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www.mathcentre.ac.uk/.../5.3Trigonometric%20ratios%20of%20any%20angle.mp4

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    In this video, we're going to be
    looking at the definitions of
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    signs, cosine and tangent for
    any size of angle.
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    Let's first of all recall the
    sine, cosine, and tangent for a
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    right angle triangle. There's
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    are triangle. We identify one
    angle and then label the sides.
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    The side that's the longest
    side in the right angled
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    triangle and the one that is
    opposite the right angle is
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    called the hypotenuse and we
    write HYP hype for short. The
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    side that is opposite the
    angle is called the opposite
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    side.
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    Up for short, this side the
    side that is a part of the
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    angle that runs alongside the
    angle we call the adjacent
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    side or edge for short.
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    Sign of the angle a.
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    Is defined to be the
    opposite over hypotenuse.
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    The cosine of the angle A
    is defined to be adjacent
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    over hypotenuse and the
    tangent of the angle a is
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    defined to be the opposite
    over the adjacent.
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    But this is a right angle
    triangle and so the angle a is
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    bound to be less than 90
    degrees, but more than 0.
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    In other words, it's an acute
    angle, so these so far are
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    defined only for acute angles.
    What happens if we've got an
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    angle that's bigger than 90? Or
    indeed if we've got an angle
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    that's less than 0?
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    That raises a question. To
    begin with, why? How can we
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    have an angle that's less than
    zero? So first of all, let's
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    just have a look at angles.
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    Got a set of axes there.
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    And based upon the origin, let
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    me draw. Roughly a unit.
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    Circle. Circle of radius
    one unit so that where it
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    crosses these axes, X is
    one, Y is One X is minus
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    one, Y is minus one there.
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    Let's imagine a point P on
    this circle and it moves round
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    the circle in that direction. In
    other words, it move round
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    anticlockwise, then this is the
    angle that Opie makes with OX
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    the X axis.
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    So there's an acute angle.
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    When we get around to hear we've
    come right round there and that
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    gives us an obtuse angle and
    angle between 90 and 180.
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    Come round to here and we've
    gone right the way around there.
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    An angle that is greater than
    180 but less than 270.
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    And similarly into this
    quadrant. So that's positive.
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    Is anticlockwise.
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    So we get positive angles
    if we go round in an
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    anticlockwise way.
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    We draw it again.
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    Same unit circle.
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    And we think about our radius,
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    Opie. Or if we start to move it
    around this way.
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    Then this is clockwise, and
    so this is a negative
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    angle, so negative were
    going around clockwise.
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    So of course we can go round
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    from here. To the Y axis, the
    negative part of the Y axis, and
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    that's minus 90 degrees. If we
    go right the way round to the
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    negative part of the X axis,
    that's minus 180 degrees.
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    Effectively the same as coming
    round to 180 degrees coming
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    round anticlockwise. So that's
    how we can have any size of
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    angle. The question is can we
    put these two together? Can we
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    bring together these
    definitions and these ideas
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    about having angles which are
    greater than 90 both positive
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    and negative? Well, let's take
    sign and have a look at that.
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    So draw the same diagram again.
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    Put our point P on the
    circle which is going to
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    move around that way in an
    anti clockwise direction.
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    And here Mark this angle going
    around that way.
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    OK. Sign is opposite
    over hypotenuse. Well, if
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    I complete the right angle
    triangle.
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    This would be the side that
    is opposite that angle.
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    There is the right angle, so
    this is the hypotenuse. And
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    because this here is a unit
    circle, the length of that is
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    just one. So the question is,
    how can I describe this line?
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    If I imagine got my eye here
    and I'm looking in that
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    direction, what do I see I see
    that length as though it were
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    projected onto the Y axis.
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    So I'm looking that way and I
    can see that length which is
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    Opie, as though it were
    projected onto the Y axis. So
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    perhaps a way of describing sign
    of let me call this theater?
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    Way of describing sign theater
    would be to say.
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    That it is equal
    to the projection.
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    Of OP.
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    Until OYY Axis divided by OP
    and of course Opies then just
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    one. How does it work with
    any angle? Well, think what
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    happens as we go round.
