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In this video, we're going to be
having a look at trigonometric

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ratios in a right angle
triangle, and that's a long

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phrase trigonometric ratios, so
we'll talk about them just as

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trig ratios. What I've got here
in this diagram is.

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Straight line and marked off are
the points A1A Two and a three

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and each one of these is 10
centimeters on from the next

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one. So that's 10 centimeters.
That's 10 centimeters, and

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that's 10 centimeters, and I've
made a right angle triangle by

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drawing vertically up from each
of these points.

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To meet this line here.

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So in effect what I've got is an
angle here at oh.

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An angle here at oh and I'm
extending the two arms of the

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angle like that, but the angle
staying the same even though the

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separation is getting bigger,
the angle is staying the same.

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what I want to have a look at is
what's the ratio?

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Of that length till that length.

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As we extend.

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So that length till that length.

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And then that one till the whole

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of that. I want to have a
look at that and see what it

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actually does. So going to use a
ruler, do some measurement, but

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these are fixed at 10
centimeters each. So first of

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all, let's have a look at a 1B
one divided by.

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Oh a one.

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Well, the bottom is definitely
10 centimeters. We've had that

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before. If I measure this one
with the ruler, it's.

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6.1 centimeters.
So that's not .61. Now

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let's have a look at
the same ratio for this

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extended angle is bigger
triangle, so will take A2B2

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over. 082 equals

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now this. Oh
A2 is 20 centimeters over

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20 and take our ruler
and will measure this. That's

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about 12 point 212.2 we
do the division and we've

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got nought .61 again.

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This bottom one just to check
is 30 centimeters long, so we

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want a 3B three over 08,
three, 30 centimeters. We've

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just measured it, and if we
measure again here.

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This is 18 point.

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2. So 18.2 centimeters.
And if we do, the division

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that is not.

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.6.

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Oh, and it's going to
be 6 recurring, which is

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about N .61.

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So what we see is that all of
these three ratios are the same.

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So if we take the side in a
right angle triangle which is

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opposite to an angle.

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And we divide it by the side
which is adjacent to it.

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Then opposite over
adjacent seems to

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be constant.

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For the. Given.

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Angle. And indeed
it is.

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And we have a name for it.
They call it the tangent of the

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angle. So.
Take a right angle triangle.

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Right angle there and let's look
at this angle here. Label it.

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Angle a so we've got a clear
picture that that's the angle

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we're talking about.

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Let's give names now to the
sides. The longest side of a

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right angle triangle, the one
that's opposite to the right

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angle is always called the

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hypotenuse. So then
we have

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the hypotenuse.
Usually shorten

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it till
HYP.

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This side. Is opposite
this angle. It's across from it,

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it's opposite to it, and so we
call that the opposite side,

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which we shorten till O Double

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P. Inside is a part of
this angle, so it is adjacent.

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Till the angle.

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And we shorten this to a DJ.

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If I take not this angle
but that angle, then some

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things change.

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Let's draw the diagram again.

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This time let's focus on
the angle B.

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This side is still the
longest side and is still

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opposite the right angle, so
it is still the hypotenuse.

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But it's now this side which is
opposite across from the angle

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be. So this side is now the

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opposite side. It's now this
side which is a part of the

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angle which is alongside the
angle which is adjacent to the

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angle. So if we change the
angle that we're talking

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about in a right angle
triangle, we change the

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labels.

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They're not fixed until we have
fixed appan. The angle that we

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are talking about. Let's now
take a little step back.

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Remember, at the beginning of

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the video. We said

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But if we picked our angle.

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And we took the opposite side.

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And the adjacent side.

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Then the ratio.

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The opposite to the
adjacent side was a

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constant. And that we call
that the Tangent.

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Of the angle, a well
tangents along word

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we tend to shorten
that simply to tan.

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What other ratios are there in
this right angle triangle?

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Let's label this side as

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the hypotenuse. And the other
important ratios that we need to

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know about and be able to work
with are the opposite.

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Over the

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hypotenuse. Which is the sign
of the angle a?

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And for all we say, sign.

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We shorten it in written form
till sin sin. We don't say that

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we say sign.

