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www.mathcentre.ac.uk/.../Polar%20Co-ordinates.mp4

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    In this video, we're going to be
    looking at polar.
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    Coordinates.
    Let's begin by actually
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    looking at another coordinate
    system. The Cartesian coordinate
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    system. Now in that system we
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    take 2. Axes and X axis
    which is horizontal.
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    And Y Axis which is vertical and
    a fixed .0 called the origin,
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    which is where these two points
    cross. These two lines cross.
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    Now we fix a point P in
    the plane by saying how far it's
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    displaced along the X axis to
    give us the X coordinate.
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    And how far it's displaced along
    the Y access to give us the Y
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    coordinate and so we have.
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    A point P which is uniquely
    described by its coordinates XY
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    and notice I said how far it's
    displaced because it is
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    displacement that we're talking
    about and not distance. That's
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    what these arrowheads that we
    put on the axes are all about
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    their about showings, in which
    direction we must move so that
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    if we're moving down this
    direction, it's a negative
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    distance and negative
    displacement that we're making.
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    Now that is more than one way of
    describing where a point is in
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    the plane. And we're going to
    be having a look at a system
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    called polar coordinates.
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    So in this system of polar
    coordinates, we take a poll. Oh,
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    and we take a fixed line.
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    Now, how can we describe a point
    in the plane using this fixed
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    .0? The pole and this baseline.
    Here. One of the ways is to
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    think of it as. What if we turn?
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    Around. Centering on oh
    for the moment we rotate around,
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    we can pass through a fixed
    angle. Let's call that theater.
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    And then along this radius we
    can go a set distance.
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    And we'll end up at a point P.
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    And so the coordinates of that
    point would be our theater, and
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    this is our system of polar.
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    Coordinates.
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    Now, just as we've got certain
    conventions with Cartesian
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    coordinates, we have certain
    conventions with polar
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    coordinates, and these are quite
    strong conventions, so let's
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    have a look at what these are.
    First of all, theater is
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    measured. In
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    radians. So
    that's how first convention
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    theater is measured in radians.
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    2nd convention
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    well. Our second convention
    is this that if this is our
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    initial line and this is our
    poll, then we measure theater
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    positive when we go round in
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    that direction. Anticlockwise
    and we measure theater negative.
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    When we go around in
    that direction which is
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    clockwise. So in just the same
    way as we had an Arrowhead on
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    our axes X&Y. In a sense, we've
    got arrowheads here,
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    distinguishing a positive
    direction for theater and a
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    negative direction for measuring
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    theater. We have 1/3
    convention to do with theater
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    and that is that we never go
    further round this way.
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    Number there's our poll. Oh, our
    fixed point. We never go further
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    around this way then there, so
    theater is always less than or
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    equal to pie and we never go
    round further that way than
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    there again. So theater is
    always strictly greater than
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    minus Π - Π ramped there plus Π
    round to there. And notice that
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    we include. This bit of the line
    if you like this extended bit of
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    the line by going route to their
    having the less than or equal to
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    and having strictly greater than
    Theta strictly greater than
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    minus pie there.
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    1/4 Convention 1/4
    convention is that
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    our is always
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    positive. One of the things that
    is quite important is that we be
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    able to move from one system of
    coordinates to another. So the
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    question is if we have.
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    A point. In our
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    XY plane.
    Who's coordinates are
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    X&Y? How can we
    change from cartesians into
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    pohlers? And how can we change
    back again, but one obvious
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    thing to do is to associate the
    pole with the origin, and then
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    to associate the initial line
    with the X axis.
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    And then if we draw the radius
    out to pee.
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    And that's our.
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    And that is the angle theater.
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    So we can see that in
    Cartesians, we're
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    describing it as XY, and
    in Pohlers, where
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    describing it as our
    theater. So what's the
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    relationship between them?
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    Let's drop that perpendicular
    down and we can see that this is
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    a height. Why? Because of the Y
    coordinate the point and this is
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    at a distance X because of the X
    coordinate of the point.
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    And looking at that, we can see
    that Y is equal to R sign
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    theater and X is equal to our
    cause theater. So given R and
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    Theta, we can calculate X&Y.
