Graphs of Tangent and Cotangent Functions (Including Transformations)

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

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

00:00
are you wondering what the graph of tangent looks like and cotangent
why does it look like that and why does it have isotopes in this episode we’ll
practice graphing periodic functions involving tangents and cotangents i
think you’ll find this constructive method very useful let’s do some math
[Music] hi everyone welcome back i’m dave so in this episode um we’re going to
talk about graphing functions that involve tangents and cotangents and
let’s get started um so we’re going to get started by first mentioning um how
or what we covered on the previous episodes so this episode is part of the
series trigonometry is fun step-by-step tutorials for beginners link is below in
the description and in this series here so far we’ve

00:01
covered uh trigonometric angles gradients in degrees unit circle
we talked about special angles and reference angles
and we introduced the trig functions and we start graphing we graph sine and
cosine and we graph secant and cosecant and we did transformations to them
yeah and so today we’re going to start looking at the tangent function right
here so the tangent function in order to get
a good understanding of what the tangent function looks like let’s go back to
what we know and so i’m going to i’m going to put a grid over here of kind of
the special angles and what we know so far so i’m going to put an x column
a sine x column a cosine x column and then i’m going to put a tangent x column
and a cotangent x column and we haven’t looked at those uh those rows yet
and so you remember the special angles right so we’ve got 0 degrees
or 0 radians and then we got pi over 6 radians and pi over four

00:02
and pi over three sixty degrees and then pi over two is ninety degrees
and two pi over three and three pi over four so hope you
checked out that uh episode where we practice sketching the graph of the unit
circle with all the special angles on it and you remember our goal was to get
that under five minutes so hopefully you can make a unit circle
with all four quadrants and all these special angles on it and label the points
where x is the uh cosine and y is the sign all right and so we have here um
so just to you know recognize a nice cute pattern we have square root of zero
over two square root of one over two square root of two over two
square root of three over two square root of four over two
but you know some of those simplify square root of zero over two is just
better known as zero and square root of one over two is better known as one half

00:03
and we have square root of two over two square root of three over two and square
root of four over two is better known as one and then once we
get into the second quadrant then we start to decrease so we have
square root of three over two square root two over two and one half
and then the zero down here and so remember we practice finding those
values by looking at reference angles and so now i’m going to try to beef up my
rows here so remember the cosine here is going to start at 1
and then it’s going to decrease so square root of three over two square
root of two over two one half zero and cosine is negative in quadrant two
so it’s going to repeat it’s gonna follow this right here but it’s gonna
have negative values and this looks very beautiful when you put
it on the unit circle i’m just putting in tabular form right now
um we have tangent x now so here’s two crucial identities that

00:04
we’re going to need for this vid for this episode tangent is sine over cosine
and cotangent is cosine over sine and so by knowing these values here
remember i promised earlier if you if you have these special angles memorized
for cosine sine that would be a you know a stepping stone to get a lot
more information because there’s so much symmetry in trigonometry
um in any case these are just definitions and you know if we look at here
you know if if you look at tangent it’s sine it’s the opposite over hypotenuse
adjacent over hypotenuse the hypotenuse is cancel you get tangent right there so
it’s almost by definition there but any case
and then of course you can notice that cotangent and tangent are reciprocals of
each other but anyways that’s good enough for right now so let’s build a

00:05
tangent all right so let’s look at sine over cosine so that’s zero
and one half over square root of three over two
you see the one halves cancel this gets us one over square root of three and
this is square root of two over two over
square root of two over two which is one and then we look at uh this over this
the twos cancel we get square root of three
and then right here we’re going to get one over zero
right and so one over zero is undefined and so i’ll just put a dash right there
that means undefined there’s no value there
and then i’ll put two over the twos will
cancel here we have minus square root of three
and here we’re going to have a minus 1 over square root of 3
and here we’re going to have 0 over -1 which is 0 here
now just to get an idea kind of a flavor let’s also put this over here 1 over
square root of 3 is approximately 0.58 if you were to go

