[music] What's trigonometry?
Well here's the basic idea.
So I've got a bunch of right triangles,
these are all right triangles, and this
angle is the same in all of them, but
they're all different sizes.
So the side lengths are all different but
the ratios between the corresponding sides
are the same.
What do I mean?
Well, take a look at this big triangle.
It's got some height that I'll label in,
in orange, and it's got some width that
I'll label in blue.
Now the width of this triangle isn't the
same as this triangle, isn't the same as
this triangle, isn't the same as this
small triangle.
And the height of this triangle is
different than the heights of these three
triangles.
But look at the ratio of this height to
this width.
Alright, here's the height, here's the
width, and if I double the width, Width.
It's a little bit more than the height.
In other words the height is just a little
bit less than the twice the width and
that's true for all of these triangles.
The ratio of their heights to their widths
are all the same, even though they're
different sizes.
That's the key fact that makes
trigonometry work.
If you just draw some random right
triangle with given angles, it doesn't
really make a lot of sense to ask
questions about the side lengths, it
really depends on how big the triangle you
drew.
But it does make sense to ask questions
about the ratios of side lengths, those
only depend on the angles.
So we can give names to these ratios.
We call the sine of theta as the ratio
between the height and the hypotenuse.
So if you like, the opposite side and
hypotenuse.
So y over r in this diagram.
The cosine of theta is the ratio between
the adjacent side and the hypotenuse of
the right triangle.
It's x over r in this picture.
And the tangent of theta.
That's the opposite side over the adjacent
side, it's y over x in this picture
because the sine, cosine, and tangent of
this angle theta.
And they're all just defined in terms of
ratios of side lengths.
Now that we've got a definition of sine.
We can, for instance, calculate sine of pi
over 5.
If we build a triangle, any right triangle
with an angle of measure pi over 5.
I built a right triangle.
And this angle is pi over 5.
I can use my little ruler to measure the
hypotenuse.
The hypotenuse here is about 25.2
centimeters.
And I can measure the opposite side, and
the opposite side's about 14.8
centimetres.
And sine is the opposite side over the
hypotenuse, which means sine of pi over
five is about 14.8.
Over 25.2, which is about 0.59.
.