the smallest p is called the period smallest p

is the period so we’ll keep this in mind when we start

looking at the sine graph right here so what we’re going to do is we’re going

to make a table up here of some values and we’re only going to do this the

first time we’re trying to understand what it looks like once we understand

what it looks like we’ll be able to to get to it much quicker so let’s make a

table here sine t and t and we have some values and these are

some values that we’ve been looking at so far pi over six you know the special

angles pi over six pi over four pi over three pi over two

that’s the first quadrant and pi over two and two pi over three

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and three pi over four and five pi over six that’s my ugly looking five

five pi over six and then six pi over six well let’s just

go with pi right there and the output the sine values what were zero one half

square root of two over two or 1 over square root of 2 if you whatever you want

square 3 over 2 1 and then in the second quadrant right so

sine was still positive and so we’re just going to go right back down

and so here’s the output values now in the previous

episode we talked about how to uh given an input like a t and to

evaluate the sign and we did that by using reference angles and looking where

the sign was positive and negative according to quadrants right so there’s

quadrant one in quadrant two um and then let’s go ahead and make up

the table four quadrant three and four so now i’m going to say pi seven pi over

six uh five pi over four and these are the

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angles that go on the unit circle right here so i really hope that you

like that video on the unit circle and how to compute

the uh unit circle with all the special angles on it in five minutes or less

and so now we have the sine t right here so sine team row

so we have pi and then sine of pi 0 and then we have minus one half so sine

is negative in the third quadrant here right so these are quadrant three angles

we have minus square root of three over two we have minus 1

and we have minus square root of 3 over 2 and minus square root of 2 over 2

and minus 1 half and we end up with 0 right there so there we go

so these are the values that we know how to find by the previous episodes they

represent the quadrant one quadrant two three and four special angles and they

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represent the output values right here so if we go to draw the unit circle i

won’t we’ll go through that process again you just saw that

but we have the pi over six pi over four pi over three and we keep going around

the circle and we get all those special angles in there and the point is is that

each of these points has an x and a y and these are the y values right

here remember sine t was defined as the y values right there so these y values

are right here and we’re going around the unit circle right here

and so if you count them up we put them all right here in a table format

rather than a unit circle geometric format right so this was the

xy and right there at pi over 2 it would be pi over 2

1 right there so the angle would be pi over 2

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and the height the y value would be a 1. and because we’re interested in the

graph of sine we’re only looking at the y values right now so the y value right

here is one for that for that example right there

all right and so yeah so we know how to do all that but the point what we want

to do right now is to put this on a coordinate axis x y axis and if we do that

um so we have here all these angles right here to pi and

then all the and then continue all the way to 2 pi

and what we’re going to get here at 0 0 so there’s that point right there

and then we’re going to get these three points right here

and then they’re growing up larger to one if we look at the decimals point

five point seven point eight and they’re going to come up to a height of one

and then they’re going to start going back down and notice the symmetry right

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here to see the symmetry it’s going to come back right and go right back down

and then after that and i’ll have to move if i can move out

of the way say over here and then it’ll go back down and it’ll be

it’s equally a space right to see the symmetry right here and the symmetry

right here these are the output values right here and so we have

and the highest point right here is pi is one and it happens at pi over two

and this is when it gets back to 0 at pi and this right here

the lowest it goes is minus 1 and that happens at 3 pi over 2

so this is the t axis right where the t is the angle

and then this is the output axis right here and so i kind of

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got that one off a little bit in any case once you get back to two pi right here

then it starts to repeat itself so it’s going up and it

starts to repeat itself it starts to go up again and it repeated

if you look at the negative angles and repeat it like that

so it just keeps going over and over and over again

the period of of of sine so the period is two pi

and the way that we we look at that is um you know the the first

first and second quadrants we’re going up and then we do then we’re decreasing

and then we’re decreasing and then we’re increasing and then we repeat that

pattern after that so that’s a sketch of the sine graph right there

we want to label these important points right here

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we can label this one right here pi over six pi over four

pi over three we get these three points right here

and we get those numbers right there along the x along the y axis right there

and then we get these right here by plugging in these so we get 2 pi over 3

3 pi over 4 5 pi over 6 and that gives us these

three points here so you see it’s symmetric it’s going through those

points there and symmetric it goes through those and then

we get these outputs right here from these inputs seven pi over six

five pi over four four pi over three we get those and we do the same over here

we get those right there and so i mean this is just um plotting

some points obviously it’s the special angles

and from there you can kind of get the idea of the shape of it so

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i think we should look at cosine now how do things change for the cosine

