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Extracts from chapters : 1 | 2 | 8 | 12
Lesson 1
 
Protect your nose, study trigonometry!
Definition of trigonometric functions
Problem of calculating the height of a tree
Properties of similar triangles
Finding the height of a tree and the definition of tangent
Calculating distance on a rough terrain and the definition of sine
Definitions of cosine, cotangent, secant and cosecant

 
  Lesson 1

 

 

15-year-old Michelle and 13-year-old Nick ran into their father’s office.

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)....

 

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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.

 
 

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