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# Intro to arccosine

Sal introduces arccosine, which is the inverse function of cosine, and discusses its principal range. Created by Sal Khan.

Video transcript

I've already made videos on the
arcsine and the arctangent, so to kind of complete the
trifecta, I might as well make a video on the arccosine. And just like the other inverse
trigonometric functions, the arccosine is kind of the
same thought process. If I were to tell you the arc,
no, I'm doing cosine, if our tell you that arccosine
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 immediately says, this
is saying that if I take the cosine of some angle theta,
that I'm going to get x. Or that same statement up here. Either of these should
boil down to this. If I say, 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, two c's there, I'm told to evaluate the
arccosine of minus 1/2. My brain, you know, let's say
that this is going to be equal to, it's 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 if I
can draw a little straighter. 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. OK, so that is my y-axis,
that is my x-axis. Not the most neatly drawn
axes ever, but it'll do. Let me draw my unit circle. Looks more like a unit ellipse,
but you get the idea. And the cosine of an angle as
defined on the unit circle definition is the x-value
on the unit circle. So if we have some angle,
the x-value is going to be equal a minus 1/2. So we got a minus
1/2 right here. And so the angle that we have
to solve for, our 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
need to determine. So how can we do that? So 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 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, 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 the 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, just call this a. So you'd get a squared,
plus 1/2 squared, which is 1/4, which 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 know this
is a 30, 60, 90 triangle. And you know that because the
sides of a 30, 60, 90 triangle, if the hypotenuse is 1, are 1/2
and square root of 3 over 2. And you also know that the side
opposite the square root of 3 over 2 side is 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 arccosine, or the
inverse cosine, let me write that down. The arccosine of minus 1/2
is equal to 120 degrees. Did I 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 right 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 over 3 radians. Now, just like we saw in the
arcsine and the arctangent videos, you probably say, hey,
OK, if I have 2 pi over 3 radians, that gives me
a cosine of minus 1/2. And I can write that.
cosine of 2 pi over 3 is equal to minus 1/2. This gives you the same
information as this statement up here. But I can just keep going
around the unit circle. For example, I could, how
about 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 arccosine function can take on. So we're essentially
restricting it's range. We're restricting it's range. What we do is we restrict it's
range to this upper hemisphere, the first and second quadrants. So if we say, if we make the
statement that the arccosine 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 0 and less than or equal to 102 pi. Less, oh sorry, not 2 pi. Less than or equal
to pi, right? Where this is also 0
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 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
arccosine 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 go check our work. Let's see if the value I got
here, that the arccosine of minus 1/2 really is 2 pi over
3 as calculated by the TI-85. We turn it on. So i need to figure out the
inverse cosine, which is the same thing as the arccosine
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 as 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, well, it's not a useless number. It's a valid, that
is the answer. But 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, actually let
me ask you, let me just finish this up with an
interesting question. And this applies
to all of them. If I were to ask you, you know,
say I were to take the arccosine of x, and then I were
to take the cosine of that, what is this going
to be equal to? Well, this statement right here
can be said, well, let's say that the arccosine of x is
equal to theta, that means that the cosine of theta is
equal to x, right? So if the arccosine 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 get
confuse you there, right? I'm saying look, arccosine
of x, just call that theta. Now, 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, it 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 ask you
what the arccosine of the cosine of theta is? What is this going
to be equal to? My answer is, 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 a range for, kind of, our range
for the product of the arccosine, then this
will be equal to theta. If this is true for theta. But what if we take some
theta out of that range? Let's try it out. Let's take, so let me do one
with theta in that range. Let's take the arccosine of
the cosine of, let's just do one of them that we know. Let's take the cosine
of, let's stick with cosine of 2 pi over 3. Cosine of 2 pi over 3 radians,
that's the same thing as the arccosine 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 solved this. We said, oh, this is
equal to 1 pi over 3. So for in the range of thetas
between 0 and pi it worked. And that's because the
arccosine function can only produce values
between 0 and pi. But what if I were to ask you,
what is the arccosine of the cosine of, I don't
know, of 3 pi. So if I were to draw the unit
circle here, let me draw the unit circle, a 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 1 1/2
times the unit circle. So this is 3 pi. What's the x-coordinate here? It's minus 1. So cosine of 3 pi
is minus 1, right? So what's arccosine of minus 1? Arccosine of minus 1. Well remember, the range, or
the set of values, that arccosine can evaluate to is
in this upper hemisphere. It's between, this can
only be between pi and 0. So arccosine of negative 1
is just going to be pi. So this is going to be pi. Arccosine of negative, this
is negative 1, arccosine of negative 1 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, you're at 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. Well, actually, let
me write it up here. This one is a useful one. The cosine of the arccosine of
x is always going to be x. I could also do that with sine. The sine of the arcsine 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 should
just think a little bit about it, and you'll never forget It.