so let’s let’s get rid of this and look at the cosine graph here now okay so

cosine here we go all right so now we’re going to look at the um

difference here we’re going to have 0 and we’re going to have pi over 6

and we’re going to have pi over four and pi over three

and pi over two those are our special angles in quadrant one

and let’s go ahead and look at quadrant two so two pi over three

and three pi over four and five pi over six and six pi over six

and so let’s look at sine and cosine what are the differences

so the sine which we just put in here a second ago and

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so this is 1 right there and then square root of 3 over 2 square root 2 over 2

one half and then 0. so it goes up to 1 and then down to zero and then

you know the quadrat three and four i won’t write them out again but

all right so cosine of t so this starts at one

so sine would start at zero went up to one cosine is going to start at one

remember these are cofunctions um and so you know

the pi over six is complementary with pi over three

pi over four is complementary with pi over four pi over six and then pi um

and then pi over two wait anyways um cosine so we have square root of three

over two square root two over two one one-half and zero and so

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what does cosine do in the second quadrant so cosine is negative in the

second quadrant so this is going to be minus one-half

minus square root of two over two minus square three over two and then minus one

so we can use these right here to sketch the graph of cosine so

cosine starts up here at one and then we get these three here

at these special angles right here it’s going down to zero so it’s going to go

down to zero right here and this is at pi over two

and we get these three right here point and so this is one

this is point eight point seven point five approximately

um and then we get to zero and then we have negative values so now

it’s going to go like this we’re going to get these three negative

values right here and we’re going to end up at negative 1 down here

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so this will be like negative 0.5 approximately negative 0.7

approximately negative 0.8 and then when we get to the quadrat 3 angles

in quadrant 3 cosine is still negative right so we’ll still have these negative

outputs right here that’ll correspond to these right here but just in reverse

and we’ll get those and then we get back to zero so

if we continue on here we’ll get um you know 2 pi over three five pi um

three pi over four five pi over six uh that was the quadrant two so we

already d quadrant two sorry um so we’ll get like uh pi and then we’ll

get like seven pi over six um five pi over four four pi over three

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and 3 pi over 2 and cosine in quadrant 3 is negative so

we’re going to get negative square root of 3 over 2

negative square root of 2 over 2 negative 1 half and then here right here

we get zero again so that’s these values right here we get zero again and

right so i think they’re lined up okay right you can you can kind of tell them

so you see the difference between the sine and the cosine if you and this is

the cosine graph cosine t so t axis and the y axis right and so it’s just

going to come back up here in quadrant four the cosine is positive

and so we’ll come up here and get a height of one again

and so i guess i’ll just put them down here if i can

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so we’re going to get 5 pi over 3 and 7 pi over 4

these are the special angles in quadrant 4 and then 2 pi

and so i’ll just say that’s a t and then cosine t and this will be

you know 5 pi over 3 will be like right here and that’ll be the one half

and this will be the 1 over square root of 2 or let’s say square root of 2 over

2 square 3 over 2 and then a 1. and so at 2 pi here this says 2 pi

so 2 pi here gets back to 1. so we have this value right here again point five

about a point seven and about a point eight

and so the period of cosine is also two pi so period is two pi

and this keeps repeating itself and the reason why is this at two pi right here

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and so like if we wanted to find out two pi plus pi over six

if we wanted to find out that output right there

we would just say oh it’s the same thing as

the reference angle so we would take two pi way and we would use that value and

we would say cosine is positive in quadrant one so we would get

the same output right there okay so here we have decreasing and

decreasing and then decreasing and then increasing and increasing and there’s a

rough sketch of the cosine graph right there so if we look at the domain and

range of the cosine so the domain and range for both so we’ll say domain [Music]

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and so the domain for both sine and cosine is all real numbers

um i’ll put it right here all real numbers for the domain for both

of them and the range for both of them also sorry

the domain i’ll put it right here so domain

is all real numbers and the range here is the same for both also

it’s minus one to one and that’s because the height for both

sine and cosine goes up to a one it does

it at different places cosine goes up to one at zero and two pi and four pi and

so on but yeah so we’ll look at some more properties of

sines and cosines but there’s the basic shape of them and the domain and range

and so now let’s go on and look at some transformations like what is the

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amplitude so let’s talk about the amplitude now

so we’re going to do that by looking at some examples so what if we have y

equals 2 sine x what does the 2 do so we’ll compare that with the graph of

y equals sine so whenever we’re going to do these

examples here i always recommend looking at a basic graph first so the basic

graph will be sine and cosine so what does the sign look like

goes up and then down and it has a period of 2 pi right there

so this is pi halfway and then halfway again is pi over two

and so now i’m looking at my pi over twos one pi over two two pi over two and

then this one right here is three pi over two and four pi over two

and the height is one and uh minus one so if i want to graph the sketch of sine

for one period right so that’s just one period right because it keeps going it

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keeps going over and over and over over and over and over so

