1
00:00:00,012 --> 00:00:04,862
[music] What's trigonometry?
Well here's the basic idea.

2
00:00:04,862 --> 00:00:11,368
So I've got a bunch of right triangles,
these are all right triangles, and this

3
00:00:11,368 --> 00:00:16,848
angle is the same in all of them, but
they're all different sizes.

4
00:00:16,848 --> 00:00:23,741
So the side lengths are all different but
the ratios between the corresponding sides

5
00:00:23,741 --> 00:00:25,767
are the same.
What do I mean?

6
00:00:25,767 --> 00:00:30,777
Well, take a look at this big triangle.
It's got some height that I'll label in,

7
00:00:30,777 --> 00:00:34,559
in orange, and it's got some width that
I'll label in blue.

8
00:00:34,559 --> 00:00:39,126
Now the width of this triangle isn't the
same as this triangle, isn't the same as

9
00:00:39,126 --> 00:00:42,259
this triangle, isn't the same as this
small triangle.

10
00:00:42,259 --> 00:00:46,549
And the height of this triangle is
different than the heights of these three

11
00:00:46,549 --> 00:00:49,434
triangles.
But look at the ratio of this height to

12
00:00:49,434 --> 00:00:52,490
this width.
Alright, here's the height, here's the

13
00:00:52,490 --> 00:00:57,281
width, and if I double the width, Width.
It's a little bit more than the height.

14
00:00:57,281 --> 00:01:02,151
In other words the height is just a little
bit less than the twice the width and

15
00:01:02,151 --> 00:01:07,327
that's true for all of these triangles.
The ratio of their heights to their widths

16
00:01:07,327 --> 00:01:10,841
are all the same, even though they're
different sizes.

17
00:01:10,841 --> 00:01:14,111
That's the key fact that makes
trigonometry work.

18
00:01:14,111 --> 00:01:18,675
If you just draw some random right
triangle with given angles, it doesn't

19
00:01:18,675 --> 00:01:22,836
really make a lot of sense to ask
questions about the side lengths, it

20
00:01:22,836 --> 00:01:25,867
really depends on how big the triangle you
drew.

21
00:01:25,867 --> 00:01:30,683
But it does make sense to ask questions
about the ratios of side lengths, those

22
00:01:30,683 --> 00:01:34,712
only depend on the angles.
So we can give names to these ratios.

23
00:01:34,712 --> 00:01:40,455
We call the sine of theta as the ratio
between the height and the hypotenuse.

24
00:01:40,455 --> 00:01:44,133
So if you like, the opposite side and
hypotenuse.

25
00:01:44,133 --> 00:01:49,069
So y over r in this diagram.
The cosine of theta is the ratio between

26
00:01:49,069 --> 00:01:53,523
the adjacent side and the hypotenuse of
the right triangle.

27
00:01:53,523 --> 00:01:57,616
It's x over r in this picture.
And the tangent of theta.

28
00:01:57,616 --> 00:02:03,254
That's the opposite side over the adjacent
side, it's y over x in this picture

29
00:02:03,254 --> 00:02:07,246
because the sine, cosine, and tangent of
this angle theta.

30
00:02:07,246 --> 00:02:11,811
And they're all just defined in terms of
ratios of side lengths.

31
00:02:11,811 --> 00:02:17,727
Now that we've got a definition of sine.
We can, for instance, calculate sine of pi

32
00:02:17,727 --> 00:02:22,202
over 5.
If we build a triangle, any right triangle

33
00:02:22,202 --> 00:02:27,486
with an angle of measure pi over 5.
I built a right triangle.

34
00:02:27,486 --> 00:02:33,284
And this angle is pi over 5.
I can use my little ruler to measure the

35
00:02:33,284 --> 00:02:37,379
hypotenuse.
The hypotenuse here is about 25.2

36
00:02:37,379 --> 00:02:42,894
centimeters.
And I can measure the opposite side, and

37
00:02:42,894 --> 00:02:47,645
the opposite side's about 14.8
centimetres.

38
00:02:47,645 --> 00:02:55,651
And sine is the opposite side over the
hypotenuse, which means sine of pi over

39
00:02:55,651 --> 00:03:07,366
five is about 14.8.
Over 25.2, which is about 0.59.

40
00:03:07,366 --> 00:03:12,065
.