The Inverse Cosine and Inverse Tangent Functions (Step-by-Step)

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

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

00:00
you’re now ready to discover what the inverse cosine and inverse tangent
functions are we’ll go through this step by step and we’ll work on some examples
let’s do some math [Music] hi everyone welcome back we’re going to
begin with the question what is the arc cosine function and so um
i want to point out here at the beginning that
cosine function is going to be very similar to the previous episode we
talked about the inverse sine function so the first thing that we need to know
is that one-to-one functions so one-to-one functions have inverses and
i have a link below in the description for an episode
that explains what that means uh what one-to-one functions are and
what inverse functions are and that episode has nothing to do with

00:01
trigonometry so i put that in a separate episode for you to go and check out
and also we also talked about a horizontal line test in a previous
episode so definitely want to check out those episodes there now in this series
here we talked about how to graph how to sketch the graph of cosine so this uh
series is called uh trigonometry is fun step-by-step tutorials for beginners and
in previous episodes we talked about sketching cosine graph and different
transformations of it so i’m going to begin by sketching a
quick sketch of cosine so cosine starts up here and then it starts to repeat
and this is a height of one and this is two pi and then halfway here is pi
and then pi over two and then three pi over two and then it just keeps repeating

00:02
so let’s see if we can get a good nice uh something like that and um
yeah just keeps repeating on and on and on so this is pi over two this is uh
minus pi over two and this is just you know x y axis
just a quick sketch of y equals cosine x so again we’ve covered this in a
previous episode definitely if that was too quick of a sketch for you and didn’t
didn’t follow that definitely want to check out that um episode where we
sketch the graph of cosine all right so what we’re going to say here is that
cosine if you notice doesn’t pass the horizontal line test and so what i mean
by that is i can find a horizontal line right here that crosses twice or more
and so this right here the graph of cosine x right here this
fails the horizontal line test spells the horizontal line test
so what that means is that the cosine function doesn’t have an inverse

00:03
function it’s not one-to-one it doesn’t have an inverse function and so what
we’re going to have to do like we did for the sine function is we’re going to
have to make a restriction here so that we can get a part of the graph of cosine
that is going to have um a one to one it’s going to have an
inverse it’s going to pass the horizontal line test now there’s lots of
different pieces that we could choose but we’re going to go with convention
and we’re going to choose this part right here
so i’ll put this part blue let’s say if let’s see if blue stands out blue
right here and i’m going to go from 0 to pi right here so
i don’t know that blue stood out enough here let’s try orange on top of that and
so there’s the sketch of the cosine graph right here i’ll put it over

00:04
here let’s say y equals cosine of x and we’re going to be on the interval
right here we’re going to work from zero to pi
and the reason why we chose zero to pi is number one because it passes the
horizontal line test so that part right there is not not part of the graph
anymore just from zero to pi this orange part right here
and so this passes the horizontal line test this restriction of cosine right
here passes horizontal line test so it has an inverse function
the restricted cosine inverse so let’s call this the restricted cosine function
just to give it a name the restricted cosine function as compared to y equals
cosine x or just the cosine function which just keeps repeating over and over
and over again now we could have chose uh 2 pi
to 3 pi and we could have chosen that part as the restricted cosine graph but

00:05
it’s convention to choose 0 to pi which represents quadrants 1 and quadrant 2
angles and so this is going to be the restricted cosine and
past horizontal line test so now let’s sketch the uh graph of the inverse
relation so the inverse relation so if we have y equals cosine x and then
we switch the x’s and y’s cosine y here so what does this right here look like
so um oh and by the way i i forgot to label these right here 1 and -1
and so the restricted cosine i’d like to put these over here
um so before we start doing this real quick here is the domain here is going
to be domain is zero to pi okay i’ll put the domain
there let me mention the range the range is what
for the restricted cosine it’s the same thing as cosine isn’t it it’s going to