what does tucson do now right so now when i input a zero sign will output a one

and so this will be two times a one so this one right here has a height of two

so the graph of this one right here looks like this y equals two sine x

so it still has the same zero right there at pi though doesn’t it because if

you input pi into here you get sine of pi which is zero

and here if you put in pi you get sine of pi is 0 2 times 0 so it still goes

through the 0 right here it’s the same thing with 2 pi that’s

also 0. so it’s just going to come up higher now and then

down lower so this is a two and this is a minus two so

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i’ll try to sketch it more symmetric there so that’s part two pi and pi

and this is still pi over two and this is three pi over two

and the height here now is two though so when we input pi over two here we get

just output of one which is about right there and here we

get output of two because it’s two times that one

so even though the graphs look the same they’re not the same height so sine

would look something like that whereas sine

sine 2x would look something like this right so it’s going up higher

and and that’s what i did here but i just labeled these as 2 and minus 2.

all right so there’s one example there and and um two is called the amplitude

of the uh trig function there two sine x so let’s look at the next one

what if we have four sine x all right so i’m just going to grab

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because we just graph sine x so now i’ll just graph 4 sine x right here

and again it’s going to have the same shape i’ll try to extravagate uh make it

look more um extravagant there i’ll try again and the height here

and the period so the period is two pi so cut it in half cut that in half

and then count up your pi over twos one pi over two two pi over two three pi

over two the height here is now four and this is a minus four right there

all right so next one what if we have a minus three sine x so

the minus is going to reflect we’re going to reflect through the

x-axis here and whenever this is a positive output this is going to

multiply it by -3 so now it’s going to be a negative output so all these

positive outputs right here that are positive along the y-axis

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they’re going to get now negative so i’ll sketch that one now

so instead of going up now because this is a minus we’re going

to go down it’s going to still have the same shape though it’s just that it’s

going to be reflected so it’s going to go down and then it’s going to go up

and so this right here the period is 2 pi so we’ll chop that in half

we’ll chop that in half and then count up our two pies

count up our pie over twos and the height here is three

and the lowest it goes is minus three so it goes down to minus -3

and then it goes up up up and all the way up to 3 right there

and then it goes back down and that’s one period there

so there we go minus 3 sine x so the amplitude the amplitude is actually the

absolute value so the amplitude is three and three means that the the range here

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is between minus uh between minus three and three that’s the range

all right so let’s do just minus cosine x this one this is a cosine

so let’s look at that so let’s recall cosine real quick

so cosine started up here and this is two pi and so if we cut it in half

and then cut that in half and then count up our pi over twos

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

and four pi over two and it gets back to a height of one

and so i make it look like it comes back down a little bit there all right so

there’s our basic cosine graph right here

and now what about the minus what is the minus going to do so whatever i input

for the x i’m going to go calculate it and then

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i’m going to put a minus sign on it and then my sign is going to flip it

whatever positive number i got as an output i then multiply it by negative so

this is going to reflect it so now instead of going down now we’re

going to start here at -1 and go up and it starts to go back down and so here’s

2 pi about right here period is 2 pi and so cut it half pi cut it in half pi

over two count up our pi over twos we have three pi over two

and this right here is a minus one and the height here is a one

and so the amplitude here is minus one oh sorry the amplitude is

absolute value of minus one absolute value of minus one the

amplitude is the amplitude is still one it’s just been reflected

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so we have a height of one and we have a all right

so there we go um let’s look at one more so what happens if we do a one-fifth

well here’s the negative so this is y equals minus cosine x

and here’s cosine x right so how does this graph differ than this one right so

to sketch the graph really all we need to do is to put a one-fifth in front and

change these to one-fifths and then that’s that’s the same

that’s the same graph right there so that’s it’s going gonna start at a

height of one-fifth and go down and this lowest one right here is minus

one-fifth right there as the height and then it goes back up to to one-fifth

again now to be honest though let me sketch

these on the same axis so you can kind of see the difference there

so a regular cosine graph would look something like this and by regular i mean

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um the amplitude is 1 and it hasn’t been reflected something like that

now if it’s at one fifth then it’s not going to go up as high it’s

only going to go up a fifth high so it’s going to start down here but it’s still

going to go through that 0 right there it’s still going to go through at pi

over 2 it’s still going to be 0 right there

because when you input pi over 2 you’re going to get out 0 and 1 5 times 0. so

it’s just going to zero slower it’s decreasing slower this is decreasing all

the way up here starting at one it’s going to go fast because it has to get

to the zero at the same the same pi over two here

all right and then the lowest here is only one fifth here

so it’s going to be like that well and then the height right here

so it’s going to be lower you know just lower whereas the regular sine graph

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goes up and down between -1 and 1 and this is just between

