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Inverse Trig Functions: Arccos

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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.
Title:
Inverse Trig Functions: Arccos
Description:

Understanding the inverse cosine or arccos function

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Video Language:
English
Duration:
13:38

English subtitles

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