00:06
approximate that using calculator and square root of 3 is approximately 1.7
right here and so knowing these approximations in fact let’s put them over here
so yeah this is just to help us sketch graphs to have an estimation of these
numbers right here um so you know by now you hopefully have a
feeling for what square root of three over two is approximately um
and square root of two over two you know
you know have those values because those are the sines and cosines and you’ve
probably been working with those especially since we’ve been making
graphs with those but anyways tangent and then we can look at the
cotangent row here also so now instead of sine over cosine now we’re looking at
cosine over sine so now we’re looking at one over zero and that’s undefined and
so here now we’re looking at this one over this one the two still cancel but
now we’re looking at square root of three over one so that’s square root of

00:07
three and here we’re still looking at a one and
you know and so you know one way to look at this is just cotangent is one over
tangent so to get cotangent you just need to take
the reciprocal tangent so you know the reciprocal of square root of three
is one over square root of three and then here we’re going to look at
zero over one which is zero and then so again i’ll just you know
take the reciprocal minus one over square root of three and
here we’re going to have a minus 1 again and now when i’m looking at this one
over this one i should get here minus square root of 3. and
here reciprocal of zero of course there’s no reciprocal zero so we get a
dash there so knowing these two rows here now uh helps us make a sketch
but what also is going to help us make a sketch are the graphs of sines and

00:08
cosines so let’s recall what those look like here so i’m going
to sketch the graph of sine right here just one quick period right here you’ve
probably done this 100 times and for the previous episodes so
let’s see if we can get that in there and so the graph of sign looks like this
something like that and this is two pi and this is pi and so on
we could label more but what i’m interested in is where sine is hitting zero
that’s going to be important when we sketch the graph of tangent and when we
sketch the graph of cotangent and let’s also sketch the graph of cosine so
cosine is going to start up here at 1 come back down go back up and here’s 2 pi
and we’re now we’re interested in where cosine hits 0.
and the reason why is because these are ratios right so if we’re going to be
looking at tangent we’re going to be needing to stay away

00:09
where cosine is zero and sine is not zero i mean those are going to have
isotopes there those and the same thing with cotangent where’s sine zero and
cosine is not zero that’ll be undefined that’ll lead to a vertical isotope also
so what are these tick marks here for cosine right so this was pi
and this was sorry two pi and this was pi this is pi over two
and this is three pi over two to get those tick marks right there so
sine is zero we’re looking at zero pi two pi three pi four pi five pi and also
minus pi minus two pi and so on uh cosine is zero at pi over two three pi
over two five pi over two um and then and then the negative ones
also right so it’s like uh you know odd multiples of pi over two
one three five seven okay and so those are going to be
important um and so now i think we have what we need to start looking at a

00:10
sketch of the graph of tangent so let’s do that right here
so when i’m going to sketch the graph of tangent i’m going to notice that
again because tangent is sine over cosine so i’m going to
deeply worried about where cosine is zero so for example at pi over two
that makes cosine zero and at pi over two where sine is one
right and so that’ll be one over zero that’s not defined right so
we had that right here one over zero it’s not defined
so what it is actually is an isotope so this pi over two we’re going to have a
vertical isotope which i usually like to put in red let me get a red here
and so i’m going to put a vertical isotope right here at x equals pi over 2

00:11
but also 3 pi over 2 and we can put one at minus pi over two if we’d like
and now what we’re going to do is so we’re looking at this row right here for
tangent right let’s look at this uh tangent uh row right here right and so what
we’re getting is when uh tangent when we input zero into tangent we get out zero
it’s going right through there and when we input pi over six um actually
uh you know at pi over two it’s the isotope halfway i usually like
to go halfway first and that’s pi over four and so and then at
and so we’re going to get a 1 right there and at pi over 6
we’re going to get the um 1 over square root of 3 which is about 0.58