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The third of the major
trig ratios is the adjacent.

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Over.
The hypotenuse and this is

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the cosine of the angle

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a. We shorten that to cause

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a. Now all of these have been
worked out in tables and before

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calculators mathematicians used
to work these out quite

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regularly and published books of
tables of the sines, cosines,

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and tangents of all the angles
through from North up to 90

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degrees. In doing all the
calculations, things like a

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Calculator are invaluable and
you really do need want to be

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able to work with.

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What we're going to have a look
at now are some calculations

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working out the sides and angles
of triangles when given certain

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information about them.

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When doing this, it does help
if you have a way of

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recalling or remembering these
definitions. One of the ways is

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a nonsense word.

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So a toca.

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Sign is opposite over
hypotenuse, tangent is opposite

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over adjacent and cosine is
adjacent over hypotenuse. Summer

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talker. Some people remember it
as soccer to us.

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Simply changing the syllables

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around. Still means the
same thing. Others remember it

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by a little verse Toms

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old aunt. Sat
on him.

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Cursing At him and
again, tangent is opposite

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over adjacent sign is
opposite over hypotenuse,

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cosine is adjacent over
hypotenuse. Whichever you

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learn to help you remember
these ratios, that's up to

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you, but it is helpful to
have one of them. So let's

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have some problems.

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Many of the examples that we
want to look at actually come

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from the very practical area of
surveying. The problem of

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finding out the size of
something, the length of

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something when you can't
actually measure it. Typically,

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perhaps the height of a tower,
so we're still here.

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And we want to look
at a tower over here.

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And how high is not tower? We
can't measure it. We can't get

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to the top of it. But how can we
find out how high it is? One of

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the ways is to take a sighting
of the top of the tower.

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And with an instrument surveying
instrument, we can measure that

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angle to the horizontal there.
So let's say we do make that

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measurement, and we find that
it's 32 degrees. We can also

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find how far away we were stood
from the base of the tower.

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That's a relatively easy
measurement to make as well. So

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let's say that was five meters.

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Somebody about 6 foot tall
probably measures about 1.72

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meters there or thereabouts.

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So we're set up.

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We know how far away from the
tower we stood. We know this

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angle, sometimes called the
angle of elevation of the top of

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the tower from our IO. How high

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is this? Well, here we've got a
right angle triangle and we know

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one side and we know an angle
and this is the side we want to

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find. It's the side that is
opposite to the angle, so let's

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label it as the opposite side.

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This side is 5 meters.

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It's the side that we know, and
it's also adjacent to the angle.

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So what trig ratio links
opposite and adjacent? Well,

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the answer to that is it's
the tangent, so the opposite

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over the adjacent. Is that an
of 32 degrees?

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Right, we are in a position now
to be able to solve this. To be

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able to substitute in values

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that we know. And placeholders
for those that we don't know. So

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the opposite side, let's call

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that X. So that's X for the
length of the opposite side

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over 5 meters is the length
of the adjacent side.

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Equals
1032.

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Defined X, we multiply
both sides by 5,

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so X is 5

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* 1032. And now this is
where we need our Calculator 5.

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Times. 10

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32 Equals and the
answer we've got there is 3.124

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and then some more decimal
places after it.

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One word of warning, by the way.

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Most calculators operate with
angles measured in one of two

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ways. Degrees or radians. You
need to make sure that your

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00:14:42,100 --> 00:14:46,920
Calculator is operating with the
right measurement of the angle.

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I'm using degrees here 32
degrees, so my Calculator

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Calculator needs to be set up so
as to operate in degrees.

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The answer to my question, how
high is this tower? We haven't

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quite got yet. The reason we
haven't got it is we've only got

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this length here and so the
height of the tower we can find.

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Adding on.

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The height of our eye
above ground level, so that

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gives us 4.844 meters.

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This is a very accurate
answer. It says that we know

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the height of this tower till,
well, a millimeter. This place

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00:15:37,416 --> 00:15:41,746
here is 4 millimeters. That's
far too accurate. We would

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00:15:41,746 --> 00:15:47,375
probably be happy to say this
is 4.8 meters and work to two

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00:15:47,375 --> 00:15:48,241
significant figures.