00:06
be minus one to one so the range of the restriction is still
the same thing as the range of cosine all right so yeah if we start to um
look at this going up and down along the x-axis now it’s going to go back and
forth along the y-axis so this point right here is going to be right here
right because this is when x is zero and y is one and now we’re
going to switch it and get and get uh x is one and y zero and so it’s just going
to come through like this and then like that it’s just gonna keep repeating
like that and these y values right here is going to be 1 and then minus 1.
those were the y values over here now over here they’re the x values
and we have this right here so we’re going to go through one full period
and we’re going to get back to right here and so this will be 2 pi and halfway

00:07
will be pi and then halfway right here will be 3 pi
over 2. and then halfway between these two right here will be pi over two and so
this is the inverse uh cosine relation inverse cosine
relation it’s just a relation because it’s not a function all right so um
there we go and so this would be pi and this would be pi over two
this would be minus pi over 2 and so on it just keeps going back and forth and
as you can see the original cosine function didn’t pass the horizontal line
test when we switched the x’s and y’s now we see that this does not pass the
vertical line test this is not a function so what we want to do is look
at the restriction so the the restricted cosine function
when you switch the x’s and y’s we end up with this part right here this part

00:08
right here from zero to pi so we end up with this part right here 0 to pi
and this is the cosine inverse function because if we look at just this part
right here then it passes the vertical line test and so this will be the inverse
cosine function and we’re going to say that x equals y cosine y
if and only if which i’ll abbreviate iff if and only if y equals cosine inverse
in other words we’re defining this right
here this new notation right here cosine inverse of x
by meaning that we took the regular cosine we switch the x’s and y’s
and we have the restriction so the restriction is you know domain [Music]
is what’s the domain here from minus one to one

00:09
which was the range of the restricted cosine and the range here is
so the range here is zero two pi which was the domain over here
all right there we go so that’s the cosine inverse function right here in orange
and it came from the cosine restriction function the restricted cosine function
there we go so now let’s work out some examples so uh let me go here
and say let you know let’s look at some examples but i got to get rid of this
right here so let’s see if you have any questions so far about any of this
let me know in the comments below and if you’re enjoying this video please
like and subscribe alright so now let’s look at some
examples so here we go example we’re going to plug in plug in some
numbers and and try to understand these um functions a little bit more cosine

00:10
inverse of 0 right so let’s plug in 0 into cosine inverse now what this means is
that 0 is equal to cosine y and this you know
so we have to have x’s in minus one to one and the y is in zero to pi
because that’s the domain and range for the inverse cosine right so the
so the domain of cosine inverse is minus one to one and the range is zero to pi
um right so just to go up like that this is one to minus -1 that’s the domain
and this is pi and this is pi over 2 here all right so just to refresh your
memory that what we just talked about here a second ago all right so

00:11
to solve a here y equals cosine inverse in other words to plug in zero into
cosine inverse that’s the same thing as you know if we switch the x’s and y’s
so now we’re looking at zero is equal to cosine y
and now we can rely upon our knowledge of cosine what is cosine going to be
uh cosine of what angle gets me to zero and so that says that y is equal to
um the you know pi over two but the y has to be between zero and pi which it is
and you know because there’s infinitely many values where cosine y is equal to
zero right so it’s going to be pi over two another one here right so if you
look at the cosine graph right here up and down right so pi over two we get
cosine is zero but we also get three pi over two
and then there’s another one and there’s another one and so

00:12
you know but we need the restriction right that’s the whole point of making
cosine uh have an inverse function is to restrict it from zero to pi and so the
answer is going to be just pi over 2. so
the point is is that whenever we input 0 we want to get a unique output and that
unique output here is going to be pi over 2. so this is equal to this is the y
so cosine inverse of 0 is just pi over 2. so let’s look at another one here
so now let’s use the arc cosine notation so arc cosine of
minus square root of three over two and so now we’re looking at cosine
of y equals minus three over square root of two
um and so from right here when we’re looking at the unit circle here we’re
going to say where is cosine 0 sorry where is cosine negative right and so