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    As we go round as it rotates
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    around. And you're still looking
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    this way. Then you still
    got a projection.
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    Goes down to a length of zero
    and as we come back around here.
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    We've still got a projection
    on this axis that we can see,
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    so we've still got something
    that we can measure when it
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    gets around to hear. Of
    course, it's on the negative
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    part of the Y axis, and so
    it's going to be negative.
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    Well, let's have a look
    what that might mean in
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    terms of a graph.
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    What I've got here is a
    protractor and the middle bit of
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    this protractor rotates.
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    Here I've got a black line which
    is a fixed horizontal line.
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    Along that here I've got a red
    line which is going to be my
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    opi. This is the point P moving
    around in an anticlockwise
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    direction, marking out.
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    Positive angles, and if it went
    that way around it would be
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    marking out negative angles, so
    they're going to start off
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    together there, both pointing on
    zero, the angle 0. So let's
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    recall sign theater.
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    The angle. That opi.
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    As, oh that's P makes with the
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    X axis. Is defined to be
    the projection.
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    Of OP.
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    Until a why?
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    Divided by opi.
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    But if I choose to make opie
    the measure, the unit then sign
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    Theta is just the projection.
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    Of OP.
    Until OY.
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    Now let's have a look.
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    What that means in
    terms of a graph?
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    So this is the axis.
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    Measuring marking off the
    degrees, so set that to 0 so the
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    first point on the graph is
    there because as we look along
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    there. So we look along there.
    What we see is nothing. Just see
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    a point. So the length.
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    Of Opie, the projection of Opie
    onto a Y is 0.
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    Now there.
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    Halfway round.
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    45 degrees it's about.
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    That high, so let's market
    there. That's 45. This 90. Let's
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    move that around. Do we get
    to the top?
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    Mark that across.
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    Roughly about there when we
    start to go back down here.
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    We are 135.
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    The projection is along
    there and taking that
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    through its to there.
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    When we get round to 180 again
    as we look along there, we just
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    see a point, no length.
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    As we come down here.
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    245 there, or in fact 180 +
    45, which is 225. We get a
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    point which is about.
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    There.
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    And then as we come down to 270.
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    About
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    there.
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    And as we come round here to
    315 and through their or about
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    there and then when we come
    round to their where back to 0
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    again, all having been all the
    way round 360.
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    And if we join up those points.
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    Get. Quite a nice smooth curve
    out a bit with that one if we
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    think of this going back in
    this direction, what we can
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    see is that we're going to get
    the same ideas developing.
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    Here, let's just fill in the 90.
    It will be right down there, so
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    it's there. And then at minus
    180 right around there it's
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    there and then minus 270 going
    right. The way around there we
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    back up at the top again.
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    And then minus 360 having
    come all the way around where
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    down there. And so again we
    have a nice smooth.
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    Camera.
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    Notice that this shape is
    exactly the same as that shape
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    and that we could keep on
    drawing it this block.
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    This block repeats itself, it's
    periodic. It keeps repeating
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    itself every 360 degrees from
    there to there is 360. And
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    similarly from here through
    here. Till here is also 360
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    degrees. So now we have a
    sine function if you like.
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    That we can think of as being
    defined by this graph.
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    Covers any angle that we would
    want it to cover. We can keep on
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    going for 720,000 degrees that
    way, minus 1000 that way.
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    But this is always going to give
    us a well defined function.
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    What about something like the
    cosine curve? What about that
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    one? Well, let's just have a
    look at that and see how we
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    can make a similar graph for
    our cosine function.
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    I'll just stick that down again.
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    And quickly draw a set of
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    axes. I'm not draw this
    one as accurately.
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    But it should be enough for
    us to be able to see the
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    results were wanting, so
    will mark off the.
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    Divisions as we have before.
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    Zero, 9180 two 7360
    and then minus 90
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    - 180 - 270
    and minus 360 there.
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    OK thing that we haven't done is
    made a definition.
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    What do we mean
    by cause theater?
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    Well, let's have a look.
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    We want to do the same sort of
    thing as we did for sign.
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    So that's the angle.
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    In there.
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    The adjacent side will be this
    side, which is the projection.