    What about moving the other way
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    will from Pythagoras? We can see
    that X squared plus Y squared is
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    equal to R-squared. So give now
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    X. I'm now why we can
    calculate all and we can also
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    see that if we take the
    opposite over the adjacent, we
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    have Y over X is equal to 10
    theater. So given AY in an X,
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    we can find out what theater
    is.
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    Now.
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    Always when doing these, it's
    best to draw sketches. If we're
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    converting from one sort of
    point in Cartesians to its
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    equivalent in Pohlers, or if
    we're moving back from Pohlers
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    to cartesians, draw a picture,
    see where that point actually is
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    now. Want to have a look at some
    examples. First of all, we're
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    going to have a look at how to
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    plot points. Then we're going to
    have a look at how to convert
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    from one system into the other
    and vice versa. So let's begin
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    with plotting. Plot.
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    And what I'm going to do is I'm
    going to plot the following
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    points and they're all in polar
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    coordinates. I'm going to put
    them all on the same
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    picture so we can get
    a feel for whereabouts things
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    are in the polar play
    or the plane for the
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    polar coordinates. So we put
    our poll, oh.
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    And we have our initial line.
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    First one that we've got to plot
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    is 2. Pie so we know
    that Pi is the angle all the
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    way around here, so there's pie
    to there and we want to go
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    out to units.
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    So it's there. This
    is the .2.
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    Pie.
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    Next to .1 N wealthy to is 0 so
    we're on the initial line.
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    An one will be about there,
    so there is the .1 note.
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    2 - Π by 3 - π by
    3 means come around this
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    way, and so minus π by
    three is about there, and
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    we're coming around there
    minus π by three, and we
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    want to come out a distance
    to, so that's roughly 2 out
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    there, so this would be the
    .2 - π by 3.
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    And finally, we've got the
    point. One 2/3 of Π. So we take
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    the 2/3. That's going all the
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    way around. To there.
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    And we draw out through there,
    and we want a distance of one
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    along there, which roughly
    called the scale we're using is
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    about there, and so that's the
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    .1. 2/3 of Π.
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    Notice that we've taken theater
    first to establish in which
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    direction were actually facing.
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    OK, let's now have a
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    look. Having got used to
    plotting points, let's now have
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    a look in polar coordinates.
    These points 2.
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    Minus Π by 2.
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    1.
    3/4 of
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    Π. And 2
    - π by three. Now these
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    are all in Pohlers.
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    What I want to do is convert
    them into cartesian coordinates.
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    So first a picture whereabouts
    are they? And I'll do them one
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    at a time. So let's take this
    one 2 - π by two initial point
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    poll. Oh, and initial line.
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    Minus Π by two? Well, that's
    coming down here.
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    To there. So that's minus
    π by two, and we've
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    come a distance to to
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    there. Well, we don't need to do
    much calculation. I don't think
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    to find this. If again we take
    our origin for our cartesians as
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    being the pole, and we align the
    X axis with our initial line.
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    And there's our X. There's RY
    and we can see straight away the
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    point in Pohlers that's 2 - π
    by two in fact, goes to the
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    point. In Cartesians, That's 0 -
    2 because it's this point here
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    on the Y axis, and it's 2
    units below the X axis, so it's
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    0 - 2.
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    Notice how plotting the point
    actually saved as having to do
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    any of the calculations. So
    let's take the next point now,
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    which was one 3/4 of Π.
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    1 3/4 of π. So
    again, let's plot where it
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    is. Take our initial.
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    .0 our poll and our initial line
    3/4 of π going round. It's
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    positive so it drought there be
    somewhere out along that.
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    Direction there's our angle of
    3/4 of Π, where somewhere out
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    here at a distance one unit.
    So again, let's take our X&Y
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    axes, our X axis.
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    To be along the initial line.
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    And now why access to be
    vertical and through the pole?
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    Oh So that the polo becomes our
    origin of, and it's this point.
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    But where after?
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    Now how we going to work
    this out that remember the
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    formula that we had was X
    equals our cause theater.
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    Let's have a look at that.
    Are is one an we've got
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    cause of 3/4 of Π and
    the cosine of 3/4 of Π
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    is minus one over Route 2,
    so that's minus one over Route
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    2. Why is our sign Theta?
