www.mathcentre.ac.uk/.../5.2Trigonometric%20Ratios%20in%20a%20Right%20Angled%20Triangle.mp4

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.