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

• 0:04 - 0:08
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
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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
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over hypotenuse and the
tangent of the angle a is
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defined to be the opposite
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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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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
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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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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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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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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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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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There.
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And then as we come down to 270.
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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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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
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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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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
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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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recall how we define the tangent
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to begin with, it was the
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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
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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
• 21:27 - 21:29
• 21:31 - 21:36
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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• 22:01 - 22:07
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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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
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it means that Tan Theta is of
course sine Theta divided by Cos
• 22:45 - 22:50
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
• 23:46 - 23:48
Opie on the Y axis.
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Divided by the projection of
Opie on the X axis.
• 23:53 - 23:57
So when we begin
here at feta is 0.
• 23:59 - 24:05
Then we know that the projection
onto the Y axis looking this way
• 24:05 - 24:12
is zero and onto the X axis is
one, so we've got 0 / 1 which is
• 24:12 - 24:17
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
• 24:22 - 24:28
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
• 24:40 - 24:42
bigger and bigger approaching
• 24:42 - 24:49
one. But the projection onto the
X axis is getting smaller and
• 24:49 - 24:54
smaller and smaller, and it's
that that we are dividing by, so
• 24:54 - 24:57
we're dividing something
approaching one by something
• 24:57 - 25:02
that is getting smaller and
smaller and smaller. So our
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bigger and bigger. It's becoming
• 25:06 - 25:12
infinite and we have a way of
showing that on a graph.
• 25:14 - 25:19
I've deliberately put in a
dotted line. Now a graph
• 25:19 - 25:25
approaches that dotted line, but
does not cross it, and that's an
• 25:25 - 25:30
asymptotes. That's what we call
• 25:30 - 25:36
next bit of the graph up to 180
degrees? Well, as we tip over
• 25:36 - 25:41
projection on to the X axis
• 25:41 - 25:45
becomes negative, but he's still
very, very small.
• 25:46 - 25:51
The projection onto the Y axis
is still positive, still near 1,
• 25:51 - 25:55
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
• 25:59 - 26:04
must be very, very big, but
negative, and so there's a bit
• 26:04 - 26:10
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
happen now as it comes around
• 26:47 - 26:49
here. Let's draw one in.
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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
• 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:
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