In this video, we're going to be
having a look at trigonometric
ratios in a right angle
triangle, and that's a long
phrase trigonometric ratios, so
we'll talk about them just as
trig ratios. What I've got here
in this diagram is.
Straight line and marked off are
the points A1A Two and a three
and each one of these is 10
centimeters on from the next
one. So that's 10 centimeters.
That's 10 centimeters, and
that's 10 centimeters, and I've
made a right angle triangle by
drawing vertically up from each
of these points.
To meet this line here.
So in effect what I've got is an
angle here at oh.
An angle here at oh and I'm
extending the two arms of the
angle like that, but the angle
staying the same even though the
separation is getting bigger,
the angle is staying the same.
what I want to have a look at is
what's the ratio?
Of that length till that length.
As we extend.
So that length till that length.
And then that one till the whole
of that. I want to have a
look at that and see what it
actually does. So going to use a
ruler, do some measurement, but
these are fixed at 10
centimeters each. So first of
all, let's have a look at a 1B
one divided by.
Oh a one.
Well, the bottom is definitely
10 centimeters. We've had that
before. If I measure this one
with the ruler, it's.
6.1 centimeters.
So that's not .61. Now
let's have a look at
the same ratio for this
extended angle is bigger
triangle, so will take A2B2
over. 082 equals
now this. Oh
A2 is 20 centimeters over
20 and take our ruler
and will measure this. That's
about 12 point 212.2 we
do the division and we've
got nought .61 again.
This bottom one just to check
is 30 centimeters long, so we
want a 3B three over 08,
three, 30 centimeters. We've
just measured it, and if we
measure again here.
This is 18 point.
2. So 18.2 centimeters.
And if we do, the division
that is not.
.6.
Oh, and it's going to
be 6 recurring, which is
about N .61.
So what we see is that all of
these three ratios are the same.
So if we take the side in a
right angle triangle which is
opposite to an angle.
And we divide it by the side
which is adjacent to it.
Then opposite over
adjacent seems to
be constant.
For the. Given.
Angle. And indeed
it is.
And we have a name for it.
They call it the tangent of the
angle. So.
Take a right angle triangle.
Right angle there and let's look
at this angle here. Label it.
Angle a so we've got a clear
picture that that's the angle
we're talking about.
Let's give names now to the
sides. The longest side of a
right angle triangle, the one
that's opposite to the right
angle is always called the
hypotenuse. So then
we have
the hypotenuse.
Usually shorten
it till
HYP.
This side. Is opposite
this angle. It's across from it,
it's opposite to it, and so we
call that the opposite side,
which we shorten till O Double
P. Inside is a part of
this angle, so it is adjacent.
Till the angle.
And we shorten this to a DJ.
If I take not this angle
but that angle, then some
things change.
Let's draw the diagram again.
This time let's focus on
the angle B.
This side is still the
longest side and is still
opposite the right angle, so
it is still the hypotenuse.
But it's now this side which is
opposite across from the angle
be. So this side is now the
opposite side. It's now this
side which is a part of the
angle which is alongside the
angle which is adjacent to the
angle. So if we change the
angle that we're talking
about in a right angle
triangle, we change the
labels.
They're not fixed until we have
fixed appan. The angle that we
are talking about. Let's now
take a little step back.
Remember, at the beginning of
the video. We said
But if we picked our angle.
And we took the opposite side.
And the adjacent side.
Then the ratio.
The opposite to the
adjacent side was a
constant. And that we call
that the Tangent.
Of the angle, a well
tangents along word
we tend to shorten
that simply to tan.
What other ratios are there in
this right angle triangle?
Let's label this side as
the hypotenuse. And the other
important ratios that we need to
know about and be able to work
with are the opposite.
Over the
hypotenuse. Which is the sign
of the angle a?
And for all we say, sign.
We shorten it in written form
till sin sin. We don't say that
we say sign.
The third of the major
trig ratios is the adjacent.
Over.
The hypotenuse and this is
the cosine of the angle
a. We shorten that to cause
a. Now all of these have been
worked out in tables and before
calculators mathematicians used
to work these out quite
regularly and published books of
tables of the sines, cosines,
and tangents of all the angles
through from North up to 90
degrees. In doing all the
calculations, things like a
Calculator are invaluable and
you really do need want to be
able to work with.