minus one-fifth and one-fifth all right and so the amplitude on this

one is one-fifth all right and so how can the period

change now so now let’s look at uh how the period changes so

when b is greater than 0 i’ll put this up here

when b is greater than 0 then the graph of sine bx

right so it’ll resemble still sine x but it’ll have a period 2 pi over b

and the same thing for the cosine graph if b is positive then that’s going to

change the period the period is going to be 2 pi over b

so basically to graph this i need to know what the sine graph looks like

but then i’m going to change the period so here we go let’s look at the regular

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graph of sine just to compare it so the graph goes like this

and the regular period is 2 pi and then we’re going to chop this in half and

chop this in half and then add them up 1 pi over 2 2 pi over 2 3 pi over 2

and then we have our heights all right very good so we got all these

special points in here we know where it’s zero where it’s where

it’s at the highest place and so on and it continues and it continues

but um what happens when we have a two here so this is for sine x

and now for sine 2x so now the period is going to be

2 pi over what is the b for this problem the b is 2

right so sine 2x so the period for this one the period here is two pi

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you can think about this as you know sine times sine of one times x and this

is sine of two times x so the period is 2 pi over 2 which is just pi

so i’m going to have the same shape so i’m going to come up here like this

same shape but now the period has changed here the

period is two pi and then it starts to repeat here the period is pi

and then it starts to repeat so what’s halfway halfway right here is pi over two

and then halfway again is pi over four and then now here we go one pi over four

two pi over four three pi over four so this right here is three pi over four

and then the pi is four pi over four and so now we have our special points

right here it hits the highest at one and hits the lowest right here at the

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minus one and there we go there’s the graph of y

equals sine two x the period is pi so it goes through one full

period and then it starts to repeat itself over and over again both ways just

over and over again all right so let’s look at the next one

what if we have cosine 4x let’s graph this one right here real quick

so here we go here’s the graph of cosine first

starts up there and goes down here and comes back

and it gets to the highest point right there and highest point right there one

and lowest point at minus one and the period is two pi so i chop it in half

and i chop it in half and i add up my pi over twos one pi over

two two pi over two three pi over two and this is just y equals cosine x

so what happens if we have y equals cosine four x

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so now the period is going to be i’ll put it over here

so now the period is going to be 2 pi over 4 which simplifies to pi over 2.

so we’re going to have the same shape as

this right here it’s going to come right

through like this right here and go back up and it’s going to start repeating

it’s going to hit the maximum right there and the maximum right here and the

minimum and it’s going to be a 1 and a -1 but this period right here

is going to be pi over 2. so i’m going to cut the pi over 2 in half pi over 4

and i’m going to cut that in half and hit pi over

so now i need to count up my pi over eights so what do we have here

one pi over eight two pi over eight this will be three pi over eight

and this will be four pi over eight which is pi over two there

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so there we go we have the one and minus one we got all the tick marks here

special here here here here here and we have the shape that we need uh to see

that it’s a cosine graph there so yeah that looks very nice short and sweet i

like that all right what about sine of one third of an x

so this was two this is you know the period

and you know like the period here four and that made it go a lot faster didn’t

it i mean man i was only at pi over two ninety degrees and i already went

through one full cycle this one’s going faster than the regular so one third

here should probably go slower right so let’s check that out so we have the

regular graph assigned up and then down and then we have 2 pi pi pi over two

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and then three pi over two minus one and one and now let’s graph one third x

so what’s the period going to be so the period will be 2 pi divided by

the number in front of x as long as it’s positive right so um

b so the period is 2 pi over b so this is 1 3.

and so 2 pi over 1 3 is just better known as 6 pi

so now the period is 6 pi so it still has the same shape right let’s draw it

over here and we have a nice shape going through

here and then right there there we go and so this right here is one period is

six pi and so i’m going to chop that in half whatever it is in half is three pi

and then i’m gonna chop that in half and i get three pi over two

and now i’m going to count up my 3 pi over 2’s 1 pi over 2 what sorry 1 3 pi

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over 2 and then 2 3 pi over 2’s and then 3 3 pi over 2’s which is 9 pi over 2