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    And so this is one times the
    sign of 3/4 of Π and the sign
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    of 3/4 of Π is just one over
    Route 2, and so we have one over
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    Route 2 for RY coordinate. And
    notice that these answers agree
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    with where the point is in this
    particular quadrant. Negative X
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    and positive Y, negative X and
    positive Y so.
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    Even if I've got the calculation
    wrong in the sense that I, even
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    if I've done the arithmetic
    wrong, have no, I've got the
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    point in the right quadrant.
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    Let's have a look at the
    last one of these two.
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    Minus Π by
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    3. So again, our poll.
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    Our initial line.
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    Minus Π by three is around here.
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    So we've come around there minus
    π by three, and we're out a
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    distance, two along there.
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    Take our X axis to coincide with
    the initial line.
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    And now origin.
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    Coincide with the pole.
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    Let's write down our equations
    that tell us X is
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    our cause theater, which is
    2 times the cosine of
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    minus π by three, which
    is equal to 2.
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    Times Now we want the cause
    of minus π by three and
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    the cosine of minus π by
    three is 1/2, and so that
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    gives US1.
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    Why is equal to our
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    sign theater? Which is 2 times
    the sign of minus π by three,
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    which is 2 Times Now we want the
    sign of my minus Pi π three, and
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    that is minus Route 3 over 2.
    The two is cancelled to give us
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    minus Route 3.
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    And so again, notice we know
    that we've got it in the right
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    quadrant. 'cause when we drew
    the diagram, we have positive X
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    and negative Y, and that's how
    we've ended up here.
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    What do we do about going back
    the other way?
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    Well, let's have a look at
    some examples that will do that
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    for us. What I'm going to look
    at as these points, which are
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    cartesians. The .22 point
    minus 3 four.
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    The point minus 2 -
    2 Route 3.
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    And the .1 - 1 now these
    are all points in Cartesian's.
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    So let's begin with this one.
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    Show where it is.
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    To begin with, on the cartesian
    axes so it's at 2 for X and two
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    for Y. So it's there.
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    So again.
    We'll associate the origin in
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    Cartesians with the pole in
    polar's, and the X axis, with
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    the initial line and what we
    want to calculate is what's that
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    angle there an what's that
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    radius there? Well.
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    All squared is equal to X
    squared plus Y squared.
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    So that's 2 squared +2 squared,
    keeps us 8 and so are is
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    equal to 2 Route 2.
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    When we take the square root of
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    8. What about theater? Well,
    tan Theta is equal to
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    Y over X.
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    In this case it's two over
    2, which is one, and so
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    theater is π by 4, and
    so therefore the polar
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    coordinates of this point are
    two route 2π over 4.
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    Let's have a look at this one
    now, minus 3 four.
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    Let's
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    begin.
    By establishing whereabouts it
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    is on our cartesian
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    axes. Minus 3 means it's back
    here somewhere, so there's
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    minus three and the four on
    the Y. It's up there, so I'll
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    point is there.
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    Join it up to the origin as our
    point P and we are after. Now
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    the polar coordinates for this
    point. So again we associate the
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    pole with the origin and the
    initial line with the X axis,
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    and so there's the value of
    theater that we're after. And
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    this opie is the length are that
    were after, so R-squared is
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    equal to X squared.
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    Plus Y squared, which in this
    case is minus 3 squared, +4
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    squared. That's 9 + 16, gives us
    25, and so R is the square
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    root of 25, which is just five.
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    What about finding
    theater now well?
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    Tan Theta is.
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    Y over X.
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    Which gives us.
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    4 over minus three.
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    Now when you put that into your
    Calculator, you will get.
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    A slightly odd answers. It will
    actually give you a negative
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    answer. That might be
    difficult for you to
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    interpret. It sits actually
    telling you this angle out
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    here.
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    And we want to be all the way
    around there now the way that I
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    think these are best done is
    actually to look at a right
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    angle triangle like this and
    call that angle Alpha. Now let's
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    have a look at what an Alpha is.
    Tan Alpha is 4 over 3 and when
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    you put that into your
    Calculator it will tell you that
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    Alpha is nought .9.
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    Three radians. Remember, theater
    has to be in radians and
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    therefore. Theater here is
    equal to π minus
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    Alpha, and so that's
    π - 4.9. Three,
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    which gives us 2.2
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    one radians. And that's
    theater so you can see that
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    the calculation of our is
    always going to be relatively
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    straightforward, but the
    calculation this angle theater
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    is going to be quite tricky,
    and that's one of the reasons
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    why it's best to plot these
    points before you try and
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    workout what theater is.
