## www.mathcentre.ac.uk/.../5.9Trigonometric%20Functions.mp4

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This video is going to look at
three knew trig functions. Cosec
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Zack and caught.
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However, they're not entirely
knew, because they are derived
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from the three that we know
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tan. So let's have a look.
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The first one that we want to
have a look at is cosec that is
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defined to be one over sine. So
one over sine Theta is equal to.
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Now to give it its full name it
is the cosecant.
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Home theater But we
shorten that till cosec theater.
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Second one. Follows the
same line one over 'cause
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Theater and it's full name is
the secant of Theta. But again,
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we shorten that to set theater
and the final one, one over
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10 theater. Equals and it's
full name is the cotangent of
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Theta, and again we shorten that
to caught theater.
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Now, why do we need these? Well,
first of all, they will help us
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to solve trig equations.
Secondly, there involved in
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identity's and 3rd they come up
when we do calculus,
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particularly when we do
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integration. Let's just have a
look at one example of where
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they might occur in terms of
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basic identity's. So the basic
trig identity that we've got,
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the sine squared Theta Plus Cost
Square theater, equals 1.
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And if I choose to divide
everything on both sides of this
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identity by Cos squared, then
I'll have sine squared Theta
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over cost Square theater plus
cost Square theater over Cos
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squared Theta equals one over
cost squared Theta.
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And so this one is sign over
cause all squared. So that gives
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us stand square theater plus
cost squared into Cos squared is
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one equals and then one over
cost squared is one over cause
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all squared. So that is set
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squared Theta. So there's one
of our new trick functions
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popping up in an identity this
time, and there is a similar
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one that we can get if we
divide throughout by sine
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squared, and if we do that, we
end up with one plus cot
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squared Theta is equal to
cosec squared Theta, so there
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the other two trig functions
that we've just introduced
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again pop up.
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Let's have a look at what might
happen when we reached the later
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stages of solving a trig
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equation. So let's take
cot squared Theta equals
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3 four theater between
360 and 0 degrees.
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Well, we begin to solve this by
taking the square root. So
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caught theater equals Route 3 or
minus Route 3. Remember, we take
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a square root, it has to be plus
or minus. Now we might think,
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well, let's just look this up in
some tables or let's take our
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Calculator, but do we really
need to? We know what caught
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theater is. It's one over Tan
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Theta. And that's Route 3 or
minus Route 3.
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Now we can turn this one upside
down to give us Tan Theta equals
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and we can think of each of
these as being root 3 over one
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or minus Route 3 over one, and
so we can turn these upside down
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to get one over Route 3 or minus
one over Route 3.
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And now it's in terms of Tan
Theater and this is now one of
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those special values of our trig
functions. In fact, one over
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Route 3 is the tangent of 30
degrees, so we know that this
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has one solution that is 30
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other solutions? Well, let's
have a look at those.
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Sketch of the graph.
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Tan Theta 0 up to
90 from 90 up through
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180 up towards 270.
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Stopping there at
360, so that's not.
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9180, two, 70
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and 360. And
the tangent of 30 is one over
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Route 3. So somewhere here is
one over Route 3 coming down to
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30. So of course the next one is
across there and the symmetry
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tells us if this is 30 on from
zero. This is 30 on from 180, so
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the next one is 210 degrees
minus one over Route 3. Well,
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that's going to be somewhere
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along here. And again, the
symmetry tells us if this is 30
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on this way, then this one is 30
back this way. So that gives us
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150 degrees and we've got
another value here which is
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going to be 30 back from there,
which is going to be 330
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degrees. So solving equations
that involve things like caught,
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encek and Cosec is no different
to solving equations to do with
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sign causing tan because we just
turn them into sign calls and
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tab to conclude this, we're just
going to have a look at the
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graphs of these three knew trig
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functions. And in order to do
that, we will begin each one by
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looking at the graph of the
related trig function. So to
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look at Cosec, we're going to
look at sign first. So what does
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the graph of sign?
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Look like. Will take one
complete cycle between North and
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360. So 0.
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180, three, 160 and the
peak and trough are in
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between 1970 and that goes
from one down 2 -
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1 and what we're going
to graph now is cosec
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theater, which of course is
one over sine Theta. So
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let's set up similar axes.
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So mark them off, there's 90.
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180 270
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360 Now
here at 90
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the value of
sign is warm.
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So at 90 the value of cosec must
also be one, so I'm going to
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market their one here at 270.
The value of sign is minus one.
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And so at 270, the value of
cosec must be one over minus
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one, which again is just minus
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one. So there are two points.
