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

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.