What we're going to have a look
at now are some calculations
working out the sides and angles
of triangles when given certain
information about them.
When doing this, it does help
if you have a way of
recalling or remembering these
definitions. One of the ways is
a nonsense word.
So a toca.
Sign is opposite over
hypotenuse, tangent is opposite
over adjacent and cosine is
adjacent over hypotenuse. Summer
talker. Some people remember it
as soccer to us.
Simply changing the syllables
around. Still means the
same thing. Others remember it
by a little verse Toms
old aunt. Sat
on him.
Cursing At him and
again, tangent is opposite
over adjacent sign is
opposite over hypotenuse,
cosine is adjacent over
hypotenuse. Whichever you
learn to help you remember
these ratios, that's up to
you, but it is helpful to
have one of them. So let's
have some problems.
Many of the examples that we
want to look at actually come
from the very practical area of
surveying. The problem of
finding out the size of
something, the length of
something when you can't
actually measure it. Typically,
perhaps the height of a tower,
so we're still here.
And we want to look
at a tower over here.
And how high is not tower? We
can't measure it. We can't get
to the top of it. But how can we
find out how high it is? One of
the ways is to take a sighting
of the top of the tower.
And with an instrument surveying
instrument, we can measure that
angle to the horizontal there.
So let's say we do make that
measurement, and we find that
it's 32 degrees. We can also
find how far away we were stood
from the base of the tower.
That's a relatively easy
measurement to make as well. So
let's say that was five meters.
Somebody about 6 foot tall
probably measures about 1.72
meters there or thereabouts.
So we're set up.
We know how far away from the
tower we stood. We know this
angle, sometimes called the
angle of elevation of the top of
the tower from our IO. How high
is this? Well, here we've got a
right angle triangle and we know
one side and we know an angle
and this is the side we want to
find. It's the side that is
opposite to the angle, so let's
label it as the opposite side.
This side is 5 meters.
It's the side that we know, and
it's also adjacent to the angle.
So what trig ratio links
opposite and adjacent? Well,
the answer to that is it's
the tangent, so the opposite
over the adjacent. Is that an
of 32 degrees?
Right, we are in a position now
to be able to solve this. To be
able to substitute in values
that we know. And placeholders
for those that we don't know. So
the opposite side, let's call
that X. So that's X for the
length of the opposite side
over 5 meters is the length
of the adjacent side.
Equals
1032.
Defined X, we multiply
both sides by 5,
so X is 5
* 1032. And now this is
where we need our Calculator 5.
Times. 10
32 Equals and the
answer we've got there is 3.124
and then some more decimal
places after it.
One word of warning, by the way.
Most calculators operate with
angles measured in one of two
ways. Degrees or radians. You
need to make sure that your
Calculator is operating with the
right measurement of the angle.
I'm using degrees here 32
degrees, so my Calculator
Calculator needs to be set up so
as to operate in degrees.
The answer to my question, how
high is this tower? We haven't
quite got yet. The reason we
haven't got it is we've only got
this length here and so the
height of the tower we can find.
Adding on.
The height of our eye
above ground level, so that
gives us 4.844 meters.
This is a very accurate
answer. It says that we know
the height of this tower till,
well, a millimeter. This place
here is 4 millimeters. That's
far too accurate. We would
probably be happy to say this
is 4.8 meters and work to two
significant figures.
Let's have a look
at another example.
Here we've got an equilateral
triangle. Well, not
equilateral. Sorry, let's make
it isosceles. Isosceles
triangle has 2 equal sides and
in this case will say that we
know this side is of length
10.
And we know that
this side is of
length 12. You have the
units of measurement to be
meters. Now it's an isosceles
triangle and the question is how
big is that angle? That doesn't
seem to be a right angle.
In questions like this, you've
got to look to introduce a right
angle, and here if we divide the
isosceles triangle in half down
the middle, this is our right
angle, and so here we've got a
right angle triangle.
And we know that if this is 10
and this is half of 10 five
meters. Let's look at these
sides and label them. This side
is opposite the right angle.
It's the longest side in the
right angle triangle, so it is
the hypotenuse. This side, which
we also know is alongside the
angle it's adjacent to the
angle. Let's give this
angle ylabel, let's call
it the angle a.
