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