Inverse Trig Functions: Arccos

I've already made videos on the arc 
sine and the arc tangent, so to kind

of complete the trifecta I might as well 
make a video on the arc cosine and just

like the other inverse trigonometric 
functions the arc cosine it's kind of

the same thought process if I were to 
tell you that the arc now I'm doing

cosine if I were to tell you that the 
arc cosine of X is equal to theta this

is an equivalent statement to saying 
that the inverse cosine of X is equal

to theta these are just two different 
ways of writing the exact same thing and

as soon as I see either an arc anything 
or an inverse trig function in general

my brain immediately rearranges this my 
brain regionally immediately says this

is saying that if I take the cosine of 
some angle theta that I'm going to get X

or if I get in order the same statement 
up here either of these should boil down

to this if I say that the coast you know 
what is the inverse cosine of X my brain

says what angle can I take the cosine of 
to get X so with that said let's try it

out on an example let's say that I have 
the arc I'm told no I put two CS there

I'm told to evaluate the arc cosine of 
minus 1/2 my brain as you know let's say

that this is going to be equal to its 
going to be equal to some angle and this

is equivalent to saying that the cosine 
of my mystery angle is equal to minus

1/2 and as soon as you put it in this 
way at least for my brain it becomes a

lot easier to process so let's draw our 
unit circle and see if we can make some

headway here so that's my let me see I 
could draw a little straighter actually

maybe I could actually draw put rulers 
here and if I put a ruler here maybe

I can draw a straight line let me see 
no that's too hard okay so that is my

y-axis that is my x-axis not the neatest 
most neatly drawn axes but it'll do and

let me draw my unit circle looks more 
like a unit ellipse but you get the

idea and the cosine of an angle is a 
defined on the unit circle definition

is the x-value on the unit circle so if 
we have some some angle the x-value is

going to be equal to minus 1/2 so we 
go to minus 1/2 right here and so the

angle that we have to solve for R theta 
is the angle that when we intersect the

unit circle the x value is minus 1/2 so 
let me see this is the angle that we're

trying to figure out this is theta that 
we needed to determine so how can we do

that so if this is minus 1/2 right here 
let's figure out these different angles

and the way I like to think about it is 
I like to figure out this angle right

here and if I know that angle I can 
just subtract that from 180 degrees

to get this this light blue angle that's 
kind of the solution to our problem so

let me make this triangle a little bit 
bigger so that triangle now let me do it

like this that triangle looks something 
like this where this distance right here

is 1/2 that distance right there is 1/2 
this distance right here is 1 hopefully

you recognize that this is going to be 
a 30-60-90 triangle you could actually

solve for this other side you'll get to 
square root of 3 over 2 and to solve for

that other side you just need to do the 
Pythagorean theorem actually let me just

do that let me just call this I don't 
know let me just call this a so you'd

get a squared plus 1/2 squared which is 
1/4 is equal to 1 squared which is 1 you

get a squared is equal to 3/4 or a is 
equal to the square root of 3 over 2 so

you immediately notice 30-60-90 triangle 
and you know that because the sides of

a 30-60-90 triangle if the hypotenuse is 
1 or 1/2 and square root of 3 over 2 and

you'll also know that the side opposite 
the square root of 3 over 2 side is the

60 degrees that's 60 this is 90 this is 
the right angle and this is 30 right up

there but this is the one we care about 
this angle right here we just figured

out is 60 degrees so what's this what's 
the bigger angle that we care about what

is 60 degrees supplementary to it's 
supplementary to 180 degrees so the

arc cosine or the inverse cosine let me 
write that down the arc cosine of minus

1/2 is equal to 100 and 120 degrees I'll 
write 180 there no it's 180 minus the 60

this whole thing is 180 so this is right 
here is 120 degrees right 120 plus 60

is 180 or if we wanted to write that in 
radians you just write 120 degrees times

pi Radian per 180 degrees degrees cancel 
out 12 over 18 is 2/3 so it equals 2 PI

over 3 radians so this right here is 
equal to 2 pi PI over 3 radians now

just like we saw in the arc sine and 
the arc tangent videos you probably say

hey okay if I have 2 PI over 3 radians 
that gives me a cosine of minus 1/2 and

I could write that cosine of 2 pi over 
3 is equal to minus 1/2 this gives you

the same information as the statement 
up here but I could just keep going

around the unit circle for example I 
could I will at this point over here

cosine of this angle if I were to add 
if I were to go this far would also

be minus 1/2 and then I could go 2 pi 
around and get back here so there's a

lot of values that if I take the cosine 
of those angles I'll get this minus 1/2

so we have to restrict ourselves we 
have to restrict the values that the

arc cosine function can take on so we're 
essentially restricting its range we're

