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Extracts from chapters : 1 | 2 | 8 | 12
Lesson 2
  It is a duty of every triangle to live by the laws of sine and cosine
Laws of cosine and sine
Problem of finding a side of a triangle using two other sides and the angle in between them
Generalization of the Pythagorean Theorem (law of cosines)
Problem of finding a side of a triangle using another side and two angles.
The proportion of sides and sines of angles (law of sines)
Solving of triangles

  Lesson 2


Dad: Guys, we will need to use the Pythagorean Theorem. Do you remember this theorem?

Yes, of course. It says that for any right triangle with the hypotenuse c and legs a and b the following equation is true:

c2 = a2 + b2

Nick: Nice formula! I wonder how Pythagoras came up with it?
Dad: I don’t know exactly, guys. I can only guess. But isn’t it obvious, even without any Pythagorean Theorem, that there is a relationship between the hypotenuse and the legs?

Yes, this is clear. If we construct two legs

then we’ll know both endpoints of the hypotenuse, and we can measure it. Therefore, the two legs fully determine the hypotenuse.

Dad: Absolutely correct. And when Pythagoras was thinking about this, he probably got an urge to express this relationship analytically.
Nick: Analytically? What is it?
Dad: Analytically – means in terms of some formula. And Pythagoras got very lucky, he found this wonderful formula1.
Michelle: Oh! Wait… I wonder if a similar formula exists for any triangle, not only for a right one?
N: What a strange thought, Michelle! Obviously not! Because an arbitrary triangle is not defined by two sides.
M: But I am not talking about only two sides. Why not also use the angle between them? In this case the triangle will be fully defined. So, the third side of any triangle is completely defined by the other two sides and the angle between them. Therefore, there should be some kind of formula, expressing one side in terms of the other two sides and the angle between them.
N: Oh, Michelle! Yes! You are actually right. If we could find this formula, we would beat Pythagoras! Wow! We would get a more general theorem for any triangle, not just for a right one. This would be cool!
M: But it is probably very hard to find such a formula. Isn’t it, Dad?
D: No guys, it turns out to be pretty easy. But only because we have such useful tools as trigonometric functions. …



1 Actually, the Pythagorean Theorem has been discovered in many cultures before Pythagoras (580 B.C.). Thus, the Babylonians knew this fact 1000 years earlier. There are informal proofs that were found in China and India many years before Pythagoras. The formal proof has been found in the Pythagorean society. It is unknown whether this proof belongs to Pythagoras or one of his followers.


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