00:12
and so i’m gonna kind of sketch the shape of it in here
so pi over four we hit a one and then at power three here we hit a uh
one about a 1.7 so it’s a little bit higher and at power 6 right here
we’re going to hit a 0.58 right there so it’s going to kind of go up like that
and the closer we input numbers to pi over 2 the higher up we get and the um
now when we go to these numbers over here so
you know halfway between pi over 2 and 3 pi over 2 we’re going to hit a
you know a 0 again so we hit it right here at pi
and we’re going to have the same shape right here and
so when we go with um three two pi over two pi over three

00:13
we’re getting negative one point seven so that’s about right here
and then we’re going to get um when we go to 3 pi over 4 we’re going to
get the negative 1 which is about right here and then when we get pi 5 pi over 6
we’re going to get negative about 0.58 which is about right here so you see
we’re getting these points right here and these you know repeat
and the period here as you can tell the the period here is pi
that distance right there is pi so this is pi over two and that’s minus pi over
two so the distance right there is pi that’s the period so the period is pi
the period for tangent is pi and so there’s two cycles of it
you know we get the same points over here here and here the
approximately you know square root of three one one over square root of three

00:14
and we get that shape right there and we can repeat that we can repeat it over
here and so on so usually i will sketch just one period
but tangent’s got such a small period it’s half the period of sine and cosine
maybe i’ll sketch two branches so there’s what tangent looks like and
so now let’s see if we can also see what uh cotangent looks like so
what does cotangent look like so this was tangent tangent x um
so here’s the row for cotangent right here let’s see if we can slip in here
on top here the graph for cotangent right here we keep this row right here
and let’s see if we can sketch the graph of cotangent right here

00:15
now to graph cotangent um because cotangent remember is
i’ll put it right here cotangent is cosine over sine
and so now i need to we worried about to graph cotangent i need to be worried
about where sine is 0. so 0 0 pi 2 pi and so on so let’s put some
isotopes down here so we’re going to have the isotope going
right through the origin right there and then another one here at pi and 2 pi
and let’s see if we can label these right here so put pi and 2 pi and 0
and just for fun let’s put one over here negative pi all right now
um as we can see here so remember these are our special angles

00:16
right here 0 pi over 6 pi over 4 pi over 3 pi over 2
and so what we’re getting here is it’s undefined when we input 0
and then we input pi over 6 it’s going to be up here and then we input a
a pi over 4 and we get a 1 and when we input all the way to a pi over 2
halfway in here we’re gonna get to zero so i want to start by graphing that one
first pi over two we get zero right there and so the shape that we’re
getting is this right here and so we’re getting about 1.7 and then
we’re getting about a 1 height here and we’re getting about a 0.58 for right
there and so that’s corresponding to pi over 6 pi over 4 pi over 3
and then pi over 2. and then now we go on to the next
quadrant and you see they’ve become negative right here so

00:17
this will be negative 0.5 0.58 approximately and then negative 1
for the height and then negative 1.7 approximately for the height and you see
we get that shape right there and the closer we get to pi
the steeper this is becoming down here so we input pi
into here what’s happening is we’re getting sine of zero or so we’re getting
sine of pi which is zero and we’re getting cosine of pi which is minus one
so we’re getting minus one over zero and so the closer you get to pi
the closer sign is getting to zero which means the whole thing’s blowing
up but it’s a negative because cosine of something that’s getting close to pi
is negative so that’s why it’s going down like that and it repeats
the period is pi also and so this one right here is pi over two
and this one is going to be um you know three pi over two

00:18
and so it’s going to have a zero right here at pi over 2 3 pi over 2. and so
you know you can see the pattern and this is what makes trigonometry so fun
is because where the isotopes are over here is
where the zeros are over here so there’s
an isotope right here at pi over two and over here it’s a zero
and over here just an isotope three pi over two and over here it’s a zero and
vice versa so here’s the sketch of cotangent right
here and here’s the sketch of tangent right here and let’s
look at some of the properties now so you know
what are some characteristics of tangent and cotangent and let’s just be very
specific and write them down by looking at the graphs here what can we say
so first thing is for the tangent graph over here so what’s the domain

