Graphs of Sine and Cosine Functions (Amplitude, Period, and Phase Shift)

Video Series: Trigonometry is Fun (Step-by-step Tutorials for Beginners)

(D4M) — Here is the video transcript for this video.

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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

00:32
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

00:33
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

00:34
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

00:35
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

00:36
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

00:37
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

00:38
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

00:39
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

00:40
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

00:41
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
subscribe to my channel and click the bell icon to get new video updates

About The Author
Dave White Background Blue Shirt Squiggles Smile

David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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