0:00:00.830,0:00:05.426
In this video, we're going to be[br]having a look at trigonometric

0:00:05.426,0:00:09.256
ratios in a right angle[br]triangle, and that's a long

0:00:09.256,0:00:13.086
phrase trigonometric ratios, so[br]we'll talk about them just as

0:00:13.086,0:00:17.364
trig ratios. What I've got here[br]in this diagram is.

0:00:18.370,0:00:23.739
Straight line and marked off are[br]the points A1A Two and a three

0:00:23.739,0:00:28.695
and each one of these is 10[br]centimeters on from the next

0:00:28.695,0:00:32.412
one. So that's 10 centimeters.[br]That's 10 centimeters, and

0:00:32.412,0:00:36.955
that's 10 centimeters, and I've[br]made a right angle triangle by

0:00:36.955,0:00:40.259
drawing vertically up from each[br]of these points.

0:00:40.920,0:00:42.960
To meet this line here.

0:00:43.800,0:00:49.212
So in effect what I've got is an[br]angle here at oh.

0:00:49.840,0:00:55.391
An angle here at oh and I'm[br]extending the two arms of the

0:00:55.391,0:01:00.515
angle like that, but the angle[br]staying the same even though the

0:01:00.515,0:01:04.785
separation is getting bigger,[br]the angle is staying the same.

0:01:04.785,0:01:09.909
what I want to have a look at is[br]what's the ratio?

0:01:10.500,0:01:13.770
Of that length till that length.

0:01:14.580,0:01:17.469
As we extend.

0:01:18.080,0:01:21.746
So that length till that length.

0:01:22.450,0:01:25.880
And then that one till the whole

0:01:25.880,0:01:32.812
of that. I want to have a[br]look at that and see what it

0:01:32.812,0:01:38.164
actually does. So going to use a[br]ruler, do some measurement, but

0:01:38.164,0:01:42.624
these are fixed at 10[br]centimeters each. So first of

0:01:42.624,0:01:47.530
all, let's have a look at a 1B[br]one divided by.

0:01:48.220,0:01:50.650
Oh a one.

0:01:51.600,0:01:55.460
Well, the bottom is definitely[br]10 centimeters. We've had that

0:01:55.460,0:01:59.320
before. If I measure this one[br]with the ruler, it's.

0:02:00.110,0:02:07.935
6.1 centimeters.[br]So that's not .61. Now

0:02:07.935,0:02:15.565
let's have a look at[br]the same ratio for this

0:02:15.565,0:02:22.432
extended angle is bigger[br]triangle, so will take A2B2

0:02:22.432,0:02:25.864
over. 082 equals

0:02:25.864,0:02:32.796
now this. Oh[br]A2 is 20 centimeters over

0:02:32.796,0:02:39.956
20 and take our ruler[br]and will measure this. That's

0:02:39.956,0:02:47.116
about 12 point 212.2 we[br]do the division and we've

0:02:47.116,0:02:49.980
got nought .61 again.

0:02:50.540,0:02:58.052
This bottom one just to check[br]is 30 centimeters long, so we

0:02:58.052,0:03:04.312
want a 3B three over 08,[br]three, 30 centimeters. We've

0:03:04.312,0:03:09.946
just measured it, and if we[br]measure again here.

0:03:10.610,0:03:13.578
This is 18 point.

0:03:14.340,0:03:21.866
2. So 18.2 centimeters.[br]And if we do, the division

0:03:21.866,0:03:23.858
that is not.

0:03:24.540,0:03:26.840
.6.

0:03:27.920,0:03:34.820
Oh, and it's going to[br]be 6 recurring, which is

0:03:34.820,0:03:36.890
about N .61.

0:03:37.440,0:03:44.314
So what we see is that all of[br]these three ratios are the same.

0:03:44.314,0:03:50.697
So if we take the side in a[br]right angle triangle which is

0:03:50.697,0:03:52.661
opposite to an angle.

0:03:53.220,0:03:59.148
And we divide it by the side[br]which is adjacent to it.

0:04:00.410,0:04:08.390
Then opposite over[br]adjacent seems to

0:04:08.390,0:04:11.050
be constant.

0:04:11.620,0:04:15.160
For the. Given.

0:04:15.990,0:04:21.260
Angle. And indeed[br]it is.

0:04:22.400,0:04:29.652
And we have a name for it.[br]They call it the tangent of the

0:04:29.652,0:04:37.180
angle. So.[br]Take a right angle triangle.