00:19
right so the domain is the domain is all real numbers except
so i’ll say x is not equal to and then you know what is it for tangent
we cannot be equal to the isotopes so it’s not equal to pi over 2 plus
and then multiple of pi so i’ll say k times pi where k is any um
where k is any integer so k could be zero in which case we get
pi over two we got that one right we got that one right there
but k could be minus one and then we put them together and then we get this one
or k could be one and then we put them together and get this one so over here
the domain is all real numbers except x cannot be equal to a multiple of pi
so k could be zero we get this isotope k
could be one we get this isotope k could

00:20
be two k could be minus one and so on so this is just a way to write infinitely
many var uh values and you know we’re just saying k is a
integer same thing here we’re saying k is an integer so the domain of cotangent
is all real numbers except for rule out these x values right here so
all real numbers except for these and what about the range
what is the range of both of these so the range are the y values so it’s
like you try to extend or collapse project
project this graph onto the y-axis right here and what part will be covered and
so this keeps going up and this keeps falling down so the range is all real
numbers and the range here is the same all real numbers right here that keeps
going up and that keeps falling down so we’re going to get all real numbers
there the period for both is pi for both of them the period is pi

00:21
and the isotopes let’s talk about the symmetry so what is the symmetry
is this an even or odd function is is what i’m talking about so let’s say
tangent first i guess we can look over here tangent let’s put in a minus x into
tangent and what do we get out well tangent remember is sine over cosine
so i put that angle in into sine and cosine now remember sine is an odd
function so this minus will come out and cosine’s an even function
and so this is just minus tangent x so this argument right here says that if
you put a negative angle into tangent it’s just going to be negative of
tangent of that angle right there so this says that tangent is odd

00:22
right so these are the um for tangent here um tangent x i’ll put it right here
y equals tangent x i’m sure you we’re following along here hopefully all right
um and what about cotangent right so let’s let’s do this let’s do this
argument again but now this time for cotangent
so cotangent is it even or is it odd so cotangent remember is cosine over sine
and cosine is even so it absorbs the negative
the sine is odd so the negative comes out and so this is the same thing as
negative cosine over sine which is cotangent so cotangent is also odd
so both of them are odd functions and so that’s very useful when you’re
sketching the graph so for example i’ll just do a quick example
right here for us let’s get rid of tangent over here and what would the graph of

00:23
cotangent minus x look like let’s let’s just look at that right real quick
what’s the graph of cotangent of minus x right here’s graph of cotangent
and so cotangent’s odd and so the negative comes out
and so we’re just looking at reflecting the cotangent x
so whenever you input the x and you do the output which is the what
the cotangent gives us and then we make the output negative so everything that’s
positive up here now becomes negative and everything that’s already negative
you multiply it by negative and then it becomes positive so the sketch of this
graph would look something like this it’s not going to change the isotopes
we’ll still have the same isotopes and so this is x equals zero x equals pi

00:24
and so this part right here uh so it still goes through zero right because
zero equals minus zero right so it’s still going to go through zero right
here but all these values up here are now negative
and all these negative values right here now they’re positive because these
outputs these negative outputs are being multiplied by a negative and so they
become positive and so basically we’re getting this graph right here
and then we can just put some tick marks on here for example you know what’s
halfway right here and what’s halfway right here is where we’ll hit the minus
ones and the one and this is pi so this will be three pi over two
and this will be pi over four three pi over two no sorry three pi over four
there we go right so you just find that by finding the midpoint pi over two and

00:25
plus pi and then divide by two and think of that pi as 2 pi over 2 over 2
and so that’s 3 pi over 2 over 2 so yeah 3 pi over 4. in any case
that would be the sketch there or at least one cycle right
so multiplying the outputs by negative doesn’t change the period doesn’t change
the isotopes but it changes the outputs the outputs here are positive the
outputs here are negative because we’re multiplying the output of cotangent by
minus all right and so yeah so that knowing
that cotangent tangent or odd allows us to uh you know sketch the graph pretty
nicely and so now what we want to do is some exercises
um so let’s sketch this one right here um let me get this out of the way um yeah

