| Michelle: |
Dad, Nick broke his nose!! |
| Dad: |
What?! How
did that happen? Are you okay, Nick? |
| Nick: |
Nothing serious,
Dad. I just scratched my nose a little. I
climbed our apple tree to measure its height,
but I fell. |
| Dad: |
Oh boy! To
measure the height of a tree, you don’t
need to climb it. |
| Nick: |
Really? But
how else can I do that? |
| Dad: |
Have you ever
heard of a subject called Trigonometry? |
| Michelle: |
I have only
heard its name, but have no idea what it is. |
| Nick: |
And I haven’t
even heard that name. |
| Dad: |
This subject
studies so-called trigonometric functions.
Using these functions you can measure the
height of a tree. |
| Nick: |
Does that
mean I don’t have to climb the tree? |
| Dad: |
Exactly! The
main point is that you don’t have to. |
| Nick: |
Wow! Dad,
can you tell us about trigonometric functions? |
| Dad: |
Well, I guess
I’ll have to. So, listen. The main idea
that leads to Trigonometry is the similarity
of triangles. Do you know what it is? |
| Michelle: |
Yeah, I learned
that in Geometry. Two triangles are called
similar if
they have the same angles. |
| Nick: |
So, similar
triangles are equilateral triangles? |
| Michelle: |
No, not
at all! Look at these two triangles:
As you see, these triangles are not equilateral:
in each of them the angles are different.
But the triangles are similar because  A
= A
' ,
B = B
' and
C = C
' . In other words, these two triangles
have the same corresponding
angles1. |
| D: |
Similar triangles
or similar figures in general, may be imaged
like this. Take a figure and look at it through
a magnifying or diminishing glass. What you’ll
see is a figure similar to the original. Another
image is a parent and a child. They have different
body sizes but the same traits. |
| N: |
So, you and
I are similar figures? |
| D: |
In some sense.
Now, what do you think is a very important
property of similar triangles? |
| M: |
Their sides
are proportional. |
| N: |
Hm, and what
might that be? |
| M: |
It means that
if the side AB is, for example, 10 times greater
than A'B' , then all the other sides of  ABC
will also be 10 times greater than their corresponding
sides of
A' B' C' . |
| D: |
True. It’s
like a parent’s leg is 2 times greater
than the leg of the child, then the arm of
the parent will also be (approximately, of
course) 2 times greater than the arm of the
child. Now, can you write the proportional
sides property using formulas? |
| M: |
Sure, that’s
easy:
|
| D: |
Exactly.
These formulas say that in similar figures,
the ratios of the corresponding sides are
the same. Formulas that you wrote will help
us solve the problem with the apple tree.
Look at this diagram:
Here in the right triangle ABC , vertical
side BC represents our apple tree. Also,
I picked a point A on the ground at some
distance from the tree. What can we measure
easily? |
| N: |
The distance
AC , since there is no need to climb anywhere. |
| D: |
Yes, and what
else? |
| M: |
M-m-m…,
it seems nothing else can be measured. |
| D: |
No guys, we
can also measure angle  . |
| N: |
But how can
we do that? |
| D: |
It’s
not a big deal. We can use a simple device
that looks like a pipe with a protractor
attached to it. We will stand at A and point
the pipe at B . Then we will measure the
angle
on the protractor between points C and B
. That’s it. Such devices are called
astrolabes.
They were used by early navigators in determining
the latitude by measuring the angle of the
North Star (Polaris) above the horizon:
|
| M: |
Well, we measured
angle .
So what? |
| D: |
Now, we
can use the concept of similar triangles.
Let’s draw another right
A' B' C' with exactly the same angle :
|
| N: |
And what size
did you choose for that triangle? |
| D: |
The point
is that the size
doesn’t matter. Compare triangles
ABC and A' B' C' . |
| N: |
I got it!
They are similar. |
| M: |
Yeah, it’s
completely obvious: they have the same angle
and the same right angle. Another acute angle
is also the same because the sum of acute
angles in right triangle is always 90°.
|
| D: |
So, ABC
and
A' B' C' are similar. Now, could you guess
what to do next to determine the height BC
of our tree? |
| M: |
Oh!! I think
I know! In drawn
A' B' C' we can measure whatever we want.
Let’s measure sides A'C' and B'C'
. Since
A' B' C' is similar to
A B C , their sides are proportional:
From this proportion we can find BC :
 |
(1) |
That’s all! We know AC , B'C' , and
A'C' . Therefore, the height BC of our apple
tree is found! |
| N: |
Not bad. What
a pity, I didn’t know that before. |
| D: |
Note guys,
that we used the general principle that it
is easier to measure angles than distances.
Many practical trigonometry applications are
based on this principle. |
| N: |
OK, Dad, but
where are your trigonometric functions? I
see that we solved our problem without using
them. |
| D: |
Well, let’s
take a close look at formula (1).... |
------------------------------------------------------------------------------------------------------------------------ |
1
To be precise, angles A and A' are not equal,
since they are different
geometrical figures. Two figures are called
equal, if both represent exactly the same
figure. Our A and A' only have the same
value of
angles. It means we can put one angle on
another and they will coincide. Such figures
are called congruent. However,
we will use the term “equal”
instead of “congruent”, when
it will not confuse us.
|
