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I've already made videos on the
arcsine and the arctangent, so
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to kind of complete the
trifecta, I might as well make
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a video on the arccosine.
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And just like the other inverse
trigonometric functions, the
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arccosine is kind of the
same thought process.
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If I were to tell you the arc,
no, I'm doing cosine, if our
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tell you that arccosine
of x is equal to theta.
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This is an equivalent statement
to saying that the inverse
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cosine of x is equal to theta.
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These are just two different
ways of writing the
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exact same thing.
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And as soon as I see either an
arc- anything, or an inverse
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trig function in general, my
brain immediately
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rearranges this.
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My brain immediately says, this
is saying that if I take the
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cosine of some angle theta,
that I'm going to get x.
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Or that same statement up here.
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Either of these should
boil down to this.
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If I say, you know, what is the
inverse cosine of x, my brain
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says, what angle can I take
the cosine of to get x?
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So with that said, let's
try it out on an example.
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Let's say that I have the arc,
I'm told, no, two c's there,
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I'm told to evaluate the
arccosine of minus 1/2.
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My brain, you know, let's say
that this is going to be equal
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to, it's going to be
equal to some angle.
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And this is equivalent to
saying that the cosine of
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my mystery angle is
equal to minus 1/2.
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And as soon as you put it in
this way, at least for my
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brain, it becomes a lot
easier to process.
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So let's draw our unit circle
and see if we can make
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some headway here.
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So that's my, let me see if I
can draw a little straighter.
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Maybe I could actually draw,
put rulers here, and if I put
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a ruler here, maybe I can
draw a straight line.
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Let me see.
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No, that's too hard.
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OK, so that is my y-axis,
that is my x-axis.
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Not the most neatly drawn
axes ever, but it'll do.
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Let me draw my unit circle.
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Looks more like a unit ellipse,
but you get the idea.
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And the cosine of an angle as
defined on the unit circle
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definition is the x-value
on the unit circle.
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So if we have some angle,
the x-value is going to
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be equal a minus 1/2.
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So we got a minus
1/2 right here.
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And so the angle that we have
to solve for, our theta, is the
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angle that when we intersect
the unit circle, the
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x-value is minus 1/2.
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So let me see, this is
the angle that we're
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trying to figure out.
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This is theta that we
need to determine.
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So how can we do that?
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So this is minus
1/2 right here.
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Let's figure out these
different angles.
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And the way I like to think
about it is, I like to figure
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out this angle right here.
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And if I know that angle, I can
just subtract that from 180
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degrees to get this light blue
angle that's kind of the
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solution to our problem.
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So let me make this triangle
a little bit bigger.
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So that triangle, let
me do it like this.
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That triangle looks
something like this.
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Where this distance
right here is 1/2.
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That distance right
there is 1/2.
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This distance right here is 1.
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Hopefully you recognize
that this is going to be
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a 30, 60, 90 triangle.
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You could actually solve
for this other side.
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You'll get the square
root of 3 over 2.
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And to solve for that other
side you just need to do
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the Pythagorean theorem.
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Actually, let me just do that.
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Let me just call this, I don't
know, just call this a.
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So you'd get a squared,
plus 1/2 squared, which
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is 1/4, which is equal to
1 squared, which is 1.
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You get a squared is equal to
3/4, or a is equal to the
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square root of 3 over 2.
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So you immediately know this
is a 30, 60, 90 triangle.
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And you know that because the
sides of a 30, 60, 90 triangle,
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if the hypotenuse is 1, are 1/2
and square root of 3 over 2.
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And you also know that the side
opposite the square root of
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3 over 2 side is 60 degrees.
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That's 60, this is 90.
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This is the right angle, and
this is 30 right up there.
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But this is the one
we care about.
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This angle right here we just
figured out is 60 degrees.
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So what's this?
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What's the bigger angle
that we care about?
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What is 60 degrees
supplementary to?
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It's supplementary
to 180 degrees.
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So the arccosine, or the
inverse cosine, let
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me write that down.
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The arccosine of minus 1/2
is equal to 120 degrees.
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Did I write 180 there?
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No, it's 180 minus the 60, this
whole thing is 180, so this is,
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right here is, 120
degrees, right?
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120 plus 60 is 180.
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Or, if we wanted to write that
in radians, you just right 120
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degrees times pi radian per 180
degrees, degrees cancel out.
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12 over 18 is 2/3, so it
equals 2 pi over 3 radians.
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So this right here is equal
to 2 pi over 3 radians.
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Now, just like we saw in the
arcsine and the arctangent
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videos, you probably say, hey,
OK, if I have 2 pi over 3
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radians, that gives me
a cosine of minus 1/2.
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And I can write that.
cosine of 2 pi over 3
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is equal to minus 1/2.
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This gives you the same
information as this
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statement up here.
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But I can just keep going
around the unit circle.
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For example, I could, how
about this point over here?
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Cosine of this angle, if I were
to add, if I were to go this
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far, would also be minus 1/2.
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And then I could go 2 pi
around and get back here.
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So there's a lot of values that
if I take the cosine of those
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angles, I'll get
this minus 1/2.
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So we have to
restrict ourselves.
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We have to restrict the
values that the arccosine
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function can take on.
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So we're essentially
restricting it's range.
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We're restricting it's range.
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What we do is we restrict it's
range to this upper hemisphere,
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the first and second quadrants.