00:26
so let’s graph minus 3 tangent of 2x here so oh wrong one here we go
so to sketch this graph here it’s not a bad idea to have the tangent
graph right in front of you especially if you’re just starting out
practicing these graphs you want to you know practice making tangent over
and over again that helps memorize it so remember the goal is to have a certain
small collection of data that’s memorized so
then you can apply transformations to that so that you can graph a whole lot
more so after you know doing enough exercises you should just be able to
visualize what this graph looks like because you already know what tangent
looks like right so remember that what tangent look like so we had some
isotopes right here at pi over two and minus pi over two and tangent was

00:27
increasing throughout that’s zero and then halfway between
zero and pi over two was pi over four and that’s where it hit the one and then
halfway right here is minus pi over four and that’s where it hit the minus one
all right and so now what we wanna do is transformations to this
so we talked about what the minus did to to this we haven’t talked
about the three yet and we haven’t talked about the two
so as you might imagine from sketching the graphs of sines and cosines and
secants and cosecants the 2x is going to change the period
right so the period here is pi so what we need to solve here or work out is
2x equals to the minus pi over 2 because that’s the new angle right here
and we’re asking where is this isotope moving to
so and then 2x is equal to the positive pi over 2.

00:28
and so if we solve for x here this will be minus pi over 4
and this will be pi over 4 and so those are our new isotopes right there
so um let’s sketch this graph right here and so our new isotopes are going to be
minus pi over 4 and pi over 4 and when we input 0 into this like right
here we outputted zero tangent of zero is zero when we input zero here two
times zero is still zero zero going into tangent is still zero
and minus three times zero is still zero it’s still going through zero
in fact we’re going to have the same shape but
actually the minus three does something to this
so it still goes through zero zero now halfway between zero and pi over

00:29
four is what pi over eight and that’s we’re going to output the one
but actually not really this part will be one
but we’ll have a one times a minus three and so then that height should be minus
three so actually the minus sign remember
reflects it so actually i need to erase this and it needs to go upside down
or backwards whichever way you want to call it it needs to be reflected
all right there we go so halfway right here which is pi over eight
and we’re going to get out a negative three right there
so halfway would normally give us output a one
right halfway between zero and pi over two is pi over four
and we output a one and when we output a one for this part right here then we
multiply by minus three so we get a minus three right there now halfway

00:30
right here which is the minus pi over eight now we input the minus pi over eight
we’re going to get out a minus one here right like right here halfway gives us
the minus one the periods change so that’s why that number right there is
you know different in fact what is the period the period here is pi
and the period over here is pi over 4 that distance right there plus another
pi over 4 so it’s just two pi over fours
in other words it’s just cut in half the period is pi over two
let’s put that down here period is pi over two
and so we we get this point right here on the graph
uh uh minus pi over eight and we get the point right here three there we go
and so that gives us the shape of the graph right there it’s
really close to the isotope it’s really close to the isotope it’s decreasing
it’s got some curvature to it and we got the y values right there at three and
minus three and we could repeat more cycles but

00:31
that’s good that’s one cycle right there all right so there’s our first graph of
tangent with some transformations here’s the tangent graph and here’s the
minus three tangent 2x graph right there so the minus says it’s
not increasing now it’s decreasing the 2 means it’s going twice as fast so
here it takes a whole pie and then a pie and then a pie and here
you know we would already have two cycles done in pi because the period
is pi over two all right let’s do another one here
let’s look at the graph of a cotangent so if you have any questions
let me know in the comments below i’ll be happy to answer them
all right so now let’s look at 3 cotangent 2x
so now to do this right here i recommend first sketching the graph of cotangent
so let’s do that over here so to graph sketch the graph of

00:32
cotangent we need to know the isotopes right here at zero and pi
and the graph of cotangent was decreasing so halfway in between we get our zero
and so we’re going to be decreasing like this right here
and when we hit halfway between here and here which is what um
three pi over four and halfway in here we get pi over four
and this is where we get the one and this is where we get the minus one
right there all right so there’s a sketch of cotangent and the period is pi
the period is pi right there all right so now let’s graph this one
right here cotangent 2x so as you might as you might imagine
from the fact that this is a 2x here it’s going to go twice as fast so