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    Now the next example was the
    point minus 2.
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    Minus 2 Route 3.
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    So again.
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    Let's have a look where it is in
    the cartesian plane. These are
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    its cartesian coordinates, so
    we've minus two for X. So we
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    somewhere back here and minus 2
    route 3 four Y. So where
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    somewhere down here? So I'll
    point is here.
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    Join it up to our origin.
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    Marking our point P.
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    Again, we'll take the origin to
    be the pole and the X axis to be
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    the initial line, and we can see
    that the theater were looking
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    for is around there.
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    That's our theater, and here's
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    our. So again, let's
    calculate R-squared that's X
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    squared plus Y squared.
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    Is equal to. Well, in this
    case we've got minus two all
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    squared plus minus 2 route 3
    all squared, which gives us 4
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    + 12. 16 and so R is equal
    to the square root of 16, which
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    is just 4.
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    Now, what about this? We can
    see that theater should be
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    negative, so let's just
    calculate this angle as an
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    angle in a right angle
    triangle. So tan Alpha
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    equals, well, it's going to
    be the opposite, which is
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    this side here.
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    2 Route 3 in length over the
    adjacent, which is just two
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    which gives us Route 3. So Alpha
    just calculated as an angle is π
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    by three. So if that's pie by
    three this angle in size is 2π
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    by three, but of course we must
    measure theater negatively when
  • 25:33 - 25:38
    we come clockwise from the
    initial line, and so Theta.
  • 25:38 - 25:44
    Is minus 2π by three the 2π
    three giving us the size the
  • 25:44 - 25:50
    minus sign giving us the
    direction so we can see that the
  • 25:50 - 25:56
    point we've got described as
    minus 2 - 2 route 3 in
  • 25:56 - 26:03
    Cartesians is the .4 - 2π over 3
    or minus 2/3 of Π in Pollas.
  • 26:04 - 26:08
    Now we've taken a point in this
    quadrant. A point in this
  • 26:08 - 26:12
    quadrant appointing this
    quadrant. Let's have a look at a
  • 26:12 - 26:17
    point in the fourth quadrant
    just to finish off this set of
  • 26:17 - 26:20
    examples and the point we chose
    was 1 - 1.
  • 26:21 - 26:23
    So again.
  • 26:24 - 26:29
    Let's have a look at where it is
    in our cartesian system.
  • 26:30 - 26:35
    So we've a value of one 4X
    and the value of minus one
  • 26:35 - 26:40
    for Y. So there's our point
    P. Join it to the origin.
  • 26:41 - 26:46
    And again will associate the
    origin in the Cartesian's with
  • 26:46 - 26:50
    the pole of the polar
    coordinates and the initial line
  • 26:50 - 26:56
    will be the X axis, so we're
    looking for this angle theater.
  • 26:57 - 27:00
    And this length of OP.
  • 27:02 - 27:07
    So all squared is equal to X
    squared plus Y squared.
  • 27:08 - 27:14
    So that's one squared plus minus
    one squared, and that's one plus
  • 27:14 - 27:20
    one is 2 so far is equal
    to Route 2.
  • 27:21 - 27:25
    Let's not worry about the
    direction here. Let's just
  • 27:25 - 27:29
    calculate the magnitude of
    theater well. The magnitude of
  • 27:29 - 27:34
    theater, in fact, to do that,
    I'd rather actually call it
  • 27:34 - 27:39
    Alpha, just want to calculate
    the magnitude. So tan Alpha is.
  • 27:40 - 27:46
    Opposite, which is one over the
    adjacent, which is one which is
  • 27:46 - 27:53
    just one. So Alpha is in fact
    Π by 4. That means that my
  • 27:53 - 27:58
    angle theater for the coordinate
    coming around this way is minus
  • 27:58 - 28:05
    π by 4, and so my polar
    coordinates for this point, our
  • 28:05 - 28:08
    Route 2 and minus π by 4.
  • 28:09 - 28:15
    So. We've seen here why it's
    so important to plot your points
  • 28:15 - 28:17
    before you do any calculation.