0:04:39.740,0:04:44.480
Right angle there and let's look[br]at this angle here. Label it.

0:04:44.480,0:04:49.220
Angle a so we've got a clear[br]picture that that's the angle

0:04:49.220,0:04:50.405
we're talking about.

0:04:50.930,0:04:57.782
Let's give names now to the[br]sides. The longest side of a

0:04:57.782,0:05:03.492
right angle triangle, the one[br]that's opposite to the right

0:05:03.492,0:05:06.347
angle is always called the

0:05:06.347,0:05:13.210
hypotenuse. So then[br]we have

0:05:13.210,0:05:18.970
the hypotenuse.[br]Usually shorten

0:05:18.970,0:05:23.290
it till[br]HYP.

0:05:24.390,0:05:31.332
This side. Is opposite[br]this angle. It's across from it,

0:05:31.332,0:05:37.860
it's opposite to it, and so we[br]call that the opposite side,

0:05:37.860,0:05:41.124
which we shorten till O Double

0:05:41.124,0:05:48.400
P. Inside is a part of[br]this angle, so it is adjacent.

0:05:48.920,0:05:50.360
Till the angle.

0:05:51.050,0:05:53.864
And we shorten this to a DJ.

0:05:54.810,0:06:00.574
If I take not this angle[br]but that angle, then some

0:06:00.574,0:06:01.622
things change.

0:06:02.730,0:06:05.510
Let's draw the diagram again.

0:06:07.130,0:06:11.874
This time let's focus on[br]the angle B.

0:06:13.240,0:06:18.190
This side is still the[br]longest side and is still

0:06:18.190,0:06:23.140
opposite the right angle, so[br]it is still the hypotenuse.

0:06:24.340,0:06:30.892
But it's now this side which is[br]opposite across from the angle

0:06:30.892,0:06:34.714
be. So this side is now the

0:06:34.714,0:06:40.890
opposite side. It's now this[br]side which is a part of the

0:06:40.890,0:06:45.840
angle which is alongside the[br]angle which is adjacent to the

0:06:45.840,0:06:50.879
angle. So if we change the[br]angle that we're talking

0:06:50.879,0:06:54.578
about in a right angle[br]triangle, we change the

0:06:54.578,0:06:54.989
labels.

0:06:56.850,0:07:02.754
They're not fixed until we have[br]fixed appan. The angle that we

0:07:02.754,0:07:07.674
are talking about. Let's now[br]take a little step back.

0:07:07.674,0:07:10.134
Remember, at the beginning of

0:07:10.134,0:07:12.860
the video. We said

0:07:14.750,0:07:18.260
But if we picked our angle.

0:07:19.640,0:07:23.588
And we took the opposite side.

0:07:24.300,0:07:26.928
And the adjacent side.

0:07:27.540,0:07:29.928
Then the ratio.

0:07:30.550,0:07:36.990
The opposite to the[br]adjacent side was a

0:07:36.990,0:07:42.365
constant. And that we call[br]that the Tangent.

0:07:43.910,0:07:49.574
Of the angle, a well[br]tangents along word

0:07:49.574,0:07:55.238
we tend to shorten[br]that simply to tan.

0:07:57.250,0:08:02.160
What other ratios are there in[br]this right angle triangle?

0:08:03.510,0:08:07.270
Let's label this side as

0:08:07.270,0:08:13.085
the hypotenuse. And the other[br]important ratios that we need to

0:08:13.085,0:08:17.540
know about and be able to work[br]with are the opposite.

0:08:18.060,0:08:21.226
Over the

0:08:21.226,0:08:27.256
hypotenuse. Which is the sign[br]of the angle a?

0:08:28.210,0:08:31.618
And for all we say, sign.

0:08:32.150,0:08:38.481
We shorten it in written form[br]till sin sin. We don't say that

0:08:38.481,0:08:39.942
we say sign.

0:08:41.040,0:08:48.030
The third of the major[br]trig ratios is the adjacent.