and then this would be 12 pi over 2 or better known as 6 pi all

right and so the height here is 1 and the high minimum here is -1

and so there we go there’s the graph of sine 1 3 x and the period is 6 pi

all right this is kind of fun all right so let’s do a cosine real quick how will

this change for cosine so let’s see here can we get away with

drawing this one without drawing the cosine graph first i’ll try

so here we go and the period is going to be 2 pi over b b is 1 5th

so the period is 10 pi there we go so i need to start with the cosine shape

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so cosine is going to start up here and let’s go like that and then like that

and i don’t know i don’t like it enough try again

all right i guess that’s good enough so there’s one period right there and

the period is 10 pi so right here is 5 pi and then we’ll chop that in half 5 pi

over 2. and then now i need to count up my five

pi over twos here’s one five pi over two and here’s two five pi over twos and

here’s three five pi over twos so that’s 15 pi over two

and then here would be 20 pi over 2 or 10 pi and so the height here is 1

and this right here is -1 at the lowest point right there

okay so it’s all about marking your lay or labeling your axes to get a nice

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accurate graph of what’s going on here but it’s also nice to compare this to

a regular cosine graph so this is going really slow

it took us all the way to 10 pi to get through one cycle so it’s like

it’s stretched out okay so now let’s look at um doing some mixed

so here we’re going to mix the period and the amplitude together

so let’s see what we can get for these right here minus two times sine two x so

first off what’s the period period is two pi over the b which is two

so the period is pi and i have a minus two

so i’m going to go sketch the graph of sine

so it’s going to start right here at the origin and go up like that except for

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this is a minus two so now i’m going to go and go down first so i’m going to

draw the shape of sine and this right here is pi and so this is halfway

which is pi over 2 and then halfway from that and so that’s 1 pi 1 pi over four

two pi over four and then this will be three pi over four

and then four pi over four and now the height here is the absolute

value the amplitude of minus two so the height here is two

and the minimum right here is -2 and so there’s a reasonable sketch of the sine

sine graph right there and it didn’t take us very long to do that

okay so hope you’re having fun let’s look at minus one third cosine four x

okay so now we can go and look at a cosine shape and the period is going to be

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two pi over the four so pi over 2 is the period

and we’re going to start with a cosine but it’s been reflected in

shift stretched so what do we got so cosine will normally start up here

and go down but it’s been reflected so i’m going to start my shape like this

something like that and this is the lowest point here and here

and this is the highest point here and the period is pi over two

so this is pi over two right here and then i’m gonna chop that in half and get

pi over four and then chop that in half and get pi over eight

so we have one pi over eight two pi over eight and then three pi over eight

and then four pi over eight and the height here is the amplitude the

absolute value which is one-third the height is one-third right here

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and this is minus one-third right here so there’s the sketch of

uh number two right there um minus one-third cosine 4x right there there we go

looks like a reasonable shape graph we got all the

important points right there and we got the right shape to it and this keeps

repeating but it’s usually enough to sketch one period so first step i do is

i find the period and then i start to realize the shape

it’s not the one-third that’s important except for of course labeling it that’s

incredibly important but also the minus sign right so where is it going to start

up or down you know something like that so let’s look at another one

so here we go we have a two-thirds here so this will be

two pi over the b which is two over three and so

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you know these two cancel this is really just a three pi

