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