Trigonometric ratios of any angle

In this video, we're going to be
looking at the definitions of

signs, cosine and tangent for
any size of angle.

Let's first of all recall the
sine, cosine, and tangent for a

right angle triangle.

There's our triangle. We identify one
angle and then label the sides.

The side that's the longest
side in the right angled triangle

and the one that is
opposite the right angle is

called the hypotenuse and we
write HYP hype for short.

The side that is opposite the
angle is called the opposite side.

OPP for short, this side the
side that is a part of the

angle that runs alongside the
angle we call the adjacent

side or ADJ for short.

The sine of the angle A.

Is defined to be the
opposite over the hypotenuse.

The cosine of the angle A
is defined to be the adjacent

over the hypotenuse and the
tangent of the angle A is

defined to be the opposite
over the adjacent.

But this is a right angle
triangle and so the angle A is

bound to be less than 90
degrees, but more than 0.

In other words, it's an acute
angle, so these so far are

defined only for acute angles.
What happens if we've got an

angle that's bigger than 90? Or
indeed if we've got an angle

that's less than 0?

That raises a question. To
begin with, why? How can we

have an angle that's less than
zero? So first of all, let's

just have a look at angles.

Got a set of axes there.

And based upon the origin, let

me draw. Roughly a unit circle.

Circle of radius
one unit so that where it

crosses these axes, X is
one, Y is One X is minus

one, Y is minus one there.

Let's imagine a point P on
this circle and it moves round

the circle in that direction. In
other words, it moves round

anticlockwise, then this is the
angle that OP makes with OX

the X axis.

So there's an acute angle.

When we get around to here we've
come right round there and that

gives us an obtuse angle, an
angle between 90 and 180.

Come round to here and we've
gone right the way around there.

An angle that is greater than
180 but less than 270.

And similarly into this
quadrant. So that's positive,

is anticlockwise.

So we get positive angles
if we go round in an

anticlockwise way.

We draw it again.

Same unit circle.

And we think about our radius, OP

What if we start to move it
around this way.

Then this is clockwise, and
so this is a negative

angle, so negative we're
going around clockwise.

So of course we can go round

from here. To the Y axis, the
negative part of the Y axis, and

that's minus 90 degrees. If we
go right the way round to the

negative part of the X axis,
that's minus 180 degrees.

Effectively the same as coming
round to 180 degrees coming

round anticlockwise. So that's
how we can have any size of

angle. The question is can we
put these two together? Can we

bring together these
definitions and these ideas

about having angles which are
greater than 90 both positive

and negative? Well, let's take
sine and have a look at that.

So draw the same diagram again.

Put our point P on the
circle which is going to

move around that way in an
anti clockwise direction.

And here mark this angle going
around that way.

OK. Sine is opposite
over hypotenuse. Well, if

I complete the right angle
triangle.

This would be the side that
is opposite that angle.

There is the right angle, so
this is the hypotenuse. And

because this here is a unit
circle, the length of that is

just one. So the question is,
how can I describe this line?

If I imagine I've got my eye here
and I'm looking in that

direction, what do I see? I see
that length as though it were

projected onto the Y axis.

So I'm looking that way and I
can see that length which is

OP, as though it were
projected onto the Y axis. So

perhaps a way of describing sine
of, let me call this theta.

A way of describing sine theta
would be to say:

that it is equal to the projection
of OP onto OY, the Y axis,

divided by OP and of course
OP is then just one.

How does it work with
any angle? Well, think what

happens as we go round.

As we go round as it rotates

around. And you're still looking

this way. Then you've still
got a projection.

It goes down to a length of zero
and as we come back around here.

We've still got a projection
on this axis that we can see,

so we've still got something
that we can measure when it

gets around to here of
course, it's on the negative

part of the Y axis, and so
it's going to be negative.

Well, let's have a look
what that might mean in

terms of a graph.

What I've got here is a
protractor and

the middle bit of this protractor
rotates.

Here I've got a black line which
is a fixed horizontal line.

Along that here I've got a red
line which is going to be my OP.

This is the point P moving
around in an anticlockwise

direction, marking out

positive angles, and if it went
that way around it would be

marking out negative angles, so
they're going to start off

together there, both pointing on
zero, the angle 0. So let's

recall sine theta.

The angle that OP,
there's O, there's P,

makes with the X axis, is defined to be
the projection of OP onto OY,

Divided by OP.

