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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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