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    Of OP.
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    Onto the X axis this
    time so oh, cause of the
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    angle is the projection.
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    All OP, until OX, the
    X axis divided by opi.
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    This is a unit circle,
    so Opie is one. So
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    cost theater is the projection.
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    Of Opie
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    until OX. Now we need to
    look at that and see how
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    that grows, and varies as we
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    rotate. Around so we start with
    zero. Remember RI is now looking
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    down. What we see is Opie
    itself. So what we see is a
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    point there one.
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    We start to move this around.
    Let's go to 45.
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    And. We're looking down, so we
    see that bit there, which is
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    less, which is smaller. So we
    see something about there.
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    As we go round to 90 when we
    look straight down on this red
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    line, all we see is a dot and so
    the projection is 0.
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    Let's go round and now to 145
    and now the projection is down
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    onto the negative part of the X
    axis, so this is negative. Now
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    down there. And that's 180.
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    Well, we now right round two
    sitting on top of Opie again of
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    length minus one.
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    Start to come around again to
    225 and again we get that
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    projection back, so we're
    starting to be here and then at
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    270. Clearly again, we're going
    to come round to there looking
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    down that way.
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    As we come round, the projection
    is again at 315. Now a positive
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    projection. So again we've gone
    through there to there and then
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    360. We're back to 0 again.
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    And so the projection
    is of length one, so
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    let's fill that in.
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    Join up the points and again we
    get a nice smooth curve. And of
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    course the same curve is going
    to exist on this side.
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    Let's just check we're going to
    swing it around this way and we
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    can see. Is that the projection
    is positive but getting less
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    until we get round till 90 when
    the length of the projection is
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    0? So we're going to see this
    occurring round here.
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    Nice smooth curve up through
    there to there.
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    Again, notice it's periodic.
    This lump of curve here is
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    repeated there and will be
    every 360 as we March
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    backwards and forwards along
    this axis. So again, we've got
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    a function that's well
    defined, got a nice curve,
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    periodic nice, smooth curve,
    so that's our definition for
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    cosine.
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    Notice it's contained between
    plus one and minus one.
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    That was something that we
    didn't observe with sign, but
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    that is also the case that it is
    contained between plus one and
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    minus one, because this
    projection of Opie onto a Y can
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    never be longer than Opie
    itself, which is just one.
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    The other thing to notice
    is that the two curves are
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    the same.
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    I just flipped back but
    displaced by 90 degrees. We
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    slide this sine curve back by
    90 degrees. You can see it
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    will be exactly the same as
    the cosine curve.
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    What about tangent? Well, let's
    recall how we define the tangent
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    to begin with, it was the
    opposite over the adjacent.
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    For sign we replaced the
    opposite side by the projection
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    of Opie on 20, why?
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    I'm for cosine. We replaced the
    adjacent side as the projection
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    of Opie on Tool X.
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    What does that mean then
    for our definition?
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    We draw our unit circle.
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    Take.
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    Opi moving around in that
    direction through an angle
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    theater and complete right
    angle triangle. This here
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    is the opposite side. It's
    opposite, the angle that
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    we're talking about.
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    Tan Theta equals so the
    opposite side. We have
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    replaced by the projection.
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    Alt OP.
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    Until OY.
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    Divided by now for a right
    angle triangle it would be
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    the adjacent side and that
    adjacent side has been
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    replaced by the projection of
    OP on till OX. Well let's
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    remember oh why is the why
    axis OX is the X axis?
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    This again gives us a definition
    that's going to work as that
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    radius vector runs around the
    circle like that.
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    In either direction. So again,
    it's going to give us a
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    definition that will work.
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    For any size of angle.
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    One of the things we can notice
    straight away about this is that
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    it means that Tan Theta is of
    course sine Theta divided by Cos
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    Theta and that gives us an
    identity which we need to learn
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    and remember Tan Theta is sign
    theater divided by Costita.
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    But what does the graph of
    tangent look like?
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    It's a little bit
    trickier to draw.
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    Let's see if we can justify what
    we're going to get.
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    Now we're looking
    at can theater.
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    Theater is the angle between the
    X axis and the Y Axis.