  • 28:18 - 28:22
    Having looked at what happens
    with points, let's see if we can
  • 28:22 - 28:26
    now have a look at what happens
    to a collection of points. In
  • 28:26 - 28:27
    other words, a curve.
  • 28:28 - 28:31
    Let's take a very simple curve
  • 28:31 - 28:37
    in Cartesians. X squared plus Y
    squared equals a squared.
  • 28:38 - 28:42
    Now this is a circle, a
    circle centered on the
  • 28:42 - 28:44
    origin of Radius A.
  • 28:45 - 28:47
    So if we think about that.
  • 28:50 - 28:57
    Circle centered on the origin of
    radius a, so it will go through.
  • 28:58 - 29:01
    These points on the axis.
  • 29:09 - 29:10
    Like so.
  • 29:12 - 29:17
    If we think about what that
    tells us, it tells us that no
  • 29:17 - 29:19
    matter what the angle is.
  • 29:22 - 29:25
    For any one of our points.
  • 29:27 - 29:32
    If we were thinking in Pohlers,
    the radius is always a constant.
  • 29:32 - 29:39
    So if we were to guess at the
    polar equation, it would be our
  • 29:39 - 29:45
    equals A and it wouldn't involve
    theater at all. Well, it just
  • 29:45 - 29:51
    check that we know that X is
    equal to our cause theater, and
  • 29:51 - 29:55
    we know that Y is equal to our
  • 29:55 - 30:02
    sign Theta. So we can
    substitute these in R-squared
  • 30:02 - 30:08
    cost, Square theater plus
    R-squared. Sine squared Theta is
  • 30:08 - 30:14
    equal to a squared. We
    can take out the R-squared.
  • 30:17 - 30:24
    And that leaves us with
    this factor of Cos squared
  • 30:24 - 30:26
    plus sign squared.
  • 30:26 - 30:31
    Now cost squared plus sign
    squared is a well known identity
  • 30:31 - 30:36
    cost squared plus sign squared
    at the same angle is always one,
  • 30:36 - 30:41
    so this just reduces two
    R-squared equals a squared or R
  • 30:41 - 30:46
    equals AR is a constant, which
    is what we predicted for looking
  • 30:46 - 30:50
    at the situation there now.
    Another very straightforward
  • 30:50 - 30:55
    curve is the straight line Y
    equals MX. Let's just have a
  • 30:55 - 30:56
    look at that.
  • 30:57 - 31:01
    Y equals MX is a straight line
    that goes through the origin.
  • 31:03 - 31:08
    Think about it, is it has a
    constant gradient and of course
  • 31:08 - 31:13
    M. The gradient is defined to be
    the tangent of the angle that
  • 31:13 - 31:17
    the line makes with the positive
    direction of the X axis.
  • 31:18 - 31:22
    So if the gradient is a
    constant, the tangent of the
  • 31:22 - 31:27
    angle is a constant, and so this
    angle theater is a constant. So
  • 31:27 - 31:32
    let's just have a look at that.
    Why is we know?
  • 31:32 - 31:40
    All. Sign
    Theta equals M times
  • 31:40 - 31:44
    by our cause theater.
  • 31:44 - 31:51
    The ask cancel out and so I
    have sign theater over Cos Theta
  • 31:51 - 31:58
    equals M. And so I
    have tan Theta equals M and
  • 31:58 - 32:02
    so theater does equal a
  • 32:02 - 32:08
    constant. But and here there is
    a big bot for Y equals MX.
  • 32:08 - 32:13
    That's the picture that we get
    if we're working in Cartesians.
  • 32:14 - 32:20
    But if we're working in Pohlers,
    there's our poll. There's our
  • 32:20 - 32:26
    initial line Theta equals a
    constant. There is the angle
  • 32:26 - 32:33
    Theta. And remember, we do not
    have negative values of R and so
  • 32:33 - 32:40
    we get a half line. In other
    words, we only get this bit of
  • 32:40 - 32:45
    the line. The half line there.
    That simple example should
  • 32:45 - 32:50
    warnors that whenever we are
    moving between one sort of curve
  • 32:50 - 32:54
    in cartesians into its
    equivalent in polar's, we need
  • 32:54 - 32:58
    to be very careful about the
    results that we get.