or if you want you could say two pi over one times

three over two so just three pi however you want to think about that

but it’s it’s a minus sign right so it’s going to be reflected so here we go we

have a sine graph so normally we’d go like this

but it’s been uh reflected so now i’m going to go

down like that first and then like that and

if you don’t ever like it just you know erase it and do another one all right um

i don’t like that one either oh that’s going to be good enough and so

it goes down through one cycle then it comes right here

so there’s one period right there three pi

and i’m going to chop that in half and get three pi over two

00:42

and i’m going to chop that in half and get three pi over four

so now i need to count up my three pi over fours one three pi over four

to three pi over fours and then three three pi over fours which is nine

pi over four and the height here is my 1 and the minimum right here is -1

all right very good so there’s number 3 right there and now let’s do

one more what about cosine of 1 4 here so what would be the period here so the

period is 2 pi over 1 4 which is 8 pi and it’s a cosine and there’s no

reflection so we’re just going to draw a cosine graph something like that

00:43

here this is eight pi one period the height here is one

the minimum here is minus one this is eight pi which is the period

so i’m gonna cut it in half right here and go four pi and cut that in half and

go 4 pi over 2 or 2 pi and then i count my 2 pi’s 2 pi 4 pi 6 pi there we go

so there’s the graph of cosine one fourth x and there it is all right so

now we’re going to talk about translations so we’re getting everything

with sine and cosine now in the upcoming episode we’re going

to talk about the um secant and cosecant graphs and then

we’re going to talk about the cotangent and tangent graphs

and we’re going to do all the translations and everything for all of them

00:44

but in this episode let’s continue on with the sines and cosines

so here we go we’re going to be looking at something like this now so we have

and i forgot my y equals right here let’s see if i can go and fix that real

quick y equals all right so y equals um 3 cosine of x plus pi over 4 here

so now notice it’s got an x plus pi over 4 here

um the three we already you know we know what to do with the three that changes

the amplitude but what is this right here change so we’re going to

um this is going to shift it left or right in fact since it’s a positive it’s

going to shift it to the left so let’s do something first before we

talk worry about that three let’s just come over here and worry about the

00:45

cosine of x plus pi over four what does this do

the x plus pi over four what does this do to the cosine graph

so first let’s just graph the cosine graph

and compare these two right here so the cosine graph will just look like this

right here and this is a 2 pi and this is the pi and pi over 2 and 3 pi over 2

and here’s the height of 1 and 1 and the minimum of minus 1 here all right so

what does this pi over 4 do to a plus pi over 4 that’s going to shift it to the

left by pi over 4. so in other words when i plug in 0

now i need the output for plugging in pi over four

so that’s going to shift it and so what we’re going to end up with is

a graph that looks like this instead of starting up here at

00:46

one can we see that good let me put it let me put it about right about here

so what i’m going to do is i’m going to take the same shape but instead of

starting at 0 1 now the zero has been shifted and now it’s going to be

minus pi over 4 and that’s where i’m going to start

making my graph from so i’m going to still have the same i’m going down so i’m

going down like that so i have the same same shape still

and here is the highest again and here’s the lowest here so

when i try to do x plus pi over 4 let me make that a little bit skinnier

00:47

so we can see all that that’s too skinny this height here is

you know we need the tick mark for that height

how do we find all these tick marks now and we find all these points

everything’s been shifted over so the way that we do that is we take

each one of these that we already know for the regular cosine graph and we’re

just going to shift them over so what are they that we know that we know them as

so we know them as 0 pi over 2 pi 3 pi over 2 and 2 pi

so we know all those and we’re just going to shift them so i’m going to say

minus pi over 4 and then minus pi over 4 and so i’m

shifting all these points over what are the new tick marks

zero minus pi over four that’s the minus pi over four

00:48

and that’s right there that’s where we started graphing from right there minus

pi over four now why is it minus pi over four well

when i plug in or when i substitute in minus pi over four and when i substitute

that in there we get out zero and cosine of zero is at the one

so that’s going to be the newest tick mark right there the newest x value

where we’re at the highest right there which is the one and

so i have to shift them all not just uh not just that one point i gotta shift at

all so i need to find these uh new tick marks here what is pi over

two minus pi over four right so in order to do that you know we should look at

something like think of pi over two as what uh you know

two pi over four minus pi over four which is just you know pi over four

00:49

and think of that pi as four pi over four minus pi over four which is

three pi over four this one’s three pi over four and this one was pi over four

and this one right here is three pi over two minus pi over four

so think of that as six pi over four minus pi over four so five pi over four

and then the last one right here for that 2 pi

think of that as 8 pi over 4 minus pi over 4 so that’s 7 pi over 4. so we just

found all of our new tick marks here so this this final one right here it’s

not pi over four anymore because it’s been shifted so it’s seven pi over four

and this right here is five pi over four

00:50

and this one right here is three pi over four

and this right here is the pi over four there we go

so we still have the full 90 degrees right here right power think of poverty

is 90 degrees for a moment this this full 90 degrees right and so we still

have that same behavior here it’s at the greatest spot and it goes all the way

down to zero and the difference here is 90 degrees or the you know the same

width right there is pi over 2 pi over 4 this way pi over 4 that way right so

it’s still the same graph it’s just that someone took this one right here and

shifted it over and that’s what that graph right there is

all right and now to do the 3 right there that was never a problem right so

to put a three in front of here we would just change the tick marks there same

thing here we would just change these to threes so to get this graph right here

we’re going to say the amplitude is the three and now this is a three here

00:51

and this one right down here is a minus three so

this is a reasonable graph of this one right here we have this tick marks of

all these special points here and keep in mind you know it just keeps going on

and on and on you’re not just shifting one period you’re shifting the whole

graph all right so there’s one example there let’s do another one

let’s look at 3x plus pi so this one’s going to have a change in the period

and it’s going to have a shift right here also

so what we’re going to do is we’re going to sketch this one in stages as well

especially the first time you you try to do these is it’s a good idea to do them

in stages right there and so here we go we’re going to start off with the um

i’m going to ignore the shift right now and i’m going to

because i think we’ve done enough practice now to perhaps go straight for

this one right here minus cosine of 3x let’s try to sketch that one first and

00:52

then we’ll do the um the shifting right there of that right there um

yeah let’s um so we’re going to write this first actually as y equals minus 2

cosine and then the 3 is right there next to the x but

we have a shift right there so what i want to do is write this as a 3

and then i’m going to say x plus and then i’m going to say pi over 3.