0:08:48.610,0:08:56.070
Over.[br]The hypotenuse and this is

0:08:56.070,0:08:59.930
the cosine of the angle

0:08:59.930,0:09:03.795
a. We shorten that to cause

0:09:03.795,0:09:09.878
a. Now all of these have been[br]worked out in tables and before

0:09:09.878,0:09:12.910
calculators mathematicians used[br]to work these out quite

0:09:12.910,0:09:16.700
regularly and published books of[br]tables of the sines, cosines,

0:09:16.700,0:09:21.248
and tangents of all the angles[br]through from North up to 90

0:09:21.248,0:09:25.654
degrees. In doing all the[br]calculations, things like a

0:09:25.654,0:09:30.417
Calculator are invaluable and[br]you really do need want to be

0:09:30.417,0:09:32.149
able to work with.

0:09:33.140,0:09:37.952
What we're going to have a look[br]at now are some calculations

0:09:37.952,0:09:42.363
working out the sides and angles[br]of triangles when given certain

0:09:42.363,0:09:43.566
information about them.

0:09:44.430,0:09:51.930
When doing this, it does help[br]if you have a way of

0:09:51.930,0:09:58.180
recalling or remembering these[br]definitions. One of the ways is

0:09:58.180,0:10:00.055
a nonsense word.

0:10:00.160,0:10:02.278
So a toca.

0:10:02.870,0:10:08.430
Sign is opposite over[br]hypotenuse, tangent is opposite

0:10:08.430,0:10:14.685
over adjacent and cosine is[br]adjacent over hypotenuse. Summer

0:10:14.685,0:10:20.940
talker. Some people remember it[br]as soccer to us.

0:10:20.960,0:10:23.648
Simply changing the syllables

0:10:23.648,0:10:30.486
around. Still means the[br]same thing. Others remember it

0:10:30.486,0:10:33.996
by a little verse Toms

0:10:33.996,0:10:40.900
old aunt. Sat[br]on him.

0:10:41.950,0:10:48.114
Cursing At him and[br]again, tangent is opposite

0:10:48.114,0:10:52.328
over adjacent sign is[br]opposite over hypotenuse,

0:10:52.328,0:10:56.542
cosine is adjacent over[br]hypotenuse. Whichever you

0:10:56.542,0:11:02.562
learn to help you remember[br]these ratios, that's up to

0:11:02.562,0:11:09.786
you, but it is helpful to[br]have one of them. So let's

0:11:09.786,0:11:11.592
have some problems.

0:11:12.670,0:11:16.918
Many of the examples that we[br]want to look at actually come

0:11:16.918,0:11:20.458
from the very practical area of[br]surveying. The problem of

0:11:20.458,0:11:23.644
finding out the size of[br]something, the length of

0:11:23.644,0:11:26.476
something when you can't[br]actually measure it. Typically,

0:11:26.476,0:11:30.016
perhaps the height of a tower,[br]so we're still here.

0:11:30.720,0:11:34.130
And we want to look[br]at a tower over here.

0:11:35.280,0:11:38.543
And how high is not tower? We[br]can't measure it. We can't get

0:11:38.543,0:11:42.810
to the top of it. But how can we[br]find out how high it is? One of

0:11:42.810,0:11:46.073
the ways is to take a sighting[br]of the top of the tower.

0:11:46.890,0:11:50.870
And with an instrument surveying[br]instrument, we can measure that

0:11:50.870,0:11:55.646
angle to the horizontal there.[br]So let's say we do make that

0:11:55.646,0:12:00.024
measurement, and we find that[br]it's 32 degrees. We can also

0:12:00.024,0:12:05.198
find how far away we were stood[br]from the base of the tower.

0:12:05.198,0:12:09.178
That's a relatively easy[br]measurement to make as well. So

0:12:09.178,0:12:11.566
let's say that was five meters.

0:12:12.910,0:12:17.770
Somebody about 6 foot tall[br]probably measures about 1.72

0:12:17.770,0:12:19.930
meters there or thereabouts.

0:12:20.660,0:12:22.008
So we're set up.

0:12:22.550,0:12:27.581
We know how far away from the[br]tower we stood. We know this

0:12:27.581,0:12:31.838
angle, sometimes called the[br]angle of elevation of the top of

0:12:31.838,0:12:34.547
the tower from our IO. How high

0:12:34.547,0:12:39.961
is this? Well, here we've got a[br]right angle triangle and we know

0:12:39.961,0:12:45.526
one side and we know an angle[br]and this is the side we want to

0:12:45.526,0:12:49.978
find. It's the side that is[br]opposite to the angle, so let's

0:12:49.978,0:12:52.204
label it as the opposite side.

0:12:52.850,0:12:56.010
This side is 5 meters.