  • 32:59 - 33:05
    So let's just have a look at a
    couple more examples. Let's take
  • 33:05 - 33:12
    X squared plus. Y squared is
    equal to 9. We know that X is
  • 33:12 - 33:16
    our cause theater. And why is
    our sign theater?
  • 33:17 - 33:23
    We can plug those in R-squared,
    Cos squared Theta plus
  • 33:23 - 33:26
    R-squared, sine squared Theta
  • 33:26 - 33:29
    equals 9. All squared is a
  • 33:29 - 33:35
    common factor. So we can take it
    out and we've got cost squared
  • 33:35 - 33:40
    Theta plus sign squared. Theta
    is equal to 9 cost squared plus
  • 33:40 - 33:45
    sign squared is an identity cost
    squared plus sign squared of the
  • 33:45 - 33:51
    same angle is always one, and so
    R-squared equals 9. R is equal
  • 33:51 - 33:54
    to three IE a circle of radius
  • 33:54 - 33:56
    3. Let's
  • 33:56 - 34:03
    take. The
    rectangular hyperbola XY is
  • 34:03 - 34:06
    equal to 4.
  • 34:06 - 34:13
    And again, we're going to use
    X equals our cause theater and
  • 34:13 - 34:19
    Y equals R sign theater. So
    we're multiplying X&Y together.
  • 34:19 - 34:25
    So when we do that, we're
    going to have our squared.
  • 34:25 - 34:33
    Sign theater
    Cos Theta equals 4.
  • 34:34 - 34:42
    Now. Sign Theta Cos
    Theta will twice sign tita cost
  • 34:42 - 34:45
    theater would be signed to
  • 34:45 - 34:51
    theater. But I've taken 2 lots
    there, so if I've taken 2 lots
  • 34:51 - 34:55
    there, it's the equivalent of
    multiplying that side by two. So
  • 34:55 - 34:57
    I've got to multiply that side
  • 34:57 - 34:59
    by two. So I end up with that.
  • 35:01 - 35:02
    For my equation.
  • 35:03 - 35:07
    I still some the other way round
    now, but one point to notice
  • 35:07 - 35:11
    before we do. Knowledge of
    trig identity's is very
  • 35:11 - 35:14
    important. We've used cost
    squared plus sign. Squared is
  • 35:14 - 35:19
    one and we've now used sign to
    Theta is equal to two
  • 35:19 - 35:22
    scientist accosts theater. So
    knowledge of those is very
  • 35:22 - 35:27
    important. So as I said, let's
    see if we can turn this around
  • 35:27 - 35:31
    now and have a look at some
    examples going the other way.
  • 35:33 - 35:40
    First one will take is 2 over,
    R is equal to 1 plus cause
  • 35:40 - 35:46
    theater. I don't like really the
    way it's written, so let's
  • 35:46 - 35:52
    multiply up by R so I get R
    Plus R cause theater.
  • 35:53 - 35:58
    Now, because I've done that,
    let's just remember that are
  • 35:58 - 36:02
    squared is equal to X squared
    plus Y squared.
  • 36:03 - 36:10
    So that means I can replace
    this are here by the square
  • 36:10 - 36:13
    root of X squared plus Y
  • 36:13 - 36:20
    squared. Our costs theater.
    Will, our Cos Theta is equal
  • 36:20 - 36:25
    to X so I can replace this
    bit by X.
  • 36:26 - 36:31
    Now it looks untidy's got a
    square root in it, so naturally
  • 36:31 - 36:37
    we would want to get rid of that
    square root. So let's take X
  • 36:37 - 36:38
    away from each side.
  • 36:38 - 36:44
    And then let's Square both
    sides. So that gives us X
  • 36:44 - 36:51
    squared plus Y squared there and
    on this side it's 2 minus X
  • 36:51 - 36:58
    all squared, which will give us
    4 - 4 X plus X squared.
  • 36:58 - 37:05
    So I've got an X squared on each
    side that will go out and so I'm
  • 37:05 - 37:09
    left with Y squared is equal to
    4 - 4 X.
  • 37:10 - 37:16
    And what you should notice there
    is that actually a parabola.
  • 37:16 - 37:22
    So this would seem to be the
    way in which we define a
  • 37:22 - 37:24
    parabola in polar coordinates.
Title:
www.mathcentre.ac.uk/.../Polar%20Co-ordinates.mp4
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