so i factored out a 3 from these two right here

and this is what we’re looking at right here right now so we actually need to do

the shift first because if we distribute the three back

then we get this one right here back and the point is is that we when we

00:53

input an x we’re shifting it first um and then we can change the period on

it so when i change the period on it i can do that but i need to do that on

the tick marks that will be shifted already here so

let’s actually not sketch that one first here let’s shift it over first and so

you know when we start by looking at the original right here

we’re going to look at some inequality like 3x plus pi

is less than or equal to 2 pi because the normal cosine graph

is going to be one period it’s going to be between 0 and 2 pi

and so now if i subtract the pi’s everywhere then we’re going to get minus pi 3x

00:54

and then pi right so subtract a pi subtract a pi subtract a pi

and then divide the three and we’re going to get minus pi over 3

and x and pi over three so the this says the x’s is between the

minus pi over three and the pi over three so we need to divide

this interval right here into four equal parts and join them

so we can take that approach or we can take this approach first

it really just depends upon what you want kind of want to do there but

so let’s give that an experiment let’s let’s leave this on the board here and

we’ll think about this here in a minute but i like the approach that i was going

to take originally which was -2 cosine and just ignore the

pi for a moment and just sketch this one and we know how to sketch this one

00:55

remember that was actually kind of fun so let’s look at that right there real

quick so the period is 2 pi over the b which is the 3. so the period is 2

pi over 3. and this is a cosine graph so i’m going

to start up here except it’s been reflected so actually i’m going to start

down here and i’m going to make my cosine shape

now that goes too straight going to be curvy so start right down here

all right something like that and this right here is a minus 2 down here

and the height of the maximum here is a 2

and so we have these tick marks here so that the period is 2 pi over 3 so this

one’s going to be 2 pi over 3 and if i chop that in half it’ll just be pi over 3

00:56

and then i chop this in half again it’ll be pi over six

so i got one pi over six two pi over six and then three pi over six which is pi

over two and then two pi over three there all right so

this is the graph of this one right here um and so now we’re tempted to say okay

let’s just shift it over it says plus pi so we’re going to shift it to the left

and so we can do that by looking at the tick marks here so zero pi over six

pi over three three pi over two oh sorry pi over two and then two pi over three

and so i’m going to shift it to the left so i’m going to take off a pi

just like in the last example we shifted

by pi over four so we subtracted pi over four so i want to do the same here so

00:57

minus pi so it’ll be minus pi i’m going to subtract a minus pi

off each one of these just a pi and so what are our new tick marks so

think about this as pi over 6 minus six pi over six

so that’s minus five pi over six and think of this as three pi over three

or six pi over six no 3 pi over 3 is better

because we have a pi over 3 here so pi over 3 minus 3 pi over 3

so minus 2 pi over 3 and then we have pi over 2

so think about this as pi over two and two pi over two

so that’s minus pi over two and so that’s two pi over three so think

about this as two pi over three minus three pi over three which is just

00:58

minus pi over three and so now i’m going to try to sketch this graph right here

by using these new tick marks here but using the same period here

so it’s been shifted minus pi so it’s been shifted way over here right

so let’s see if we can sketch this graph here now and

so it never hits a positive angle here does it so um

yeah okay the period is two pi over three which is um minus pi okay yeah okay so

00:59

let’s um see here we’re going to have this over here really and

let’s go with the shape right here so i’m going to shape it right here like this

and these got these new tick marks here in here and here

and so this one right here is the minus pi minus pi and minus five pi over six

and minus two pi over three and minus pi over two and minus pi over three

and the height here is um a two the amplitude is two and the minimum

right here is minus two all right and so we’ve taken this shape

01:00

right here and we’ve shifted it over um by minus 180 degrees by minus pi

so we’ve shifted it over so instead of starting right here this point right

here has been shifted over to minus pi and there it is right there and then we

have the exact same shape right there that we had up here which is we have new

tick marks right here which we found over here all right and so um

so another approach would have been to shift first

and then found those tick marks and then chop the period um into

you know like we did over here we chopped the period into thirds

and so that would have been basically the same approach you’d get

the same answer right here at the end uh provided you looked over the same

interval right here um so it’s really a matter of do you

stretch or shrink first you know do you do the period first you stretch it or

01:01

shrink it that’s the period and then you shift

or you can shift first and then stretch or shrink the period either way you’re

going to get the same ending graph as long as you look on the same interval

you’ll get the same tick marks there so i’m not going to graph that one a second

time let’s do let’s do one more example though so let’s do um like

any kind of shift or any kind of transformation you can think

of let’s look at something like this right here