But if I choose to make OP
the measure, the unit then

sine theta is just the projection
of OP onto OY.

Now let's have a look,

what that means in
terms of a graph?

So this is the axis.

Measuring marking off the
degrees, so set that to 0 so the

first point on the graph is
there because as we look along

there. So we look along there.
What we see is nothing. Just see

a point. So the length.

Of OP, the projection of OP
onto OY is 0.

Now there.

Halfway round.

45 degrees it's about

that high. So let's mark it
there. That's 45. This 90. Let's

move that around. Do we get
to the top?

Mark that across.

Roughly about there. And we
start to go back down here.

There we are 135.

The projection is along
there and taking that

through its to there.

When we get round to 180 again
as we look along there, we just

see a point, no length.

As we come down here.

245 there, or in fact 180 +
45, which is 225. We get a

point which is about.

There.

And then as we come down to 270.

About

there.

And as we come round here to
315 and through there or about

there and then when we come
round to there we're back to 0 again

or, having been all the
way round, 360.

And if we join up those points.

We get quite a nice smooth curve
out a bit with that one if we

think of this going back in
this direction, what we can

see is that we're going to get
the same ideas developing.

Here, let's just fill in the 90.
It will be right down there, so

it's there. And then at minus
180 right around there it's

there and then minus 270 going
right. The way around there we

are back up at the top again.

And then minus 360 having
come all the way around we're

down there. And so again we
have a nice smooth curve.

Notice that this shape is
exactly the same as that shape

and that we could keep on
drawing it. This block,

This block repeats itself, it's
periodic. It keeps repeating

itself every 360 degrees from
there to there is 360. And

similarly from here through
here. Till here is also 360

degrees. So now we have a
sine function if you like.

That we can think of as being
defined by this graph.

It covers any angle that we would
want it to cover. We can keep on

going for 720, 1000 degrees that
way, minus 1000 that way.

But this is always going to give
us a well defined function.

What about something like the
cosine curve? What about that one?

Well, let's just have a
look at that and see how we

can make a similar graph for
our cosine function.

I'll just stick that down again.

And quickly draw a set of

axes. I'm not draw this
one as accurately.

But it should be enough for
us to be able to see the

results that we're wanting, so
we'll mark off the divisions

as we have before.

0, 90, 180, 270, 360
and then -90,

-180, -270 and -360 there.

OK the thing that we haven't
done is made a definition.

What do we mean
by cos theta?

Well, let's have a look.

We want to do the same sort of
thing as we did for sine.

So that's the angle.

In there.

The adjacent side will be this
side, which is the projection.

Of OP onto the X axis, this time,

so O cos of the angle is the
projection of OP onto OX,

the X axis, divided by OP.

This is a unit circle,
so OP is one.

So cos theta is the projection
of OP onto OX

Now we need to
look at that and see how

that grows, and varies as we
rotate around.

So we start with zero. Remember
our eye is now looking down

What we see is OP itself.
So what we see is

a point there, one.

We start to move this around.
Let's go to 45.

And. We're looking down, so we
see that bit there, which is

less, which is smaller. So we
see something about there.

As we go round to 90 when we
look straight down on this red

line, all we see is a dot and so
the projection is 0.

Let's go round and now to 145
and now the projection is down

onto the negative part of the X
axis, so this is negative. Now

down there. And at 180.

Well, we now right round to
sitting on top of OP again of length -1.

Start to come around again to
225 and again we get that

projection back, so we're
starting to be here and then at

270. Clearly again, we're going
to come round to there looking

down that way.

As we come round, the projection
is again at 315. Now a positive

projection. So again we've gone
through there to there and then

360. We're back to 0 again.

And so the projection
is of length one, so

let's fill that in.

Join up the points and again we
get a nice smooth curve. And of

course the same curve is going
to exist on this side.

Let's just check we're going to
swing it around this way and we

can see. Is that the projection
is positive but getting less

until we get round till 90 when
the length of the projection is

0? So we're going to see this
occurring round here.

Nice smooth curve up through
there to there.

Again, notice it's periodic.
This lump of curve here is

repeated there and will be
every 360 as we march

backwards and forwards along
this X axis. So again, we've got

a function that's well
defined, got a nice curve,

periodic nice, smooth curve,
so that's our definition for cosine.

Notice it's contained between
plus one and minus one.

