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.