so first thing i’m going to do is write it like this right here i’m going to say

y is equal to let me move it over here y is equal to a cosine of

and let’s write this as a um k and then x minus b here and so i’m going to

write it looking like that so here we we can see what the a is right so the

01:02

amplitude is absolute value of a which is in this case 3 4 and the period is

2 pi over over the k or b if you want to use

b right here and k right there whichever

whichever way you want to do it but it’s 2 pi over that number right there in

this case it’s 2 pi over 2 so the period is pi

and i’ll move up here for a moment so the period is pi right there

and the horizontal shift i’ll abbreviate that right there horizontal shift is b

equals minus pi over three and so because we have a minus and a minus

and so that gives us the horizontal shift so we’re going to shift to the left

01:03

shift to the left again okay and so here’s the shape that we end up with here

we end up with um i need more here we’re going to end up with something

like this right here and it’s going to come down and go like that and then back

up and this highest place right here is going to be minus pi over 3

right because we’re shifting to the left and this is a cosine so it’s going to

start up here so we shift it to the left and then the period right here

is going to be pi but we’re shifting it to the left so

it’s going to be 2 pi over 3 and we can get this tick mark right here

and a stick mark right here corresponding to the minimum right there

the minimum is going to be the minus three force right there

01:04

so i’ll just label that minus three fourths and this height here is

minus three fourths and so you know how do we get all of

these tick marks in here that one right there this one right here and we need

this point right here so we need all those points right in there so

you know we can find those um by you know you can’t take this right here

and chop it in half anymore because it’s all been shifted over

so what you really what we can really do is you know try to um find the

halfway right here so this is one pi over three uh

one pi over three so that’s going to be minus pi over six

and then we have a another one right there so we’re going to get to the

01:05

pi over 6 right here and then we get to the next one which will be

right there that’ll be 5 pi over twelve right

and then the last one will be six pi over twelve so

you know pi over three to po pi over six um yeah okay so that one right there is

actually let me move that out of the way minus three fourths this one here is a

minus pi over 12. because we’ve shifted it um

and what we shifted was right so this is the cosine graph so

we’re shifting this one right here and this one right here and this was 2 pi

01:06

3 pi over 2 and pi over 2 and um pi over two

uh three pi over two and now let’s see here pi over two pi three pi over two

for the cosine graph and for just the cosine graph here and

so you know when we shift them that’s how we’re going to get these points in

here because we’re going to be shifting them by minus pi over three

but we also have to change the period so the period here is

i didn’t yeah just pi two pi over three so the period is pi right there

so halfway would just be pi over two so i need to

start right here and add a pi over two and you know half of way here here pi

over two would be pi over four so i need to take this right here and add power

01:07

four to it that’s how we get that and then i add another pi over four and we

get that and then i add another pi over four we get that and then i add the last

pi over four and so and then we get from there to there and then that would be

one period right there for the last one right there so that gives us the

horizontal shift the period and the amplitude what if we wanted to go and make a

vertical shift also what if we wanted to say say plus i don’t know say a plus

plus a five out here what would that do so that would take this graph right here

and it would shift it up by five units so the lowest right here now would be

minus three fourths plus five and this one right here would be

positive three fourths and then plus five so we could sketch the graph of

that one right here and so we would get three-fourths plus

01:08

five what’s three-fourths plus five so three-fourths plus five think of five is

20 over four so it’s 23 over four so that would be the new height would be 23

over 4 so this new height right here so we’d

have the same shape right here like this right here and we’d come down

something like like that right there and we’d come down

and through and we come back up and we keep repeating

but this new height right here would be uh what do we say was uh 23 pi over four

uh or just 23 over four and this minimum right here would be minus three over

four plus five so minus three over four plus twenty over four that’s just 17

over four so that would be 17 over four as the new minimum but we would have the

same tick marks so we would have the same minus pi over three right here

01:09

and all these tick marks would be the same right here this tick mark would be

the same right here minus pi over 12 and so on that tick mark right there

would still be 2 pi over 3. so giving a vertical shift up wouldn’t change the

uh tick marks along the x-axis there it would just shift it up there

so just to keep that in mind it’s that’s just a fun little caveat there of

shifting it up you’ve probably seen something like that when you did algebra

uh or early in a pre-calculus class all right well that’s it for uh this episode

uh i want to say thank you for watching and i hope you are enjoying trigonometry

uh see you in the next episode if you enjoyed this video please like and

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