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00:15:50,350 --> 00:15:54,760
Let's have a look
at another example.

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Here we've got an equilateral
triangle. Well, not

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equilateral. Sorry, let's make
it isosceles. Isosceles

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triangle has 2 equal sides and
in this case will say that we

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00:16:08,892 --> 00:16:12,105
know this side is of length
10.

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00:16:13,160 --> 00:16:20,496
And we know that
this side is of

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00:16:20,496 --> 00:16:26,202
length 12. You have the
units of measurement to be

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00:16:26,202 --> 00:16:30,701
meters. Now it's an isosceles
triangle and the question is how

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00:16:30,701 --> 00:16:35,609
big is that angle? That doesn't
seem to be a right angle.

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00:16:36,770 --> 00:16:41,558
In questions like this, you've
got to look to introduce a right

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angle, and here if we divide the
isosceles triangle in half down

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the middle, this is our right
angle, and so here we've got a

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right angle triangle.

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00:16:53,380 --> 00:16:59,665
And we know that if this is 10
and this is half of 10 five

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00:16:59,665 --> 00:17:05,450
meters. Let's look at these
sides and label them. This side

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00:17:05,450 --> 00:17:10,455
is opposite the right angle.
It's the longest side in the

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00:17:10,455 --> 00:17:13,185
right angle triangle, so it is

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00:17:13,185 --> 00:17:20,290
the hypotenuse. This side, which
we also know is alongside the

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00:17:20,290 --> 00:17:23,290
angle it's adjacent to the

213
00:17:23,290 --> 00:17:28,391
angle. Let's give this
angle ylabel, let's call

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00:17:28,391 --> 00:17:30,483
it the angle a.

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00:17:31,840 --> 00:17:38,368
What trig ratio of the angle a
connects the adjacent side and

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00:17:38,368 --> 00:17:44,352
the hypotenuse? Well, that's the
cosine, so the cosine of A.

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Is equal to the adjacent.

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00:17:49,640 --> 00:17:53,420
Over. The hypotenuse.

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00:17:54,510 --> 00:18:00,659
And we can substitute the
numbers in the adjacent is 5.

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The hypotenuse is 12.

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Now I've got a problem. Here it
says cause I is equal to five

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00:18:10,918 --> 00:18:15,670
over 12 and I want the angle a
angle A is.

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00:18:16,380 --> 00:18:21,896
The angle whose cosine is 5 over
12. I've got a problem about how

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00:18:21,896 --> 00:18:23,866
I might write that down.

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00:18:24,410 --> 00:18:27,966
Do it. I write it like this.

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00:18:28,550 --> 00:18:35,030
Calls to the minus one
of five over 12.

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00:18:35,530 --> 00:18:42,130
What that bit there this
caused to the minus one says

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is the angle whose cosine
is. So let's put an arrow

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00:18:48,730 --> 00:18:54,130
on to that and write that
down the angle.

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00:18:56,420 --> 00:18:58,595
Who's

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00:18:58,595 --> 00:19:05,109
cosine? Is that's
what this piece of notation

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00:19:05,109 --> 00:19:08,811
means. Now I've got to work

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00:19:08,811 --> 00:19:12,768
that out. So I need to look at

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00:19:12,768 --> 00:19:16,675
my Calculator. And we're
looking for the angle whose

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00:19:16,675 --> 00:19:21,415
cosine is, and we can see
that as cost to the minus

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00:19:21,415 --> 00:19:26,155
one, which is just along the
top of the cause button, and

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00:19:26,155 --> 00:19:30,500
it's on the shift key there.
For the second function key.

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00:19:30,500 --> 00:19:34,845
So in order to calculate it
we have to press shift.

239
00:19:36,650 --> 00:19:40,406
Cosine key, we see that we get
cost to the minus one.

240
00:19:41,220 --> 00:19:48,660
Bracket 5 / 12 closed
bracket and that gives us

241
00:19:48,660 --> 00:19:56,100
the angle of 65.3756 etc.
Degrees and I probably give

242
00:19:56,100 --> 00:20:03,540
this to the nearest 10th
of a degree. In other

243
00:20:03,540 --> 00:20:07,260
words to one decimal place.