00:33
it’s going to have one cycle and it doesn’t need to go all the way to pi to
get to it it’s just going to be pi over 2. but just to you know
be complete i guess 2x equals 0 and 2x equals pi
so that’s this isotope and this isotope and the angle is 2x now so what will be
the new x’s i’ll divide by zero or sorry i’ll divide by two
and here i’ll divide by two and get pi over two
and so here we go with our new sketch and we’re going to have the isotope
right here is still at zero x equals zero and here it’s pi over two now
you know it’s not important that you use a ruler to measure the pie
you know it almost looks like the same it’s what’s the mat what’s important is
you get the shapes right and you get the labels right so now halfway in between

00:34
0 and pi over 2 is going to be pi over 4 so pi over 4 here
and this is a positive so it’s going to have the same decreasing shape to it so
it’s going to be decreasing right here through there and then right here
yeah so right here we’re going to have pi over eight
and we’re going to come up here and get a one and then we’re going to multiply
that one by three so then the height here is three and
halfway right here so that’s one pi over eight two pi over eight and then three
pi over eight and that’s for this one right here
three pi over eight and that one’s going to give us a minus one
then we’re going to multiply it by three and so then we’ll get a minus three out
right there so this one’s been scaled so this one’s going up faster

00:35
as we move to the left here this one’s going up faster because of the three
right there so when we move the same um you know here we get just a one and here
we’re going to get an a uh three also the distance is shorter
horizontally so this one’s a lot skinnier right there yeah so there’s a
reasonable sketch of this these two right here
now sometimes you might like to sketch them both on the same axes
uh to kind of distinguish the behavior right
so if we are to sketch this one over here on this on this axis over here
right so this is already power two so we would have a vertical isotope
going right through here and at pi over four we would have a zero
so let’s see if we can just sketch that graph right here say in purple or
something and i’ll put the isotope in orange

00:36
so this isotope right here is right here and halfway where it’s zero
so halfway is about right here we’re going up like that like that
and so that’s how the shape of it would look compared
this one compared to the regular cotangent graph and we would have
another one right here so this is going up faster and it’s
falling down faster here um and it’s you know twice as you know you
get two cycles already in the spear in the period of of just one
cycle for cotangent so yeah even though this is a
graph all by itself of 3 cotangent 2x it’s often nice to to sketch them
together so you can kind of help you understand what the transformations are
actually doing um okay so let’s look at another one here now

00:37
let’s look at this one right here so now we’re going to have a horizontal shift
so let’s see what that does uh so by the way to help support these
videos um it would be um if you you know if you’re enjoying it if you like it
please subscribe it really helps the channel
um and i would really appreciate that thank you
um and so yeah let’s sketch this graph right here now and so now i want to
sketch the graph of tangent i always want to have this graph right
here in front of me and so let’s put down the isotopes real quick
so we have pi over 2 and sorry we don’t go too fast here and minus pi over two
and we have the shape of tangent coming right through here just roughly
so this is just for my benefit here or hopefully your
benefit but you know you don’t really need to do this this isn’t required to

00:38
sketch the graph of this right here but i think it helps the period here is
you know pi and we got the shape and we got this
right here at pi over four that gives us our one and minus pi over four
that gives us our minus one right there all right so make sure to make it curvy
too don’t don’t just make a straight line and then bend it right make it nice
and curvy okay so now let’s grab this right here
now we’ve dealt one where we had a 2x uh
i believe so we kind of know that that’s going to chop the period in half
but what we also have is a shift horizontal shift
so you know what happens if we sketch this graph right here next tangent of 2x
so let’s just sketch that one right here real quick
and now we’re going to have these isotopes right here x equals pi over four