0:12:56.540,0:13:02.676
It's the side that we know, and[br]it's also adjacent to the angle.

0:13:03.830,0:13:08.996
So what trig ratio links[br]opposite and adjacent? Well,

0:13:08.996,0:13:15.310
the answer to that is it's[br]the tangent, so the opposite

0:13:15.310,0:13:20.476
over the adjacent. Is that an[br]of 32 degrees?

0:13:22.100,0:13:26.435
Right, we are in a position now[br]to be able to solve this. To be

0:13:26.435,0:13:27.880
able to substitute in values

0:13:27.880,0:13:32.802
that we know. And placeholders[br]for those that we don't know. So

0:13:32.802,0:13:34.642
the opposite side, let's call

0:13:34.642,0:13:40.620
that X. So that's X for the[br]length of the opposite side

0:13:40.620,0:13:45.210
over 5 meters is the length[br]of the adjacent side.

0:13:46.650,0:13:51.890
Equals[br]1032.

0:13:53.040,0:13:59.616
Defined X, we multiply[br]both sides by 5,

0:13:59.616,0:14:02.904
so X is 5

0:14:02.904,0:14:09.990
* 1032. And now this is[br]where we need our Calculator 5.

0:14:11.170,0:14:14.010
Times. 10

0:14:14.510,0:14:21.977
32 Equals and the[br]answer we've got there is 3.124

0:14:21.977,0:14:26.801
and then some more decimal[br]places after it.

0:14:28.520,0:14:30.417
One word of warning, by the way.

0:14:31.110,0:14:36.240
Most calculators operate with[br]angles measured in one of two

0:14:36.240,0:14:42.100
ways. Degrees or radians. You[br]need to make sure that your

0:14:42.100,0:14:46.920
Calculator is operating with the[br]right measurement of the angle.

0:14:46.920,0:14:51.258
I'm using degrees here 32[br]degrees, so my Calculator

0:14:51.258,0:14:57.042
Calculator needs to be set up so[br]as to operate in degrees.

0:14:57.760,0:15:02.356
The answer to my question, how[br]high is this tower? We haven't

0:15:02.356,0:15:07.335
quite got yet. The reason we[br]haven't got it is we've only got

0:15:07.335,0:15:12.314
this length here and so the[br]height of the tower we can find.

0:15:12.340,0:15:15.210
Adding on.

0:15:16.570,0:15:24.270
The height of our eye[br]above ground level, so that

0:15:24.270,0:15:27.350
gives us 4.844 meters.

0:15:27.890,0:15:32.653
This is a very accurate[br]answer. It says that we know

0:15:32.653,0:15:37.416
the height of this tower till,[br]well, a millimeter. This place

0:15:37.416,0:15:41.746
here is 4 millimeters. That's[br]far too accurate. We would

0:15:41.746,0:15:47.375
probably be happy to say this[br]is 4.8 meters and work to two

0:15:47.375,0:15:48.241
significant figures.

0:15:50.350,0:15:54.760
Let's have a look[br]at another example.

0:15:56.040,0:15:59.712
Here we've got an equilateral[br]triangle. Well, not

0:15:59.712,0:16:02.925
equilateral. Sorry, let's make[br]it isosceles. Isosceles

0:16:02.925,0:16:08.892
triangle has 2 equal sides and[br]in this case will say that we

0:16:08.892,0:16:12.105
know this side is of length[br]10.

0:16:13.160,0:16:20.496
And we know that[br]this side is of

0:16:20.496,0:16:26.202
length 12. You have the[br]units of measurement to be

0:16:26.202,0:16:30.701
meters. Now it's an isosceles[br]triangle and the question is how

0:16:30.701,0:16:35.609
big is that angle? That doesn't[br]seem to be a right angle.

0:16:36.770,0:16:41.558
In questions like this, you've[br]got to look to introduce a right

0:16:41.558,0:16:46.346
angle, and here if we divide the[br]isosceles triangle in half down

0:16:46.346,0:16:51.533
the middle, this is our right[br]angle, and so here we've got a

0:16:51.533,0:16:52.730
right angle triangle.