That was something that we
didn't observe with sine, but

that is also the case that it is
contained between plus one and

minus one, because this
projection of OP onto OY can

never be longer than OP
itself, which is just one.

The other thing to notice
is that the two curves are

the same.

I just flipped back but
displaced by 90 degrees. We

slide this sine curve back by
90 degrees. You can see it

will be exactly the same as
the cosine curve.

What about tangent? Well, let's
recall how we define the tangent

to begin with, it was the
opposite over the adjacent.

For sine we replaced the
opposite side by the projection

of OP onto OY.

And for cosine, we replaced the
adjacent side as the projection

of OP onto OX.

What does that mean then
for our definition?

We draw our unit circle.

Take.

OP moving around in that
direction through an angle

theta and complete right
angle triangle. This here

is the opposite side. It's
opposite, the angle that

we're talking about.

tan theta equals, so the
opposite side, we have

replaced by the projection
of OP onto OY

divided by, now for a right
angle triangle it would be

the adjacent side and that
adjacent side has been

replaced by the projection of
OP on till OX. Well let's

remember OY is the Y
axis OX is the X axis.

This again gives us a definition
that's going to work as that

radius vector runs around the
circle like that,

in either direction. So again,
it's going to give us a

definition that will work
for any size of angle.

One of the things we can notice
straight away about this is that

it means that tan theta is of
course sine theta divided by cos theta

and that gives us an
identity which we need to learn

and remember. tan theta is sine
theta divided by cos theta.

But what does the graph of
tangent look like?

It's a little bit
trickier to draw.

Let's see if we can justify what
we're going to get.

Now we're looking
at tan theta.

where theta is the angle between
the X axis and the Y Axis.

And we know that the tangent is
defined to be the projection of

OP on the Y axis,

divided by the projection of
OP on the X axis.

So when we begin
here at theta is 0.

Then we know that the projection
onto the Y axis looking this way

is zero and onto the X axis is
one, so we've got 0 / 1 which is

0. So we start there, for the
angle zero, we start there. As we

move around. When we come to 45
degrees then the projections are

equal. So let's just mark 45 and
if the two projections are equal

then that must be 1.

What happens is we come up

towards 90. Well, the projection
on to the Y axis is getting

bigger and bigger approaching

one. But the projection onto the
X axis is getting smaller and

smaller and smaller, and it's
that that we are dividing by, so

we're dividing something
approaching one by something

that is getting smaller and
smaller and smaller. So our

answer is getting bigger and
bigger and bigger. It's becoming

infinite and we have a way of
showing that on a graph.

I've deliberately put in a
dotted line. Now a graph

approaches that dotted line, but
does not cross it, and that's an

asymptote. That's what we call
an asymptote. What about the

next bit of the graph up to 180
degrees? Well, as we tip over

into this quadrant, then the
projection on to the X axis

becomes negative, but he's still
very, very small.

The projection onto the Y axis
is still positive, still near 1,

so we're dividing something
that's positive and near 1 by

something that's negative but
very very small. So the answer

must be very, very big, but
negative, and so there's a bit

of graph. Down there on the
other side of the asymptote, and

now we run this round to 180
degrees and what happens then?

Well, when we've got round to
180, the projection onto the X

axis is then of length one, the
same as OP, but the projection

onto the Y axis is 0, so we've
got 0 / 1 again, which gives us

0 at 180 degrees. So this comes
up to there like that.

Think about what's going to
happen now as it comes around

here. Let's draw one in.

And we can see that the
projections on to both axes are

both negative. And a
negative divided by a

negative gives a positive.

You can also see that when we
get here again, we've got

exactly the same problems as we
had when we got here. We're

dividing by something very, very
small into something that's

around about one, so again I'll
answer is going to be very, very

big, and so again, we're going
to get a climb like that.

As we move through this, it's
the same as if we move

through there, so again we
will start back down here in

the graph, will climb and
then off up again there.

What about back this way?

When we've got negative

angles what's going to happen
then we must have the same

things occuring in terms of
our asymptotes. And so we're

going to have the same things
occur in with the graph there.

There and so on. So again,
notice we get a periodic

function. That bit of graph is
repeated again there.

Every 360 degrees we get a
periodic function. We get a

repeat of this section of the

graph. So that's our
function tangent.

And we can think of the
function as being defined if

you like by that particular
graph.