restricting its range what we do is 
we restrict the range to this upper

a hemisphere the first and second 
quadrants so if we say if we make

the statement that the arc cosine 
of X is equal to theta we're going

to restrict our range theta to that top 
so theta is going to be greater than or

equal to zero and less than or equal 
to 102 PI less oh sorry not 2pi less

than or equal to PI right or this is 
also zero degrees or 180 degrees we're

restricting ourselves to this part of 
the hemisphere right there and so you

can't do this this is the only point 
where the cosine of the angle is equal

to minus 1/2 we can't take this angle 
because it's outside of our range and

what are the valid values for X well 
any angle if I take the cosine of it

it can be between minus 1 and plus 
1 so X the domain for the the domain

for the our cosine function is going 
to be X has to be less than or equal

to 1 and greater than or equal to minus 
1 and once again let's just check our

work let's see if if the value I got 
here that the arc cosine of minus 1/2

really is 2 PI over 3 as calculated by 
the ti-85 let me turn it on so I need

to figure out the inverse cosine which 
is the same thing as the arc cosine of

minus 1/2 of minus 0.5 it gives me that 
decimal that strange number let's see

if that's the same thing is 2 PI over 
3 2 times pi divided by 3 is equal to

that exact same number so the calculator 
gave me the same value I got but this is

kind of a useless what's not a useless 
number it's it's a valid that's that is

the answer but it's it doesn't it's not 
a nice clean answer I didn't know that

this is 2 PI over 3 radians and so when 
we did it using the unit circle we were

able to get that answer so hopefully 
and actually let me ask you let me

just finish this up with an interesting 
question and this applies let's do all

of them if I were to ask you you know 
say I were to take the arc arc cosine

of X and then I were to take the cosine 
of that what is this what is this going

to be equal to well this statement right 
here could be said well let's say that

the arc cosine of X is equal to theta 
that means that the cosine of theta is

equal to X right so if the arc cosine 
of X is equal to theta we can replace

this with theta and then the cosine 
of theta well the cosine of theta is

X so this whole thing is going to be 
X hopefully I didn't confuse you there

right I'm just saying look R cosine 
of X let's just call that theta now

it by definition this means that the 
cosine of theta is equal to X these

are equivalent statements these are 
completely equivalent statements right

here so if we put a theta right there 
we take the cosine of theta has to be

equal to X now let me ask you a bonus 
slightly trickier question what if I

were to ask you and this is true for any 
X that you put in here this is true for

any X any value between negative 1 and 
1 including those two endpoints this is

going to be true now what if I were to 
ask you what the arc arc cosine of the

cosine of theta is what is this going 
to be equal to my answer is it depends

it depends on the theta so if theta is 
in the if theta is in the range if theta

is between if theta is between 0 and pi 
so it's in our valid range for for kind

of our range for the product of the arc 
cosine then this will be equal to theta

if this is true for theta but what if we 
take some data out of that range let's

try it out let's sake so let me do it 
1 with theta in that range let's take

the arc cosine of the cosine of let's 
just do some one of them that we know

let's take the cosine of let's take 
the cosine of 2 pi over 3 cosine of 2

pi over 3 radians that's the same thing 
as the arc cosine of minus 1/2 cosine

of 2 pi over 3 is minus 1/2 we just saw 
that in the earlier part of this video

and then we solve this we said oh this 
is equal to 2 PI over 3 so if we're in

the range if theta is between 0 and 
pi it worked and that's because the

arc cosine function can only produce 
values between 0 and PI but what if

I were to ask you what is the arc arc 
cosine of the cosine of I don't know

of 3 PI of 3 PI so if I were to draw 
the unit circle here let me draw the

unit circle real quick one and that's my 
axes what's 3 pi 2 pi is if I go around

once and then I go around another pi so 
I end up right here so I've gone around

one and a half times the unit circle so 
this is at 3 pi what's the x-coordinate

here it's minus one so cosine of 3 pi is 
minus one all right so what's what's arc

cosine of minus one arc cosine of minus 
one well remember the the range or the

set of values that are cosine can be can 
evaluate to is in this upper hemisphere

it's between this can only be between PI 
and 0 so arc cosine of negative one is

just going to be PI so this is going 
to be PI our cosine of negative this

is this is negative one our cosine 
of negative one is PI and that's

a reasonable statement because the 
difference between 3 PI and PI is just

going around the unit circle a couple 
of times and so you get an equivalent

it's kind of your the equivalent point 
on the unit circle so I just thought I

would throw those two at you this one I 
mean this is a useful one if I actually

let me write it up here this one is a 
useful one the cosine of the arc cosine

of X is always going to be X I can so do 
that with sign the sign of the arc sine

of X is also going to be X and these are 
just useful things to you shouldn't just

memorize them because obviously you 
might memorize it the wrong way but

you just think a little bit about 
it and it you'll never forget it