00:39
and x equals minus pi over four and then we still have the same shape right here
so it’s going to come up here like this the tick marks are changed now that’s
still zero but halfway between zero and pi over four is pi over eight
and that’s we get our 1 and at minus pi over 8 that’s we get our
minus 1 right there all right so there’s tangent x there’s tangent 2x
now the x you see the parentheses around the x
so that’s different let me just point this out tangent of 2x
minus pi over 4 obviously these are not the same so this is not
the same as that right you see the parentheses are in a different place
so we’re not sketching that one right there we’re sketching this one right here
so to sketch this one i need to it’s a minus pi over four so we need to shift
everything to the right by pi over four so i’m looking at all of these um

00:40
tick marks here right this one this one 0 pi over 8 and pi over 4.
let’s just write them down over here so minus pi over 4 and then minus pi over 8
and then 0 and then power 8 and then pi over 4.
and since this is pi over 4 here i’m going to add a pi over 4 to all of them
this is the shift and so we’re shifting all of the tick marks
and we’re going to calculate all of the new tick marks in particular
the ones on the outside are the isotopes so this right here is zero
right and that kind of makes sense because we’re shifting it to the right
by pi over four and this right here was minus pi over four
so this uh vertical isotope is going to get shifted now to the zero
let’s continue finding the new tick marks this is minus pi over eight plus
pi over four and so let’s put that over eight and so that’s going to be

00:41
a minus one right so let’s just think about that right here
so pi over four let’s think of that as two pi over eight
and so then that’s pi over eight and this will be pi over four
and this one right here will be what pi over eight
plus and then two pi over eight so three pi over eight three pi over eight
and then pi over four plus pi over four is two pi over four which is just pi
over two so we got pi over eight two pi over
eight three pi over eight four pi over eight is one way to look at it
in any case i think we can sketch the graph here now um
let me move up here actually let’s see if we can sketch it right here yeah
actually no um because i’m only going to graph one
cycle and it’s going to start zero so i’m going to leave myself more room over

00:42
here and let’s put in the isotopes we have one this isotope now becomes the
uh y axis x equals zero and let’s label it up here
and this isotope now becomes pi over two
and so halfway in between is the pi over four here
and we’re gonna have the same shape because because the minus pi over four
all that’s doing is shifting this tangent 2x we’re just shifting it over
so we’re not changing the shape or we’re not reflecting it we’re not doing
anything like that so we’re going to keep the same shape
and then this point right here at three pi over eight
is where we’re going to get the one and this point right here x is minus
three uh sorry minus uh sorry pi over eight i’ll label it right here pi over

00:43
eight then we’re going to get the minus 1 right there okay and so this was
x equals pi over 2. all right yep there we go and so that is the sketch of
the tangent 2x i’ll just write it right here
tangent of 2 times x minus pi over 4. all right so x minus pi over 4. okay so
all right let’s do one more let’s do a tangent let’s uh let’s do a cotangent
so let’s erase this real quick and do a cotangent yeah all right now this time
we have a two out here we have three in here and this three here

00:44
is you know we need the hor to be a horizontal shift we need to have it as x
plus or minus something and right now we got a 3x
so what we’re going to do is rewrite this as 2
cotangent and i’m going to put brackets here just to help me
and so i’m going to factor out a 3 and i’m going to say x minus
and i’m going to say pi over 12. now did i do that right did i do that
factoring right let’s check 3 times x 3x and then 3 times pi over 12 the 3’s
can’t the 3’s cancel we just opened up with pi over 4 there so this is the
sketch this is the graph i’m going to sketch
first i’m going to sketch cotangent then i’m going to sketch the cotangent of 3x
and then i’m going to sketch the one with the horizontal by pi over 12 there
so that shouldn’t take long let’s practice cotangent again
so here’s the cotangent graph right here and

00:45
just a quick real just a quick doodle we got pi right here
and we got zero right here and tangent is going to be decreasing and halfway
is the pi over two and then we have pi over four to get the to the one
and then we have 3 pi over 4 to get us to the minus 1.
all right so there’s a quick cotangent graph right there
and so now let’s do cotangent of 3x so let’s do that up here
so y equals cotangent of 3x and so now you know instead of x equals 0 and x
equals pi we’re going to have 3x equals 0 and 3x equals pi
in other words x equals 0 and x equals pi over 3
and so there’s our new period is pi over 3 and we can sketch