What trig ratio of the angle a
connects the adjacent side and
the hypotenuse? Well, that's the
cosine, so the cosine of A.
Is equal to the adjacent.
Over. The hypotenuse.
And we can substitute the
numbers in the adjacent is 5.
The hypotenuse is 12.
Now I've got a problem. Here it
says cause I is equal to five
over 12 and I want the angle a
angle A is.
The angle whose cosine is 5 over
12. I've got a problem about how
I might write that down.
Do it. I write it like this.
Calls to the minus one
of five over 12.
What that bit there this
caused to the minus one says
is the angle whose cosine
is. So let's put an arrow
on to that and write that
down the angle.
Who's
cosine? Is that's
what this piece of notation
means. Now I've got to work
that out. So I need to look at
my Calculator. And we're
looking for the angle whose
cosine is, and we can see
that as cost to the minus
one, which is just along the
top of the cause button, and
it's on the shift key there.
For the second function key.
So in order to calculate it
we have to press shift.
Cosine key, we see that we get
cost to the minus one.
Bracket 5 / 12 closed
bracket and that gives us
the angle of 65.3756 etc.
Degrees and I probably give
this to the nearest 10th
of a degree. In other
words to one decimal place.
Let's take another
example.
This time will take the
isosceles triangle again.
15 meters for that side,
and it costs an isosceles
triangle with those two sides
equal. That one will also
be 15 meters and will
say that's 72 degrees.
Question, how high is the
triangle here is the height.
That's what we're after. The
height of that triangle.
So let's label our sides this
side, the one we want to find is
across from the angle. It is
opposite to the angle, so it
becomes the opposite side. This
side here is opposite the right
angle. It's the longest side in
the right angle triangle, and so
it's the hypotenuse.
What trig ratio connects
opposite and hypotenuse?
The answer is sign and so
the opposite. Over the
hypotenuse. Is equal
to the sign of 72
degrees.
H. Is the length
of the opposite side the height
of the triangle were trying to
find over the length of the
hypotenuse? That's 15 is equal
to sign 72 degrees, so H
is 15 times the sign of
72 degrees. Multiplying both
sides of this equation by 15.
Again, without a stage where we
need our Calculator and so we
want 15. Times sign
of 72. And
the answer we have is 14.3
and here we work to three
significant figures.
Let's take one more example in
this area and the example it
will take his where we're
finding aside, but perhaps it's
not the one that we might
expect. So there's
the right angle.
Let's say this angle up here is
35 degrees. This side is of
length 6 and what we want to
know. Is how long is the
hypotenuse? The side is the
hypotenuse 'cause it's opposite
the right angle. It's the
longest side in the right angle
triangle. This side, that's a
blend six, let's say 6
centimeters since lens have to
have units with them, this is
the adjacent side. It's adjacent
till this angle.
What's the trig ratio that we
need that connects adjacent and
hypotenuse? It's going to be the
cosine and so we have adjacent
over hypotenuse is equal to the
cosine of 35 degrees.
Adjacent side is 6.
It's called the hypotenuse X.
So six over X is equal to
the cosine of 35 degrees.
So 6 must be X times the
cosine of 35 degrees,
multiplying both sides of this
equation by X.
This sometimes gives people a
little bit of trouble. The next
step we've got 6 equals X times
cost 35 and it's the ex that I
want. The thing that you have to
remember is cost 35 is just a
number, just a number like two
or three. So if this said six
was equal to two X, then we know
that X is equal to three,
because we'd be able to divide
the six by two.
Cost 35 is no different,
and so we divide both
sides by cause of 35
degrees that will be
equal to X.
So we need the Calculator
again. We have 6 divided by
the cosine of 35 degrees and
the answer we get for X
layer is 7.324. We shorten
that to three significant
figures that street 7.32
centimeters to three
significant figures.
We've looked at some problems
and there are some more on the
example sheet that accompanies
the video. One of the things
that we do need to have a look
at, though, are some quite
specific angles and their signs.
Cosines and tangents. And it's
important that these are
learned, learned because they
are exact and because
Mathematicians, scientists,
engineers use these a great deal
in their work.
The angles that we're talking
about, the angles 0.
30
45
60 And 90.
We're going to have a look at
their signs. Their cosines
and their tangents.