00:13
we’re going to say that arc cosine is five pi over six
and the reason why is we look at five pi over six here
and then the reference angle right here is pi over six
but uh this is in quadrant two here and we need here this angle pi over six
because it needs to be between zero and pi and
when we’re looking at minus square three over two um that’s going to be
uh where cosine is negative right here in this quadrant right here
so we’re gonna use the uh reference angle right there
but we don’t need it actually so we’re just going to get that this is

00:14
5 pi over 6 you know because if we looked in quadrant 1 or quadrant 2 [Music]
and we’re going to get um okay so probably the best way to think
about this is along the unit circle here and so when we’re going to get this
cosine right here the you know negative square root of three over two
and one half right so positive square three over two and right here
this will be the negative square root of 3 over 2 right there
right so because of the uh y right here has to be between 0 and
pi has to be in quadrant 1 or quadrant 2 so this is going to be the angle right
here 5 pi over 6 and that’s how we get this angle right here okay so for part c
we’re going to look at y equals cosine inverse of

00:15
pi so to plug in pi into cosine inverse we’re going to switch it to cosine
statement so cosine of y equals sorry um cosine of pi equals y
um yeah cosine y equals pi and we’re asking here um what angle y
will get us to pi now remember that cosine is between 1 and -1
so there is no y there is no input there is no input where we can get the
output right there to be equal to 3.14 right so no matter what you try to input
into cosine um there is no output to get to the pi uh so differently

00:16
to input something into cosine inverse this input this x here has to between be
between minus one and one and so this right here does not exist
or another way to write that is pi is not in the domain pi is not in the domain
of cosine inverse i’ll just say arc cosine so you know part c is a
question or an example where it’s not in the domain you can’t you can’t use that
value into the function there function is not defined there
all right so now let’s look at doing some composition now so let’s look at that
so in order to do the composition here we’re going to need to
remember what composition of functions are
uh there’s link below in the description if you’re not familiar with that

00:17
but what we’re going to say is that we have two functions here so f and g
so the function f here is just the cosine and this is the cosine inverse
and this is the restricted cosine and this is going to be the cosine
inverse that word that we’re defining here
so in this case we’re going to say that f composed of g of x
where when you put inputted x into inverse functions you should output the
x again so this will be cosine of cosine inverse of x and this will be
equal to x so for x is in so x here is in the restriction of the
cosine inverse right here so x is in minus one to one
and g is composed of f of x is going to be equal to
cosine inverse of cosine of x and this is remember this is the
restricted function here and so when i input this x here into

00:18
this restricted cosine you know what was the restriction so x is in and this is
the zero zero to pi we restricted cosine to be just from zero to pi and so
these are the um composition properties for arc cosine or for sine
sorry arc cosine and sine so they’re inverses of each other that’s
what this is saying right here these are inverses functions of each
other right here and they’re good on these uh you know domain right here
for these values of x whatever you input you get out again in other words these
two are undoing each other when you put an x cosine inverse does something to
that and then cosine will undo that all right so let’s look at some
examples using this right here so for a here i’ll just say here what if we have

00:19
cosine of cosine inverse of 0.73 0.73 so we’re going to pull like 0.73
into cosine inverse now one way to do that is to go use your calculator and
try it out but actually don’t need to this is cosine of cosine inverse will
they undo each other well it depends is this x right here is this x in the
domain is it between -1 and 1 and the answer is yes this is part of the
function cosine inverse so the answer is just 0.73 that’s all there is to it
um so you don’t need to go use the calculator to start cranking that away
on that you just don’t need to so what about if we try this other one
here let’s say all right we have arc cosine arc cosine of
let’s say here we have cosine of pi over 12 cosine of pi over 12.
now again we don’t have pi over 12 on a unit circle it’s not like a special

00:20
angle that we just know off the top of our head um
and so you know we can go use your calculator and then use the arc cosine
function to to undo that but actually if this angle right here cosine pi over 12
if it’s between zero and pi then these two functions are going to be
inverses of each other and so whatever you input is what you get out
all right and so now let’s try c cosine inverse of cosine of four pi over three
all right and so now let’s check do these undo each other are
they inverse functions um for this x right here so this x right here
um is it between 0 and pi and the answer is no if we go look at 4 pi over 3
let’s say let’s look at it over here where is where is 4 pi over 3
well it’s pi over 3 past 4 3 pi over 3 right so here’s pi and then 4 pi over 3
i’ll label it right here 4 pi over 3 and

