www.mathcentre.ac.uk/.../Trigonometric%20functions.mp4

In this video, we'll use what we
know about the trig functions

sine X, Cos X and Tinix to
define the functions.

F of X.

Equals sign X.

F of X equals cause.

X. And F of
X equals 10 X.

You've probably seen how to
define cynex using a circle

diagram. I'll just remind you
how to do that we draw.

An angle.

The line at X degrees from
the horizontal axis.

That line there. Is it X
degrees from horizontal?

And then sign X is the vertical
axis coordinate, so it's.

That barely there on
the vertical axis.

So this here.

Is sign X.

And we can use this picture to
see how changing X changes the

value of sine X.

When X equals
0 sign X is 0.

As we increase X up to 90, where
X is measured in degrees, sign X

increases up to one.

As we increase X further
sign X decreases.

It becomes zero when X equals

180. And then continues
to decrease.

It becomes minus
one when X is 270.

After that sign, X increases and
becomes zero when X reaches 360.

We have now come back to where
we started on the circle. So

as we increase X further, the
cycle just repeats.

We can also use this picture
to see what happens when X

is less than 0.

Also from this picture we can
see that whatever value X takes,

cynex must always be between
minus one and one.

Now that we have an idea of how
X an cynex are related.

We can try to plot a
graph of Cynex.

Let's workout some values
for sine X first.

Here I've got a table that's
ready to be filled in with

values for sine X. You can use
your Calculator to workout these

values. I'll fill in the
table now.

We know that sign of 0.

Is 0.

Using your Calculator.

You can see that sign of 45.

Is not .712 decimal places.

Sign of 90 is one.

Again, use a Calculator to see
that sign of 135.

Is no .71.

Sign of 180 zero.

You Calculator will tell you
that sign of 225.

Is minus not .71?

Sign of 270 is minus one.

Sign of 315 with your Calculator
is minus North Point 7 One.

And sign of 360 finally
is 0 again.

Now what we can do is plot those
values on a graph.

And then use our knowledge from
the circle diagram to fill in

the rest of the graph.

And that's what I've done here.

I've plotted sign of 45 and 90
and 135 and so on.

And then filled in rest of the
graph from note 360.

Now a circle diagram told us
that as we increase dex up from

360, the cycle just repeated, so

between 360. And 720 we just get
another copy of the sine graph.

And if we were to increase X

further. We just get the cycle
repeating itself again, so this

will carry on forever.

Now as we decrease X from zero.

It was like going backwards
around the circle.

So from zero down to minus 360,
it just follows the same shape.

And that would go on
forever as well.

Now, since this cycle
repeats itself every 360

degrees, we can say.

That sign X.

Equals sign X.

Plus 360.

We say that Psyonix is periodic
with periodicity 360.

Sometimes we want to work in
radians instead of degrees.

Sonnichsen radiance is very
different from Psyonix in

degrees. Here's an example.

We've seen that sign of
90 degrees.

Is equal to 1.

But sign of 90.

In radians.

Is about nought
.894.

Here I have a table of values
which shows X in degrees.

And then X and gradients and the
values of sign of X.

I can also show you.

A graph of sign of X in radians.

Again, here we have the graph
repeating itself three times.

Now looking at this graph, it
looks very very similar to sign

of X in degrees.

But really, you must remember
that sign of X in degrees is a

different function from sign of
X and radiance and you can see

this if I show you the two
graphs plotted together.

On this pair of axes I have sign
of X in degrees.

As the green line.

Now X is going between North and
just over 60 here.

So sine of X and agrees his.

Come up not very far.

It hasn't reached 1 yet.

But sign of X in radians, which
is the red line, has repeated

itself lots of times has gone up
to one back down to minus one

lots of times. So you can see
here that these are

definitely different
functions.

Sometimes we want
to work backwards.

And what I mean by that is
if we have our function.

F of X equals sign X.

We want to define a new
function.

Cool the inverse function of
sign of X, which we're going to

write as F minus one.

Now what we want F of
minus one to do is to

say F minus one over X.

Is equal to Y.

Where F of Y.

Is equal to X.

So an example of this working
for sign is that if.

Sign of X.

Is 3/4 that's the same as
saying that the inverse

sine of 3/4?

Is X.

So X is the angle
who sign is 3/4.

Now, defining a function of
like this gives us a small

problems, you'll see.

Because here we have a
graph of Cynex.

And there's 3/4.

So if we look at this, we can
see that around about.

Down here.

There's an angle who sign is

3/4. But also.

Over here.

There's an angle
who sign is 3/4.

There will be another two
over here and loads more

over to the far side of the
graph, which we can't see.

So we need to restrict where we

look. To find the angle who sign
gives us 3/4.

So what we do is we restrict
where we define sign.

We want to restrict this graph
so that every sign value has

only one angle giving it.

And we can do that. You see, we

cut. A graph down.

So that X goes between minus 90
and plus 90.

That gives one angle
for each sign value.

So we say then if we
define our function as

F of X equals sign X.

With the Main.

Minus 90.

90

then F of X.

Has an inverse.

You can use your Calculator to
workout inverse signs and later

inverse cosine inverse tangents.

In the sign is sometimes
also called.

AC sine X.

Now let's look at Cosine X.

Again, we can use our circle

diagram. To see how cosine
X&X are related.