Let's have a look at a right
angled isosceles triangle.
To begin with, there
is it right angle.
Isosceles each of these sides
will be the same length.
So if we use Pythagoras 1
squared plus, one squared is one
plus one, which gives us two. So
the length of the hypotenuse
must be Route 2.
And each of these angles
because it's isosceles are
45 degrees.
So from that triangle, can we
read off the sign cause and
tan of 45 degrees?
Let's concentrate on this angle.
This is the angle we're going to
concentrate on that 45 degrees.
The sign of 45 is opposite.
Over hypotenuse, and so that's
the opposite is one over the
hypotenuse is Route 2, so that
is one over Route 2.
The cosine, the cosine is the
adjacent. Over the hypotenuse.
The adjacent is one the
hypotenuse is Route 2 and so the
cosine is one over Route 2.
The tangent is the opposite
over the adjacent and so the
opposite is one over the
adjacent is one 1 / 1 is just
one. So we've completed that
middle column of our table.
Let's now take an
equilateral triangle.
All the sides are the same
length, all 2 units long and
that means that all the angles
will be the same as well, so
they will all be 60 degrees.
Haven't drawn in the top angle
because I'm going to split my
equilateral triangle in half, so
each of these will be 30.
There is a right angle down
there. I've split it in half, so
each piece of this is one unit.
So there my angles.
All the lengths are marked
except that one the height of
this equilateral triangle.
Let's workout how long that is.
Pythagoras tells us that that
squared plus that squared should
give us that squared.
Well, that squared is just one
that squared is 4.
To add 1 to something and get 4,
the answer has to be 3. So that
means the length of this is
Route 3. The square root of 3.
So now we can read off from
this right angle triangle signs,
cosines and tangents for 30 and
for 60. So let's look at 30.
The sign is the opposite over
the hypotenuse. The opposite is
of length one, the hypotenuse of
length 2. So that is one over
two half for the cosine, the
cosine is the adjacent over the
hypotenuse. And so that is Route
3 over 2.
Three over 2.
For the tangent, that's the
opposite over the adjacent, so
that's one over Route 3.
Let's have a look at this one.
60 Starting with the
sign sign is opposite over
hypotenuse, so that is Route
3 over 2.
Cosine is the adjacent
over the hypotenuse, so
that is one over
2A half.
The tangent is the opposite over
the adjacent and so that's Route
3 over one or just Route 3.
What about these bits, not 90?
OK, let's take any right
angle triangle.
And we'll fix
on this. Angle.
So that's the hypotenuse.
That's the adjacent.
That's the opposite.
OK. Sign is opposite over
hypotenuse. If we imagine this
angle going down towards 0 like
that, it's clear that the
opposite side is becoming zero
itself, and so we have zero or a
number approaching 0 being
divided by the hypotenuse, which
is pretty large. That would
suggest that the sign of 0 is 0.
What about the cosine? As this
comes down, the hypotenuse
becomes almost the same length
as the adjacent, which would
suggest that the cosine of 0 is
one. The tangent well, the
tangent is the opposite over the
adjacent. So as this side goes
down. The opposite side becomes
zero over a fixed side, so again
0. What about 90?
Let's imagine this angle begins
to get bigger and bigger and
bigger so that it looks like
it's going to become 90. And
what we can see is that the
opposite in the hypotenuse
become almost the same,
suggesting that the sign would
be 1. But as that is happening,
the adjacent is becoming 0.
And the cosine is the adjacent
over the hypotenuse, and so
again, that goes to 0.
Finally, the Tangent.
That adjacent side is becoming
zero, becoming very, very small,
but this opposite side is
staying fixed, so we've got a
fixed number and we're dividing
it by something which is getting
very, very small. So the answer
is becoming very, very big. In
other words, it's becoming
Infinite. We write that as an 8
on its side and call it
Infinity. So this is
a table of sines, cosines,
and tangents that are all
exact.
And for very specific angles,
not thirty 4560 and 90,
these need to be learned.
They are important and.
Textbooks. Scientists, engineers
in their everyday working and
everyday calculations take these
as being read. They assume that
they are known, and so you must
learn them and make sure that
you have them ready to hand to
work with immediately. You hear
sign 45 that's one over route
two 1060 that's Route 3.