244
00:20:07,680 --> 00:20:12,420
Let's take another
example.

245
00:20:16,670 --> 00:20:21,518
This time will take the
isosceles triangle again.

246
00:20:22,110 --> 00:20:29,430
15 meters for that side,
and it costs an isosceles

247
00:20:29,430 --> 00:20:36,750
triangle with those two sides
equal. That one will also

248
00:20:36,750 --> 00:20:43,338
be 15 meters and will
say that's 72 degrees.

249
00:20:43,350 --> 00:20:48,510
Question, how high is the
triangle here is the height.

250
00:20:49,520 --> 00:20:55,235
That's what we're after. The
height of that triangle.

251
00:20:56,520 --> 00:21:02,848
So let's label our sides this
side, the one we want to find is

252
00:21:02,848 --> 00:21:08,272
across from the angle. It is
opposite to the angle, so it

253
00:21:08,272 --> 00:21:13,244
becomes the opposite side. This
side here is opposite the right

254
00:21:13,244 --> 00:21:18,668
angle. It's the longest side in
the right angle triangle, and so

255
00:21:18,668 --> 00:21:20,024
it's the hypotenuse.

256
00:21:20,570 --> 00:21:24,609
What trig ratio connects
opposite and hypotenuse?

257
00:21:25,300 --> 00:21:29,230
The answer is sign and so

258
00:21:29,230 --> 00:21:32,480
the opposite. Over the

259
00:21:32,480 --> 00:21:39,407
hypotenuse. Is equal
to the sign of 72

260
00:21:39,407 --> 00:21:40,128
degrees.

261
00:21:41,140 --> 00:21:48,331
H. Is the length
of the opposite side the height

262
00:21:48,331 --> 00:21:55,399
of the triangle were trying to
find over the length of the

263
00:21:55,399 --> 00:22:01,878
hypotenuse? That's 15 is equal
to sign 72 degrees, so H

264
00:22:01,878 --> 00:22:07,768
is 15 times the sign of
72 degrees. Multiplying both

265
00:22:07,768 --> 00:22:11,302
sides of this equation by 15.

266
00:22:11,330 --> 00:22:17,678
Again, without a stage where we
need our Calculator and so we

267
00:22:17,678 --> 00:22:21,660
want 15. Times sign

268
00:22:21,660 --> 00:22:28,668
of 72. And
the answer we have is 14.3

269
00:22:28,668 --> 00:22:33,260
and here we work to three
significant figures.

270
00:22:34,490 --> 00:22:39,698
Let's take one more example in
this area and the example it

271
00:22:39,698 --> 00:22:44,038
will take his where we're
finding aside, but perhaps it's

272
00:22:44,038 --> 00:22:46,642
not the one that we might

273
00:22:46,642 --> 00:22:50,180
expect. So there's
the right angle.

274
00:22:51,900 --> 00:22:59,271
Let's say this angle up here is
35 degrees. This side is of

275
00:22:59,271 --> 00:23:03,240
length 6 and what we want to

276
00:23:03,240 --> 00:23:06,765
know. Is how long is the

277
00:23:06,765 --> 00:23:11,556
hypotenuse? The side is the
hypotenuse 'cause it's opposite

278
00:23:11,556 --> 00:23:16,198
the right angle. It's the
longest side in the right angle

279
00:23:16,198 --> 00:23:21,953
triangle. This side, that's a
blend six, let's say 6

280
00:23:21,953 --> 00:23:27,310
centimeters since lens have to
have units with them, this is

281
00:23:27,310 --> 00:23:31,206
the adjacent side. It's adjacent
till this angle.

282
00:23:31,320 --> 00:23:38,448
What's the trig ratio that we
need that connects adjacent and

283
00:23:38,448 --> 00:23:46,224
hypotenuse? It's going to be the
cosine and so we have adjacent

284
00:23:46,224 --> 00:23:52,704
over hypotenuse is equal to the
cosine of 35 degrees.