Like before we draw a line at X

degrees. To the horizontal.

And this time Cos X.

Is the horizontal axis
coordinate?

We can use this diagram to
see how cause exchanges is

exchange is.

When X equals 0 Cos, X is one.

As X increases up to 90.

Cos X decreases to 0.

As X goes up to 180.

Cos X continues to
decrease to minus one.

As X goes up to 270.

Cause X increases.

As X increases to 360 Cos X

increases. And that X equals
360 Cos X equals 1.

As we increase X further, the
cycle just repeats itself.

We can also see what happens
when X is negative.

Again, you can see that
cosine X must lie between

minus one and one.

Here I've got a table already
filled in with values of X and

cosine X. You can use Calculator
to do this.

Here's a graph of cosine X that
I've created by putting on the

values from the table and then
using my knowledge from the

circle diagram to fill in the
rest of the graph.

Again, you can see that cosine X

is periodic. And again it has
periodicity 360 or 2π. If you're

working in radians.

The night before we like to
define an inverse cosine.

But again. We have a problem.

Because there is more than
one angle that gives us

each cosine value.

Unlike before, we can't
restrict our graph to minus

9090, and that's because there
are still more than one angle,

giving each cosine value.

And worse than that, there are
some cosine values, IE the ones

below 0 which don't have an

angle at all. In this
range, so we need to choose

a different range.

A good range to choose.

Is X between 0 and 180.

Because as you can see.

There was only one angle giving
each cosine value here.

And every cosine value has an
angle to give it.

So we can say.

That if we define F of X
equals cosine of X.

For X between 0.

180

then F inverse which is
inverse cosine of X exists.

And inverse cosine.

Is sometimes called our cause.

Finally, will deal with Tan X.

The circle diagram isn't
quite so useful in this case,

so will go straight to making
a table of values for tonics.

Now we know that tanks.

Equals sign X.

Over cause X.

And we already know something
about sines and cosines.

So we can use what we know to
fill in the values for titanics.

When X equals 0, Silex is zero
and cosine X is one. So tinix

must be 0.

When X is 45.

Son calls the same thing, so tan
must be 1.

When X is 90, Cos X is 0.

Since any fraction with zero
denominator is not defined,

tinix doesn't have a value
at X equals 90.

And the same thing
applies when X 270.

And any other time when Cos XO.

1X is 135.

Sonics is not .71.

Call sex is minus that satanic
says minus one.

A tax is 180 sign. XO Cos X is
minus one, so tanks must be 0.

FedEx is 225 sign in
cars are the same thing

again, so tinix is one.

At X equals 315, Scion X is
minus 9.71 cause X is plus 9.71,

so tanks must be minus one.

And then X equals 360,
sign X is zero, Kha zix

is one, so tinix is 0.

Now that we have this table
of values, we can use our

identity tan X equals sign
X over cause X it's filling

the rest of the graph.

Alright, the identity up here.

As X goes from note 90.

Paul's ex was going down to 0.

In fact, Codex gets very, very
small as X gets close to 90.

Where is Sonics? Just
went up to one.

So sine X over cause X gets very

very big. So we can fill in
the graph from X is between

notes and 90.

It starts at 0 as we saw
in a table and then get

very very big indeed.

When X goes between 90 and 180.

Psyonix is at one
and goes down to 0.

Cos X is negative but
very very small.

So the size of sign X over
because X is very very big.

But because causes
negative, 10X is negative.

So just after 90 tanks is
right down here.

And as X goes to 180, it
comes up to 0.

When X is between 180 and 270,
both sign X and Cossacks were

negative, so tinix is positive.

Also. As X got close to
270, cause X became very close

to 0. So the size of
10 X must get big.

So between 180 and 270.

X starts off at zero
and then shoots up.

It's a very big values
when X is near 270.

Now, between 270 and 360.

We have that Kha zix as
positive, but sign X is

negative.

So tinix must be negative.

Again, we're starting off with X
being really tiny, so the size

of Tinix is very, very big.

So we're starting
down here again.

And coming up to 0.

When X is at 360.

Now, since psyonix and cause X
repeat themselves every 360

degrees, tinix must also
repeat itself every 360

degrees. So now we can just
fill in the rest of the graph

by copying what we've done
already.

We can do this for negative X2.

And this graph will repeat
itself, however large, let X

get or have a negative, you
get X get.

Notice also that tinix
repeats itself more often

than sign and cause.

It repeats itself
every 180 degrees.

So we say tannic, says periodic
with periodicity 180.

Unlike sign and cause, tinix
doesn't have to lie between

minus one and one.

If you look at the graph, you
can see that annex can take

any value you like, have a
larger, have a negativities.

Now we'd like to define an
inverse tan as well.

Here we need to be twice as
careful as we were before

because not only do we have to
make sure there's only one

angle that gives us each tan
value, we have to make sure we

don't define the function over
a bit where we have an

undefined tan function.

That is over a bit where
Cos X equals 0.

A good choice to make is the
graph being between minus 90 and

90, but not actually including

those values. We can't
include those values

because Cos XO there.

If we take tanic stubi RF of X.

And X to be strictly greater
than minus 90 and strictly less

than 90. Then we can
define or inverse tan.

And inverse tan of X is
sometimes called arc tonics.