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So if we say, if we make the
statement that the arccosine
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of x is equal to theta,
we're going to restrict our
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range, theta, to that top.
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So theta is going to be greater
than or equal to 0 and less
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than or equal to 102 pi.
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Less, oh sorry, not 2 pi.
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Less than or equal
to pi, right?
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Where this is also 0
degrees, or 180 degrees.
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We're restricting ourselves
to this part of the
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hemisphere right there.
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And so you can't do this, this
is the only point where the
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cosine of the angle
is equal minus 1/2.
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We can't take this angle
because it's outside
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of our range.
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And what are the
valid values for x?
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Well any angle, if I take
the cosine of it, it can be
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between minus 1 and plus 1.
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So x, the domain for the
arccosine function, is going
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to be x has to be less than
or equal to 1 and greater
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than or equal to minus 1.
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And once again, let's
just go check our work.
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Let's see if the value I got
here, that the arccosine of
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minus 1/2 really is 2 pi over
3 as calculated by the TI-85.
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We turn it on.
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So i need to figure out the
inverse cosine, which is the
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same thing as the arccosine
of minus 1/2, of minus 0.5.
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It gives me that decimal,
that strange number.
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Let's see if that's the
same thing as 2 pi over 3.
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2 times pi divided by
3 is equal to, that
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exact same number.
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So the calculator gave me
the same value I got.
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But this is kind of a
useless, well, it's
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not a useless number.
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It's a valid, that
is the answer.
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But it doesn't, it's not
a nice clean answer.
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I didn't know that this
is 2 pi over 3 radians.
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And so when we did it using
the unit circle, we were
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able to get that answer.
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So hopefully, actually let
me ask you, let me just
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finish this up with an
interesting question.
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And this applies
to all of them.
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If I were to ask you, you know,
say I were to take the
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arccosine of x, and then I were
to take the cosine of that,
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what is this going
to be equal to?
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Well, this statement right here
can be said, well, let's say
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that the arccosine of x is
equal to theta, that means that
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the cosine of theta is
equal to x, right?
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So if the arccosine of x
is equal to theta, we can
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replace this with theta.
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And then the cosine of theta,
well the cosine of theta is x.
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So this whole thing
is going to be x.
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Hopefully I didn't get
confuse you there, right?
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I'm saying look, arccosine
of x, just call that theta.
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Now, by definition, this
means that the cosine
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of theta is equal to x.
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These are equivalent
statements.
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These are completely equivalent
statements right here.
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So if we put a theta right
there, we take the cosine of
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theta, it has to be equal to x.
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Now let me ask you a bonus,
slightly trickier question.
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What if I were to ask you,
and this is true for any
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x that you put in here.
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This is true for any x, any
value between negative 1 and 1
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including those two endpoints,
this is going to be true.
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Now what if I were ask you
what the arccosine of
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the cosine of theta is?
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What is this going
to be equal to?
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My answer is, it
depends on the theta.
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So, if theta is in the, if
theta is in the range, if theta
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is between, if theta is between
0 and pi, so it's in our valid
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a range for, kind of, our range
for the product of the
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arccosine, then this
will be equal to theta.
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If this is true for theta.
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But what if we take some
theta out of that range?
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Let's try it out.
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Let's take, so let me do one
with theta in that range.
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Let's take the arccosine of
the cosine of, let's just do
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one of them that we know.
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Let's take the cosine
of, let's stick with
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cosine of 2 pi over 3.
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Cosine of 2 pi over 3 radians,
that's the same thing as
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the arccosine of minus 1/2.
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Cosine of 2 pi over
3 is minus 1/2.
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We just saw that in the
earlier part of this video.
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And then we solved this.
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We said, oh, this is
equal to 1 pi over 3.
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So for in the range of thetas
between 0 and pi it worked.
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And that's because the
arccosine function can
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only produce values
between 0 and pi.
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But what if I were to ask you,
what is the arccosine of the
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cosine of, I don't
know, of 3 pi.
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So if I were to draw the unit
circle here, let me draw the
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unit circle, a real quick one.
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And that's my axes.
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What's 3 pi?
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2 pi is if I go around once.
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And then I go around another
pi, so I end up right here.
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So I've gone around 1 1/2
times the unit circle.
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So this is 3 pi.
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What's the x-coordinate here?
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It's minus 1.
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So cosine of 3 pi
is minus 1, right?
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So what's arccosine of minus 1?
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Arccosine of minus 1.
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Well remember, the range, or
the set of values, that
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arccosine can evaluate to is
in this upper hemisphere.
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It's between, this can
only be between pi and 0.
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So arccosine of negative 1
is just going to be pi.
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So this is going to be pi.
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Arccosine of negative, this
is negative 1, arccosine
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of negative 1 is pi.
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And that's a reasonable
statement, because the
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difference between 3 pi and pi
is just going around the unit
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circle a couple of times.
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And so you get an equivalent,
it's kind of, you're at the
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equivalent point on
the unit circle.
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So I just thought I would
throw those two at you.
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This one, I mean this
is a useful one.
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Well, actually, let
me write it up here.
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This one is a useful one.
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The cosine of the arccosine of
x is always going to be x.
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I could also do that with sine.
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The sine of the arcsine of
x is also going to be x.
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And these are just useful
things to, you shouldn't just
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memorize them, because
obviously you might memorize it
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the wrong way, but you should
just think a little bit about
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it, and you'll never forget It.