285
00:23:53,780 --> 00:23:56,348
Adjacent side is 6.

286
00:23:57,090 --> 00:24:00,610
It's called the hypotenuse X.

287
00:24:01,470 --> 00:24:07,722
So six over X is equal to
the cosine of 35 degrees.

288
00:24:08,560 --> 00:24:14,698
So 6 must be X times the
cosine of 35 degrees,

289
00:24:14,698 --> 00:24:19,162
multiplying both sides of this
equation by X.

290
00:24:20,760 --> 00:24:24,577
This sometimes gives people a
little bit of trouble. The next

291
00:24:24,577 --> 00:24:29,782
step we've got 6 equals X times
cost 35 and it's the ex that I

292
00:24:29,782 --> 00:24:34,640
want. The thing that you have to
remember is cost 35 is just a

293
00:24:34,640 --> 00:24:36,722
number, just a number like two

294
00:24:36,722 --> 00:24:42,098
or three. So if this said six
was equal to two X, then we know

295
00:24:42,098 --> 00:24:45,890
that X is equal to three,
because we'd be able to divide

296
00:24:45,890 --> 00:24:47,154
the six by two.

297
00:24:48,010 --> 00:24:52,680
Cost 35 is no different,
and so we divide both

298
00:24:52,680 --> 00:24:56,883
sides by cause of 35
degrees that will be

299
00:24:56,883 --> 00:24:58,284
equal to X.

300
00:24:59,390 --> 00:25:06,474
So we need the Calculator
again. We have 6 divided by

301
00:25:06,474 --> 00:25:14,202
the cosine of 35 degrees and
the answer we get for X

302
00:25:14,202 --> 00:25:19,998
layer is 7.324. We shorten
that to three significant

303
00:25:19,998 --> 00:25:24,506
figures that street 7.32
centimeters to three

304
00:25:24,506 --> 00:25:25,794
significant figures.

305
00:25:27,830 --> 00:25:32,942
We've looked at some problems
and there are some more on the

306
00:25:32,942 --> 00:25:37,202
example sheet that accompanies
the video. One of the things

307
00:25:37,202 --> 00:25:42,740
that we do need to have a look
at, though, are some quite

308
00:25:42,740 --> 00:25:47,000
specific angles and their signs.
Cosines and tangents. And it's

309
00:25:47,000 --> 00:25:50,408
important that these are
learned, learned because they

310
00:25:50,408 --> 00:25:52,964
are exact and because
Mathematicians, scientists,

311
00:25:52,964 --> 00:25:56,798
engineers use these a great deal
in their work.

312
00:25:57,010 --> 00:26:02,257
The angles that we're talking
about, the angles 0.

313
00:26:03,330 --> 00:26:04,500
30

314
00:26:05,660 --> 00:26:06,870
45

315
00:26:07,890 --> 00:26:11,140
60 And 90.

316
00:26:11,700 --> 00:26:13,996
We're going to have a look at

317
00:26:13,996 --> 00:26:20,750
their signs. Their cosines
and their tangents.

318
00:26:21,690 --> 00:26:27,170
Let's have a look at a right
angled isosceles triangle.

319
00:26:27,680 --> 00:26:30,256
To begin with, there
is it right angle.

320
00:26:31,630 --> 00:26:35,920
Isosceles each of these sides
will be the same length.

321
00:26:36,650 --> 00:26:43,202
So if we use Pythagoras 1
squared plus, one squared is one

322
00:26:43,202 --> 00:26:49,754
plus one, which gives us two. So
the length of the hypotenuse

323
00:26:49,754 --> 00:26:51,938
must be Route 2.

324
00:26:54,200 --> 00:27:00,509
And each of these angles
because it's isosceles are

325
00:27:00,509 --> 00:27:01,911
45 degrees.

326
00:27:02,960 --> 00:27:10,016
So from that triangle, can we
read off the sign cause and

327
00:27:10,016 --> 00:27:12,368
tan of 45 degrees?

328
00:27:12,910 --> 00:27:18,202
Let's concentrate on this angle.
This is the angle we're going to

329
00:27:18,202 --> 00:27:20,407
concentrate on that 45 degrees.