00:46
sketch a graph real quick so we’re still having the x equals 0 for an isotope
and now the pi over 3 here and so we’ll say x equals pi over three x equals zero
there’s our isotopes there and now how do we get the zero is halfway
so how do we get to zero right here halfway between zero and pi over three
is pi over six and we’re not changing the sign we’re
not reflecting or nothing like that so it’s going to have the same decreasing
shape and i usually like to label something
along the y axis so halfway right here is pi over 12
and that’s where i get output of one we’re not multiplying it by anything so
it’s just going to be an output of one and then
we need to find this tick mark right here we’re counting up pi over 12 1 pi

00:47
over twelve two pi over twelves this is three pi over twelves which reduces to
pi over four so this one right here is pi over four
and that’s where we get the minus one right there
so we have 1 pi over 4 12 2 pi over 12 3 pi over 12 and then 4 pi over 12 which
of course is pi over 3. so there’s the sketch of cotangent 3x right there and
so now we can sketch the final one here and let’s see if i can
make it right here now so now what we’re
going to do is we’re going to take these tick marks right here these labels
and we’re going to do the shift and remember the shift is pi over 12. so
we’re going to take 0 pi over 12 pi over 6 pi over 4 and pi over 3.
did i get them all one two three four five one two three four five and we’re
going to add in the shift this is a negative

00:48
so we’re shifting to the right which means i’m going to add a pi over 12 to
all of them so plus pi over 12 plus pi over 12 plus pi over 12 plus pi over 12.
and let’s add up all those fractions this one is pi over 12
and so we get pi over 12 plus pi over 12 which is pi over six
and so this one will be three pi over twelve
three pi over twelve is of course pi over four and pi over four
pi over four is 3 pi over 12 plus pi over 12 is 4 pi over 12 which is pi over 3
so this one is pi over 3 and this one’s pi over 3
so pi over 3 will be 4 pi over 12 plus pi over 12 so we’re getting 5 pi over 12.

00:49
so we should be able to get to all of them by adding up pi over 12s
1 pi over 12 2 pi over 12 3 pi over 12 4 pi over 12 and then 5 pi over 12.
and so those all check out fine and so now we’re going to take this cotangent
3x graph and we’re going to shift it to the right and let’s do that right here
now when we shift this to the right this isotope at zero now becomes isotope
at pi over 12. so i’ll sketch that right here x equals pi over 12 and then
the pi over 3 isotope it gets shifted and now it’s 5 pi over

  1. and so there’s our new isotopes and all we’re doing is shifting just
    like take this up and shift it so it’s going to keep the same shape right here
    so halfway is going to be pi over 4 and we’re going to have the same shape

00:50
coming right through here ah how do you make it curvy enough
there we go a little bit better and so we got these check marks here and
so now let’s get some tick marks along the y-axis
and so halfway right here is the pi over six and this one’s going to be pi over
three and so that gives us the one and actually we have a two here don’t we
so to get our final graph we need to scale it by two so this tick mark is
going to be two i could have done the 2 up here too
but it’s okay to do it down here at the same time actually
and then right here at pi over 3 we’re going to get a minus 1 out but
that minus 1 is going to be times two so
we’re going to get a minus two out there we go

00:51
so these so this one isn’t just shifted because i actually forgot about the two
so i shifted it and i stretched it at the same
time so if we had sketched a 2 here all i would do is change this to 2 and a
minus 2 and that would stretch it out vertically
and so anyways here’s the final sketch right here this is the sketch of y
equals 2 cotangent of and then i’ll just say 3 and then x minus pi over 12.
so that’s one way to write it right there 3 right 3
3 and then x minus pi over 12. so i think we can not necessarily use
the square brackets there and just write it out like that without the awkward
space all right there we go so there we go there’s the sketch right there and
if you enjoyed this video i hope that you enjoy the next one also

00:52
so the next question is what about vertical shifts
and to find out what happens with vertical shifts well the video starts
right now

About The Author
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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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