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Here at zero the sign of
0 is 0.
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So the value of Cosec would be
one over 0.
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But we're not allowed to divide
by zero, but we can divide by
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something a little bit away.
What we can see is that would be
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a very very tiny positive number
that we were dividing by. So if
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we divide 1 by a very tiny
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to be very big, but still
positive. So with a bit of curve
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there, let's have a look at 180.
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Well, at 180 sign of Theta is
again 0 so cosec is one over
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0 at this 180 degrees. Let's go
a little bit this side here of
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180 and the value of sign is
really very small. It's very
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close to 0.
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So again, 1 divided by something
very small and positive.
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Is again something very large
and positive, so let me put in
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an asymptotes. And we've got a
piece of curve there.
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Now this curve goes like that.
What we're seeing is that this
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curve is going to come down and
up like that, and it's going to
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do the same here, except because
what we're dividing by are
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negative numbers, it's going to
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be like. That
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So there's our graph of cosec
derived from the graph of sign.
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Let's take now calls feta.
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Do the same.
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Will take the graph of
costita between North and 360.
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At the extreme, values will
be minus one plus one
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9180 two 7360. Just make
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that clearer. And so let's have
a look here.
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Mark off the same
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points. And we're
graphing SEK this
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time sex theater,
which is one
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over 'cause theater.
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So again, let's Mark some
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points. Here when theater is 0.
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Costita is one Soucek
Theater is one over one
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which is one. So will mark
the one there here at 180.
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Cost theater is minus one.
Soucek Theater is 1 divided
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by minus one and so will
mark minus one here.
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Here at 90 we got exactly the
same problems we have before the
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value of Cos theater at 90 zero.
So 1 / 0 is a very big number.
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Well, in fact we're not allowed
to do it, so we have to go a
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little bit away from 90 to get a
value of Cos Theta which is very
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small, close to 0 but positive.
And if we divide 1 by that small
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we get is very big and.
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Positive so we have a bit of
curve going up like that. What
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side of 90 where dividing by
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something which the value of Cos
Theta is very small but
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definitely negative. So the
answer is going to be very big
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in size when we divide it into
one but negative. So a bit of
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the curve here coming down to
their same problem again at 270
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so we can see the curve is going
to go round.
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And back like that. And then
here again at 360, we're going
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to be able to mark that point.
We're going to have that one
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coming down at that.
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So there we've managed to get
the graph of SEK.
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Out of the graph, of course.
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Let's now have a look at the
graph of Tan Theater.
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These
off
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9180,
two,
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70
and
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360.
And now we'll have a look at
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caught theater, which is one
over Tan Theater.
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So we'll take the same graph and
I'll do the same as I've done
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before. Mark these off.
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So we're using the same
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scale. OK, let's have a look
what's happening here. This bit
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of curve between North and 90.
We begin with something for tan
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that is very small but positive.
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Just above 0 and then it gets
bigger and bigger and bigger as
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it rises. The value of Tan Theta
rises towards Infinity.
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Well down here divide the value
of theater is very near to zero
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and so tan Theta is very small
but positive. So when we divide
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into one we're going to get
something very big and positive
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self. But if curve there.
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Up here, the value of Tan
Theater is enormous. It's huge.
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So if we divide something huge
into one, the answer is going to
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be very nearly zero. And the
closer we get to 90, the closer
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it would be to 0.
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So now if we look here, we can
see we've got something very,
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very big, but negative. So the
answer is going to be very, very
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small, but also negative. This
is going to be coming out of
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that point there. Here 180.
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Got a problem at 180. Tan
Theater is 0 one over 10 theater
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is there for something very very
big so we can put in an acid
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tote and we can see we've got
exactly the same problem here at
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360. So if I join up what I've
got in the direction of what's
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happening, we're getting a very
similar curve and repeat it over
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here, 'cause the curves are
repeated. We're getting a very
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similar curve, except the other
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way around. So we've seen again
how we can derive the graph of
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coffee to directly from the
graph of Tan.
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So remember these three
new functions.
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Co sack sack and caught.
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Respectively, they are one
over sign, one over cosine
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and one over Tangent.
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We can use them to solve
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equations. But each time we
can get back to using sign
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cause and tab to help us
workout the angles.
Title:
www.mathcentre.ac.uk/.../5.9Trigonometric%20Functions.mp4
Video Language:
English
 mathcentre edited English subtitles for www.mathcentre.ac.uk/.../5.9Trigonometric%20Functions.mp4