00:21
then so what’s the reference angle right here
the reference angle is pi over three and this is in quadrant three so
let’s remember that i guess i can put over here cosine of four pi over three
equals two cosine of the reference angle actually let me put it down here
cosine of four pi over three is equal to cosine pi over 3 because that’s the
reference angle but what goes in front of it positive or
negative this is in quadrant three cosine is negative in quadrant three so
i’m going to say negative right there so um yeah so
i don’t really need to actually find this value right here which we could
it’s a special angle we we should have that memorized right but the point is is

00:22
that this um cosine 4 pi over 3 here we can substitute it in here and
so we can get this right here cosine 4 pi over 3 is right here and so the
cosine um pi over three here we can um yeah this isn’t um this
is a quadrant two angle here quadrant uh three angle here um so let’s use
actually let’s use cosine four pi over three

00:23
is equal to cosine two pi over three this is a quadrant two angle this is
between zero and pi and this doesn’t have the minus sign in it
so this is better to actually use because they’re straight up equal to
each other so actually i prefer to use this right here
so this will be cosine inverse and now we don’t need the minus sign to get in
the way it’s just cosine of two pi over three and so two pi over three
is in between zero and pi so now they undo each other and we just gonna get
two pi over three yeah so that’s a very nice way there one two three there we go
all right so now let’s look at the um arc tangent function here
and so let’s let’s uh figure that out now so with arc tangent here
let’s recall the graph of tangent so let’s say we have a tangent of x and

00:24
remember tangent is sine over cosine so we’re looking where the cosine graph
is zero so where’s cosine zero so cosine goes like that right and then
it gets to two pi and halfway is pi and halfway is pi over two
and halfway three pi over two and then also we hit here negative pi over two
and so there’s some there’s where cosine hits zero
right so when i start to graph tangent i’m going to graph me some vertical
isotopes where the where the cosine is 0 so pi over 2 right here
and minus pi over 2 right here and now we have the shape of tangent
we’re going to come in through here with the shape of tangent
now i just want to remember i remind you that if you’ve never graphed tangent

00:25
before go back and check out that episode where we sketch the graph of tangent
so now i’ll just erase this and erase this and just say yeah here’s the graph of
tangent now of course tangent repeats there’s another isotope right here at x
equals 3 pi over 2 and you can see that this is not going to pass the horizontal
line test it does not have an inverse function
and it does the same here it repeats here so we can easily find a horizontal
line where it fails i’ll just draw a one right through here
and it fails it passes through tangent infinitely many times because this just
keeps repeating over and over again the period of tangent is pi so in order to
have an inverse tangent function we’re going to need to make the restriction
now keep in mind these are isotopes so they don’t actually ever reach the pi
over 2. it’s just getting closer and closer to it and the same thing here as

00:26
we’re going this way right here we’re just getting closer and closer minus pi
over two so i’m going to put the restriction
um to be just one period here and i’m gonna put the restriction to be and i’m
going to leave this as open so not include that and then pi over 2 to pi over 2.
so let’s put the tangent the restricted tangent graph
let’s put that in blue let’s say and so that’s the
this is the function right here that does pass so this does pass the
horizontal line test so this one right here all by itself
just this branch right here it has a uh inverse function right here
and so we’re going to define that right now so we’re going to say we’re going to
switch the x’s and y’s to get the inverse so we’re going to say that um you know

00:27
y equals tangent inverse and then we’re going to say if and only
if in other words we’re making a definition right here
you know we’re going to say if and only if x equals tangent of y and
and then we have the restriction right here so the domain is minus pi over 2
to the power 2 and the range is um so sorry this right here has domain and
this here has range where can i put the range right here range is
so here the range for the restricted tangent is just all real numbers here
and so here so the domain is the range over here which is all real numbers
all real numbers and the domain the range here is the

