Dad: 
Guys, we will need to use the Pythagorean
Theorem. Do you remember this theorem? 
Michelle: 
Yes, of
course. It says that for any right triangle
with the hypotenuse c and legs a and b the
following equation is true:
c^{2} = a^{2}
+ b^{2} 
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? 
Michelle: 
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
formula^{1}. 
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.