330
00:27:21,210 --> 00:27:24,360
The sign of 45 is opposite.

331
00:27:24,930 --> 00:27:31,178
Over hypotenuse, and so that's
the opposite is one over the

332
00:27:31,178 --> 00:27:37,426
hypotenuse is Route 2, so that
is one over Route 2.

333
00:27:38,850 --> 00:27:41,874
The cosine, the cosine is the

334
00:27:41,874 --> 00:27:45,030
adjacent. Over the hypotenuse.

335
00:27:45,680 --> 00:27:51,272
The adjacent is one the
hypotenuse is Route 2 and so the

336
00:27:51,272 --> 00:27:54,068
cosine is one over Route 2.

337
00:27:55,090 --> 00:28:00,502
The tangent is the opposite
over the adjacent and so the

338
00:28:00,502 --> 00:28:06,898
opposite is one over the
adjacent is one 1 / 1 is just

339
00:28:06,898 --> 00:28:11,818
one. So we've completed that
middle column of our table.

340
00:28:13,150 --> 00:28:16,438
Let's now take an
equilateral triangle.

341
00:28:18,030 --> 00:28:22,866
All the sides are the same
length, all 2 units long and

342
00:28:22,866 --> 00:28:28,105
that means that all the angles
will be the same as well, so

343
00:28:28,105 --> 00:28:32,941
they will all be 60 degrees.
Haven't drawn in the top angle

344
00:28:32,941 --> 00:28:37,374
because I'm going to split my
equilateral triangle in half, so

345
00:28:37,374 --> 00:28:39,792
each of these will be 30.

346
00:28:40,360 --> 00:28:43,360
There is a right angle down

347
00:28:43,360 --> 00:28:50,548
there. I've split it in half, so
each piece of this is one unit.

348
00:28:51,430 --> 00:28:53,198
So there my angles.

349
00:28:53,710 --> 00:28:59,958
All the lengths are marked
except that one the height of

350
00:28:59,958 --> 00:29:01,662
this equilateral triangle.

351
00:29:02,330 --> 00:29:04,868
Let's workout how long that is.

352
00:29:05,620 --> 00:29:09,890
Pythagoras tells us that that
squared plus that squared should

353
00:29:09,890 --> 00:29:11,598
give us that squared.

354
00:29:12,510 --> 00:29:17,410
Well, that squared is just one
that squared is 4.

355
00:29:18,140 --> 00:29:25,548
To add 1 to something and get 4,
the answer has to be 3. So that

356
00:29:25,548 --> 00:29:31,567
means the length of this is
Route 3. The square root of 3.

357
00:29:32,650 --> 00:29:38,686
So now we can read off from
this right angle triangle signs,

358
00:29:38,686 --> 00:29:45,225
cosines and tangents for 30 and
for 60. So let's look at 30.

359
00:29:45,225 --> 00:29:50,758
The sign is the opposite over
the hypotenuse. The opposite is

360
00:29:50,758 --> 00:29:57,297
of length one, the hypotenuse of
length 2. So that is one over

361
00:29:57,297 --> 00:30:03,333
two half for the cosine, the
cosine is the adjacent over the

362
00:30:03,333 --> 00:30:09,044
hypotenuse. And so that is Route
3 over 2.

363
00:30:09,080 --> 00:30:11,819
Three over 2.

364
00:30:12,410 --> 00:30:18,290
For the tangent, that's the
opposite over the adjacent, so

365
00:30:18,290 --> 00:30:21,230
that's one over Route 3.

366
00:30:21,240 --> 00:30:24,866
Let's have a look at this one.

367
00:30:25,790 --> 00:30:33,088
60 Starting with the
sign sign is opposite over

368
00:30:33,088 --> 00:30:38,736
hypotenuse, so that is Route
3 over 2.

369
00:30:38,750 --> 00:30:45,934
Cosine is the adjacent
over the hypotenuse, so

370
00:30:45,934 --> 00:30:51,322
that is one over
2A half.

371
00:30:53,290 --> 00:30:59,698
The tangent is the opposite over
the adjacent and so that's Route

372
00:30:59,698 --> 00:31:03,436
3 over one or just Route 3.