0:16:53.380,0:16:59.665
And we know that if this is 10[br]and this is half of 10 five

0:16:59.665,0:17:05.450
meters. Let's look at these[br]sides and label them. This side

0:17:05.450,0:17:10.455
is opposite the right angle.[br]It's the longest side in the

0:17:10.455,0:17:13.185
right angle triangle, so it is

0:17:13.185,0:17:20.290
the hypotenuse. This side, which[br]we also know is alongside the

0:17:20.290,0:17:23.290
angle it's adjacent to the

0:17:23.290,0:17:28.391
angle. Let's give this[br]angle ylabel, let's call

0:17:28.391,0:17:30.483
it the angle a.

0:17:31.840,0:17:38.368
What trig ratio of the angle a[br]connects the adjacent side and

0:17:38.368,0:17:44.352
the hypotenuse? Well, that's the[br]cosine, so the cosine of A.

0:17:45.130,0:17:49.120
Is equal to the adjacent.

0:17:49.640,0:17:53.420
Over. The hypotenuse.

0:17:54.510,0:18:00.659
And we can substitute the[br]numbers in the adjacent is 5.

0:18:00.850,0:18:03.570
The hypotenuse is 12.

0:18:04.870,0:18:10.918
Now I've got a problem. Here it[br]says cause I is equal to five

0:18:10.918,0:18:15.670
over 12 and I want the angle a[br]angle A is.

0:18:16.380,0:18:21.896
The angle whose cosine is 5 over[br]12. I've got a problem about how

0:18:21.896,0:18:23.866
I might write that down.

0:18:24.410,0:18:27.966
Do it. I write it like this.

0:18:28.550,0:18:35.030
Calls to the minus one[br]of five over 12.

0:18:35.530,0:18:42.130
What that bit there this[br]caused to the minus one says

0:18:42.130,0:18:48.730
is the angle whose cosine[br]is. So let's put an arrow

0:18:48.730,0:18:54.130
on to that and write that[br]down the angle.

0:18:56.420,0:18:58.595
Who's

0:18:58.595,0:19:05.109
cosine? Is that's[br]what this piece of notation

0:19:05.109,0:19:08.811
means. Now I've got to work

0:19:08.811,0:19:12.768
that out. So I need to look at

0:19:12.768,0:19:16.675
my Calculator. And we're[br]looking for the angle whose

0:19:16.675,0:19:21.415
cosine is, and we can see[br]that as cost to the minus

0:19:21.415,0:19:26.155
one, which is just along the[br]top of the cause button, and

0:19:26.155,0:19:30.500
it's on the shift key there.[br]For the second function key.

0:19:30.500,0:19:34.845
So in order to calculate it[br]we have to press shift.

0:19:36.650,0:19:40.406
Cosine key, we see that we get[br]cost to the minus one.

0:19:41.220,0:19:48.660
Bracket 5 / 12 closed[br]bracket and that gives us

0:19:48.660,0:19:56.100
the angle of 65.3756 etc.[br]Degrees and I probably give

0:19:56.100,0:20:03.540
this to the nearest 10th[br]of a degree. In other

0:20:03.540,0:20:07.260
words to one decimal place.

0:20:07.680,0:20:12.420
Let's take another[br]example.

0:20:16.670,0:20:21.518
This time will take the[br]isosceles triangle again.

0:20:22.110,0:20:29.430
15 meters for that side,[br]and it costs an isosceles

0:20:29.430,0:20:36.750
triangle with those two sides[br]equal. That one will also

0:20:36.750,0:20:43.338
be 15 meters and will[br]say that's 72 degrees.

0:20:43.350,0:20:48.510
Question, how high is the[br]triangle here is the height.

0:20:49.520,0:20:55.235
That's what we're after. The[br]height of that triangle.

0:20:56.520,0:21:02.848
So let's label our sides this[br]side, the one we want to find is

0:21:02.848,0:21:08.272
across from the angle. It is[br]opposite to the angle, so it

0:21:08.272,0:21:13.244
becomes the opposite side. This[br]side here is opposite the right

0:21:13.244,0:21:18.668
angle. It's the longest side in[br]the right angle triangle, and so

0:21:18.668,0:21:20.024
it's the hypotenuse.

0:21:20.570,0:21:24.609
What trig ratio connects[br]opposite and hypotenuse?

0:21:25.300,0:21:29.230
The answer is sign and so

0:21:29.230,0:21:32.480
the opposite. Over the

0:21:32.480,0:21:39.407
hypotenuse. Is equal[br]to the sign of 72

0:21:39.407,0:21:40.128
degrees.

