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## Inverse trigonometric functions

# Intro to arccosine

## Video transcript

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