373
00:31:04,680 --> 00:31:07,890
What about these bits, not 90?

374
00:31:09,210 --> 00:31:14,110
OK, let's take any right
angle triangle.

375
00:31:15,280 --> 00:31:19,180
And we'll fix

376
00:31:19,180 --> 00:31:23,180
on this. Angle.

377
00:31:24,740 --> 00:31:27,280
So that's the hypotenuse.

378
00:31:27,970 --> 00:31:31,990
That's the adjacent.
That's the opposite.

379
00:31:33,320 --> 00:31:39,394
OK. Sign is opposite over
hypotenuse. If we imagine this

380
00:31:39,394 --> 00:31:44,740
angle going down towards 0 like
that, it's clear that the

381
00:31:44,740 --> 00:31:51,058
opposite side is becoming zero
itself, and so we have zero or a

382
00:31:51,058 --> 00:31:55,432
number approaching 0 being
divided by the hypotenuse, which

383
00:31:55,432 --> 00:32:01,750
is pretty large. That would
suggest that the sign of 0 is 0.

384
00:32:02,570 --> 00:32:07,780
What about the cosine? As this
comes down, the hypotenuse

385
00:32:07,780 --> 00:32:12,990
becomes almost the same length
as the adjacent, which would

386
00:32:12,990 --> 00:32:16,637
suggest that the cosine of 0 is

387
00:32:16,637 --> 00:32:22,410
one. The tangent well, the
tangent is the opposite over the

388
00:32:22,410 --> 00:32:24,990
adjacent. So as this side goes

389
00:32:24,990 --> 00:32:31,732
down. The opposite side becomes
zero over a fixed side, so again

390
00:32:31,732 --> 00:32:35,680
0. What about 90?

391
00:32:36,710 --> 00:32:40,846
Let's imagine this angle begins
to get bigger and bigger and

392
00:32:40,846 --> 00:32:45,358
bigger so that it looks like
it's going to become 90. And

393
00:32:45,358 --> 00:32:49,494
what we can see is that the
opposite in the hypotenuse

394
00:32:49,494 --> 00:32:52,878
become almost the same,
suggesting that the sign would

395
00:32:52,878 --> 00:32:59,170
be 1. But as that is happening,
the adjacent is becoming 0.

396
00:32:59,670 --> 00:33:05,577
And the cosine is the adjacent
over the hypotenuse, and so

397
00:33:05,577 --> 00:33:08,262
again, that goes to 0.

398
00:33:09,270 --> 00:33:10,950
Finally, the Tangent.

399
00:33:12,210 --> 00:33:16,230
That adjacent side is becoming
zero, becoming very, very small,

400
00:33:16,230 --> 00:33:20,652
but this opposite side is
staying fixed, so we've got a

401
00:33:20,652 --> 00:33:25,074
fixed number and we're dividing
it by something which is getting

402
00:33:25,074 --> 00:33:29,898
very, very small. So the answer
is becoming very, very big. In

403
00:33:29,898 --> 00:33:34,320
other words, it's becoming
Infinite. We write that as an 8

404
00:33:34,320 --> 00:33:36,732
on its side and call it

405
00:33:36,732 --> 00:33:43,986
Infinity. So this is
a table of sines, cosines,

406
00:33:43,986 --> 00:33:48,618
and tangents that are all
exact.

407
00:33:49,710 --> 00:33:56,420
And for very specific angles,
not thirty 4560 and 90,

408
00:33:56,420 --> 00:34:02,459
these need to be learned.
They are important and.

409
00:34:03,180 --> 00:34:08,126
Textbooks. Scientists, engineers
in their everyday working and

410
00:34:08,126 --> 00:34:12,706
everyday calculations take these
as being read. They assume that

411
00:34:12,706 --> 00:34:18,660
they are known, and so you must
learn them and make sure that

412
00:34:18,660 --> 00:34:24,156
you have them ready to hand to
work with immediately. You hear

413
00:34:24,156 --> 00:34:29,194
sign 45 that's one over route
two 1060 that's Route 3.