0:21:41.140,0:21:48.331
H. Is the length[br]of the opposite side the height

0:21:48.331,0:21:55.399
of the triangle were trying to[br]find over the length of the

0:21:55.399,0:22:01.878
hypotenuse? That's 15 is equal[br]to sign 72 degrees, so H

0:22:01.878,0:22:07.768
is 15 times the sign of[br]72 degrees. Multiplying both

0:22:07.768,0:22:11.302
sides of this equation by 15.

0:22:11.330,0:22:17.678
Again, without a stage where we[br]need our Calculator and so we

0:22:17.678,0:22:21.660
want 15. Times sign

0:22:21.660,0:22:28.668
of 72. And[br]the answer we have is 14.3

0:22:28.668,0:22:33.260
and here we work to three[br]significant figures.

0:22:34.490,0:22:39.698
Let's take one more example in[br]this area and the example it

0:22:39.698,0:22:44.038
will take his where we're[br]finding aside, but perhaps it's

0:22:44.038,0:22:46.642
not the one that we might

0:22:46.642,0:22:50.180
expect. So there's[br]the right angle.

0:22:51.900,0:22:59.271
Let's say this angle up here is[br]35 degrees. This side is of

0:22:59.271,0:23:03.240
length 6 and what we want to

0:23:03.240,0:23:06.765
know. Is how long is the

0:23:06.765,0:23:11.556
hypotenuse? The side is the[br]hypotenuse 'cause it's opposite

0:23:11.556,0:23:16.198
the right angle. It's the[br]longest side in the right angle

0:23:16.198,0:23:21.953
triangle. This side, that's a[br]blend six, let's say 6

0:23:21.953,0:23:27.310
centimeters since lens have to[br]have units with them, this is

0:23:27.310,0:23:31.206
the adjacent side. It's adjacent[br]till this angle.

0:23:31.320,0:23:38.448
What's the trig ratio that we[br]need that connects adjacent and

0:23:38.448,0:23:46.224
hypotenuse? It's going to be the[br]cosine and so we have adjacent

0:23:46.224,0:23:52.704
over hypotenuse is equal to the[br]cosine of 35 degrees.

0:23:53.780,0:23:56.348
Adjacent side is 6.

0:23:57.090,0:24:00.610
It's called the hypotenuse X.

0:24:01.470,0:24:07.722
So six over X is equal to[br]the cosine of 35 degrees.

0:24:08.560,0:24:14.698
So 6 must be X times the[br]cosine of 35 degrees,

0:24:14.698,0:24:19.162
multiplying both sides of this[br]equation by X.

0:24:20.760,0:24:24.577
This sometimes gives people a[br]little bit of trouble. The next

0:24:24.577,0:24:29.782
step we've got 6 equals X times[br]cost 35 and it's the ex that I

0:24:29.782,0:24:34.640
want. The thing that you have to[br]remember is cost 35 is just a

0:24:34.640,0:24:36.722
number, just a number like two

0:24:36.722,0:24:42.098
or three. So if this said six[br]was equal to two X, then we know

0:24:42.098,0:24:45.890
that X is equal to three,[br]because we'd be able to divide

0:24:45.890,0:24:47.154
the six by two.

0:24:48.010,0:24:52.680
Cost 35 is no different,[br]and so we divide both

0:24:52.680,0:24:56.883
sides by cause of 35[br]degrees that will be

0:24:56.883,0:24:58.284
equal to X.

0:24:59.390,0:25:06.474
So we need the Calculator[br]again. We have 6 divided by

0:25:06.474,0:25:14.202
the cosine of 35 degrees and[br]the answer we get for X

0:25:14.202,0:25:19.998
layer is 7.324. We shorten[br]that to three significant

0:25:19.998,0:25:24.506
figures that street 7.32[br]centimeters to three

0:25:24.506,0:25:25.794
significant figures.

0:25:27.830,0:25:32.942
We've looked at some problems[br]and there are some more on the

0:25:32.942,0:25:37.202
example sheet that accompanies[br]the video. One of the things

0:25:37.202,0:25:42.740
that we do need to have a look[br]at, though, are some quite

0:25:42.740,0:25:47.000
specific angles and their signs.[br]Cosines and tangents. And it's

0:25:47.000,0:25:50.408
important that these are[br]learned, learned because they

0:25:50.408,0:25:52.964
are exact and because[br]Mathematicians, scientists,

0:25:52.964,0:25:56.798
engineers use these a great deal[br]in their work.

0:25:57.010,0:26:02.257
The angles that we're talking[br]about, the angles 0.