00:28
minus pi over 2 to power 2. and now it would be a good idea to actually see
a sketch of that so let me see if i can come down here over here and sketch the
graph of of the arc tan or tangent inverse so we’re going to have instead of
instead of these vertical isotopes now we’re going to have horizontal isotopes
so this is the line vertical line x equals pi over 2
so now i’m going to make the horizontal line y equals pi over 2
and now we’re going to have a horizontal line y equals minus pi over 2
and we’re going to come in here with the shape of tangent inverse right here
there we go so this is just the x and the y axis and
now if we were to sketch the inverse relation of this

00:29
we would have this piece this piece this piece but inverse relation would have
another piece another piece so they would be stacked on top of each other
but because we’re only choosing this branch right here when we do the inverse
function we’re only getting this branch right here so we’re not going to have
another branch and another branch and another branch because this is just the
inverse function right here which we’re defining right here
and so this passes the vertical line test and this you know just the restricted
and this passes the horizontal line test this has an inverse function and it’s
just tangent right here so they’re inverses of each other
so um let’s do some examples right here so let’s you know see if we can do this
right here example a let’s call it so let’s say we have tangent inverse
of minus square root of 3 right here and so you know this is when x is in r
and y is in between here and here and i’m going to try to input this

00:30
number right here into tangent inverse so can this tangent inverse go so x can
be any real number so this is this is a real number certainly and so yeah that’s
perfectly fine to input this right in here and now the question is how do we get
uh the answer out right here so what i’m going to do is i’m going to look at the
um the the if and only if part right here so i’m going to convert this tangent
inverse into a tangent problem and use my skill and knowledge of tangent to
evaluate this right here so you know y equals tangent inverse if and only if so
the x this part right here which is minus square root of 3
is equal to tangent inverse of the number we’re looking for right here um
and so let’s just call that a y and so we have to ask ourselves here
what do we take the tangent of to get minus square root of three
and so you know using our knowledge of tangent right here we know that this y

00:31
that we can substitute in right here to get tangent right there is going to be
minus pi over 3. now if you’re not sure about that remember tangent of y is
sine of y over cosine of y and tangent of y for that to be minus
square root of three we’ll go with minus square root of three over two
and one half and so we’re asking the what is the y where i get this and what
is the y when i get this and the answer of course is minus pi over three and so
this is how we get tangent inverse of negative square root of 3
is just minus pi over 3. okay so let’s do another one now let me
get some more space up here so let’s do um how about a arctan of

00:32
arctan of tangent [Music] tangent negative 0.89 okay
so to do this one right here we’re going to rely upon our inverse functions
right which i’m going to state over here now
so just like we did for sine and cosine this right here is our restricted
tangent function so i’ll just put that in parentheses here restricted
this is not just the a regular tangent function it’s the restricted tangent
we’re only taking the branch um and then this right here will be tangent inverse
and so we’ll have the composition evaluated at some input whatever that input is
so this will be tangent of tangent inverse of x and that will give us x

00:33
and that’ll be whenever whenever this x is what
and we can input it into tangent inverse so
tangent tangent inverse we’re going to need whenever x is a real number
and the second one is the reverse composition so now we have tangent inverse
of tangent of x which is the one we’re going to use right here
tangent inverse of tangent is going to be just equal to x whenever x is in
right and so now we’re going to need the restriction of the domain
of tangent that we’re restricting it to the restriction up here
so whenever x is in the restriction here
i can put it into tangent right here and
these two undo each other and we’ll just get out the original x here so is x

00:34
between minus pi over 2 and pi over 2 here and so the answer is uh yes
so the the answer is just negative 0.89 and so let’s see here let’s go to um
yeah so we just got to check you know try to
have some decimal approximation of that that’s a little bit bigger than one
that’s a little bit smaller than minus one so minus is is in there um okay so
there’s our introduction to inverse cosine and inverse tangent
and uh i look forward to seeing you in the next episode which starts right here
click right here and you get the next episode i’ll see you there bye

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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