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    And we know that the tangent is
    defined to be the projection of
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    Opie on the Y axis.
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    Divided by the projection of
    Opie on the X axis.
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    So when we begin
    here at feta is 0.
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    Then we know that the projection
    onto the Y axis looking this way
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    is zero and onto the X axis is
    one, so we've got 0 / 1 which is
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    0. So we start there for the
    angle zero. We start there as we
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    move around. When we come to 45
    degrees then the projections are
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    equal. So let's just mark 45 and
    if the two projections are equal
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    then that must be 1.
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    What happens is we come up
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    towards 90. Well, the projection
    on to the Y axis is getting
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    bigger and bigger approaching
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    one. But the projection onto the
    X axis is getting smaller and
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    smaller and smaller, and it's
    that that we are dividing by, so
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    we're dividing something
    approaching one by something
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    that is getting smaller and
    smaller and smaller. So our
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    answer is getting bigger and
    bigger and bigger. It's becoming
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    infinite and we have a way of
    showing that on a graph.
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    I've deliberately put in a
    dotted line. Now a graph
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    approaches that dotted line, but
    does not cross it, and that's an
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    asymptotes. That's what we call
    an asymptote. What about the
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    next bit of the graph up to 180
    degrees? Well, as we tip over
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    into this quadrant, then the
    projection on to the X axis
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    becomes negative, but he's still
    very, very small.
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    The projection onto the Y axis
    is still positive, still near 1,
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    so we're dividing something
    that's positive and near 1 by
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    something that's negative but
    very very small. So the answer
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    must be very, very big, but
    negative, and so there's a bit
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    of graph. Down there on the
    other side of the asymptote, and
  • 26:10 - 26:16
    now we run this round to 180
    degrees and what happens then?
  • 26:16 - 26:21
    Well, when we've got round to
    180, the projection onto the X
  • 26:21 - 26:28
    axis is then of length one, the
    same as opi, but the projection
  • 26:28 - 26:35
    onto the Y axis is 0, so we've
    got 0 / 1 again, which gives us
  • 26:35 - 26:41
    0. At 180 degrees. So this comes
    up to there like that.
  • 26:42 - 26:47
    Think about what's going to
    happen now as it comes around
  • 26:47 - 26:49
    here. Let's draw one in.
  • 26:50 - 26:55
    And we can see that the
    projections on to both axes are
  • 26:55 - 26:58
    both negative. On a
    negative divided by a
  • 26:58 - 26:59
    negative gives a positive.
  • 27:02 - 27:07
    You can also see that when we
    get here again, we've got
  • 27:07 - 27:11
    exactly the same problems as we
    had when we got here. We're
  • 27:11 - 27:14
    dividing by something very, very
    small into something that's
  • 27:14 - 27:18
    around about one, so again I'll
    answer is going to be very, very
  • 27:18 - 27:22
    big, and so again, we're going
    to get a climb like that.
  • 27:24 - 27:28
    As we move through this, it's
    the same as if we move
  • 27:28 - 27:32
    through there, so again we
    will start back down here in
  • 27:32 - 27:35
    the graph, will climb and
    then off up again there.
  • 27:37 - 27:38
    What about back this way?
  • 27:39 - 27:41
    When we've got negative.
  • 27:44 - 27:49
    Angles what's going to happen
    then we must have the same
  • 27:49 - 27:54
    things occur. ING in terms of
    our asymptotes. And so we're
  • 27:54 - 27:59
    going to have the same things
    occur in with the graph there.
  • 28:01 - 28:06
    There and so on. So again,
    notice we get a periodic
  • 28:06 - 28:11
    function. That bit of graph is
    repeated again there.
  • 28:12 - 28:17
    Every 360 degrees we get a
    periodic function. We get a
  • 28:17 - 28:20
    repeat of this section of the
  • 28:20 - 28:24
    graph. So that's our
    function Tangent.
  • 28:26 - 28:29
    And we can think of the
    function as being defined if
  • 28:29 - 28:31
    you like by that particular
    graph.
Title:
www.mathcentre.ac.uk/.../5.3Trigonometric%20ratios%20of%20any%20angle.mp4
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