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.