0:26:03.330,0:26:04.500
30

0:26:05.660,0:26:06.870
45

0:26:07.890,0:26:11.140
60 And 90.

0:26:11.700,0:26:13.996
We're going to have a look at

0:26:13.996,0:26:20.750
their signs. Their cosines[br]and their tangents.

0:26:21.690,0:26:27.170
Let's have a look at a right[br]angled isosceles triangle.

0:26:27.680,0:26:30.256
To begin with, there[br]is it right angle.

0:26:31.630,0:26:35.920
Isosceles each of these sides[br]will be the same length.

0:26:36.650,0:26:43.202
So if we use Pythagoras 1[br]squared plus, one squared is one

0:26:43.202,0:26:49.754
plus one, which gives us two. So[br]the length of the hypotenuse

0:26:49.754,0:26:51.938
must be Route 2.

0:26:54.200,0:27:00.509
And each of these angles[br]because it's isosceles are

0:27:00.509,0:27:01.911
45 degrees.

0:27:02.960,0:27:10.016
So from that triangle, can we[br]read off the sign cause and

0:27:10.016,0:27:12.368
tan of 45 degrees?

0:27:12.910,0:27:18.202
Let's concentrate on this angle.[br]This is the angle we're going to

0:27:18.202,0:27:20.407
concentrate on that 45 degrees.

0:27:21.210,0:27:24.360
The sign of 45 is opposite.

0:27:24.930,0:27:31.178
Over hypotenuse, and so that's[br]the opposite is one over the

0:27:31.178,0:27:37.426
hypotenuse is Route 2, so that[br]is one over Route 2.

0:27:38.850,0:27:41.874
The cosine, the cosine is the

0:27:41.874,0:27:45.030
adjacent. Over the hypotenuse.

0:27:45.680,0:27:51.272
The adjacent is one the[br]hypotenuse is Route 2 and so the

0:27:51.272,0:27:54.068
cosine is one over Route 2.

0:27:55.090,0:28:00.502
The tangent is the opposite[br]over the adjacent and so the

0:28:00.502,0:28:06.898
opposite is one over the[br]adjacent is one 1 / 1 is just

0:28:06.898,0:28:11.818
one. So we've completed that[br]middle column of our table.

0:28:13.150,0:28:16.438
Let's now take an[br]equilateral triangle.

0:28:18.030,0:28:22.866
All the sides are the same[br]length, all 2 units long and

0:28:22.866,0:28:28.105
that means that all the angles[br]will be the same as well, so

0:28:28.105,0:28:32.941
they will all be 60 degrees.[br]Haven't drawn in the top angle

0:28:32.941,0:28:37.374
because I'm going to split my[br]equilateral triangle in half, so

0:28:37.374,0:28:39.792
each of these will be 30.

0:28:40.360,0:28:43.360
There is a right angle down

0:28:43.360,0:28:50.548
there. I've split it in half, so[br]each piece of this is one unit.

0:28:51.430,0:28:53.198
So there my angles.

0:28:53.710,0:28:59.958
All the lengths are marked[br]except that one the height of

0:28:59.958,0:29:01.662
this equilateral triangle.

0:29:02.330,0:29:04.868
Let's workout how long that is.

0:29:05.620,0:29:09.890
Pythagoras tells us that that[br]squared plus that squared should

0:29:09.890,0:29:11.598
give us that squared.

0:29:12.510,0:29:17.410
Well, that squared is just one[br]that squared is 4.

0:29:18.140,0:29:25.548
To add 1 to something and get 4,[br]the answer has to be 3. So that

0:29:25.548,0:29:31.567
means the length of this is[br]Route 3. The square root of 3.

0:29:32.650,0:29:38.686
So now we can read off from[br]this right angle triangle signs,

0:29:38.686,0:29:45.225
cosines and tangents for 30 and[br]for 60. So let's look at 30.

0:29:45.225,0:29:50.758
The sign is the opposite over[br]the hypotenuse. The opposite is

0:29:50.758,0:29:57.297
of length one, the hypotenuse of[br]length 2. So that is one over

0:29:57.297,0:30:03.333
two half for the cosine, the[br]cosine is the adjacent over the

0:30:03.333,0:30:09.044
hypotenuse. And so that is Route[br]3 over 2.

0:30:09.080,0:30:11.819
Three over 2.

0:30:12.410,0:30:18.290
For the tangent, that's the[br]opposite over the adjacent, so

0:30:18.290,0:30:21.230
that's one over Route 3.

0:30:21.240,0:30:24.866
Let's have a look at this one.

0:30:25.790,0:30:33.088
60 Starting with the[br]sign sign is opposite over

0:30:33.088,0:30:38.736
hypotenuse, so that is Route[br]3 over 2.

0:30:38.750,0:30:45.934
Cosine is the adjacent[br]over the hypotenuse, so

0:30:45.934,0:30:51.322
that is one over[br]2A half.

0:30:53.290,0:30:59.698
The tangent is the opposite over[br]the adjacent and so that's Route

0:30:59.698,0:31:03.436
3 over one or just Route 3.

0:31:04.680,0:31:07.890
What about these bits, not 90?

0:31:09.210,0:31:14.110
OK, let's take any right[br]angle triangle.

0:31:15.280,0:31:19.180
And we'll fix

0:31:19.180,0:31:23.180
on this. Angle.

0:31:24.740,0:31:27.280
So that's the hypotenuse.

0:31:27.970,0:31:31.990
That's the adjacent.[br]That's the opposite.

0:31:33.320,0:31:39.394
OK. Sign is opposite over[br]hypotenuse. If we imagine this

0:31:39.394,0:31:44.740
angle going down towards 0 like[br]that, it's clear that the

0:31:44.740,0:31:51.058
opposite side is becoming zero[br]itself, and so we have zero or a

0:31:51.058,0:31:55.432
number approaching 0 being[br]divided by the hypotenuse, which

0:31:55.432,0:32:01.750
is pretty large. That would[br]suggest that the sign of 0 is 0.

0:32:02.570,0:32:07.780
What about the cosine? As this[br]comes down, the hypotenuse

0:32:07.780,0:32:12.990
becomes almost the same length[br]as the adjacent, which would

0:32:12.990,0:32:16.637
suggest that the cosine of 0 is

0:32:16.637,0:32:22.410
one. The tangent well, the[br]tangent is the opposite over the

0:32:22.410,0:32:24.990
adjacent. So as this side goes

0:32:24.990,0:32:31.732
down. The opposite side becomes[br]zero over a fixed side, so again

0:32:31.732,0:32:35.680
0. What about 90?

0:32:36.710,0:32:40.846
Let's imagine this angle begins[br]to get bigger and bigger and

0:32:40.846,0:32:45.358
bigger so that it looks like[br]it's going to become 90. And

0:32:45.358,0:32:49.494
what we can see is that the[br]opposite in the hypotenuse

0:32:49.494,0:32:52.878
become almost the same,[br]suggesting that the sign would

0:32:52.878,0:32:59.170
be 1. But as that is happening,[br]the adjacent is becoming 0.

0:32:59.670,0:33:05.577
And the cosine is the adjacent[br]over the hypotenuse, and so

0:33:05.577,0:33:08.262
again, that goes to 0.

0:33:09.270,0:33:10.950
Finally, the Tangent.

0:33:12.210,0:33:16.230
That adjacent side is becoming[br]zero, becoming very, very small,

0:33:16.230,0:33:20.652
but this opposite side is[br]staying fixed, so we've got a

0:33:20.652,0:33:25.074
fixed number and we're dividing[br]it by something which is getting

0:33:25.074,0:33:29.898
very, very small. So the answer[br]is becoming very, very big. In

0:33:29.898,0:33:34.320
other words, it's becoming[br]Infinite. We write that as an 8

0:33:34.320,0:33:36.732
on its side and call it

0:33:36.732,0:33:43.986
Infinity. So this is[br]a table of sines, cosines,

0:33:43.986,0:33:48.618
and tangents that are all[br]exact.

0:33:49.710,0:33:56.420
And for very specific angles,[br]not thirty 4560 and 90,

0:33:56.420,0:34:02.459
these need to be learned.[br]They are important and.

0:34:03.180,0:34:08.126
Textbooks. Scientists, engineers[br]in their everyday working and

0:34:08.126,0:34:12.706
everyday calculations take these[br]as being read. They assume that

0:34:12.706,0:34:18.660
they are known, and so you must[br]learn them and make sure that

0:34:18.660,0:34:24.156
you have them ready to hand to[br]work with immediately. You hear

0:34:24.156,0:34:29.194
sign 45 that's one over route[br]two 1060 that's Route 3.