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Pythagoras Basics June 18, 2014

Posted by anagasto in history, philosophy.
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The intention here is to create a measuring instrument valid for triangles where one of the angles is 90º.

We need to know one more angle to be able to calculate everything else.

 1º

Draw a circle with a radius of 1 centimeter or 1 inch or one mile — it doesn’t matter which — and call that radius C.

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Next,  make that radius C part of a triangle by adding two more lines, here red and blue, and call them a and b:

This is the definition of a Pythagorean triangle: a triangle with  a 90º angle.

In this drawing, the 90º angle is where the red and the blue line meet.

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800px-Athene_noctua_(portrait) best

→ Now remember the formula a²+ b² = c²

→ → Remember that we set  the  radius  as  1 

→ → →  Therefore you get  c² = 1² = 1

That’s it ! Now you got it!

The ratios change if the angle changes, but they would not change at all if the size of the triangle changed. — Enlarge your triangle as much as you want, the proportions remain the same. — Think of a street map. Enlarge the map 1000 times, the proportions stay the same.

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This is all you need to know.

Below are some related definitions and  examples of use.

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Vocabulary

Some people get panicky because of the words that are used in trigonometry (and also in grammar ! and in philosophy!  and in many other places!) to denote very simple things.

1.

Be brave.
Let the worst
come first.

Look at the line opposite the right angle. The convention is to mark it C. It is called hypotenuse.   Some people never get beyond that. It is just too odd  a name.  But think of hypo(po)tanus…. …

hippopotamus

(In Greek, hippo means horse and hypo means under; …..!)

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2. The angle that is being measured is mostly marked with a Greek letter called Theta θ inside a curve. — I was not able to add the Theta θ :-(

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3. The side of the triangle which is close to Theta is called Adjacent, and in the drawing above it is marked red.

4. The other side which is opposite to Theta is called Opposite  and it is blue.

Now remember how we defined that C was going to be 1.

5. Cosine = a = the red line measured against C = a : 1 
Sine = b = the blue line measured against C = b : 1

(In Latin sinus means curve. Here it is the curve often used to mark the angle that one is talking about.)

These names  are

not necessary

to understand how a sine or a cosine are generated and how they work, but, yes, I know, you might have to know the names to do a test. :-(

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http://pages.uoregon.edu/klio/maps/gr/science</pythagoras-cartoon.jpg

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Wideband communictions satellite; photo released into public domain though stipulating attribution of copyright held by Boeing.

What is triangulation  for?

The drawing is from Isaac Azimov‘s illustrated story on the shape of the Earth in public domain at http://www.arvindguptatoys.com/arvindgupta/earthpix.pdf

See?  A minute rectangular triangle constructed using the sunray as a hypotenuse?

Maybe the Egyptians and the Chinese knew about the laws governing a rectangular triangle  before Pythagoras formulated them, because knowledge often precedes formulation by hundreds of years.

The Egyptians may also have used these laws to build the pyramids and also to restore their holdings after the yearly Nile floods.

Now trigonometry is used everywhere in navigation, architecture, acoustics, engineering, optics,
statistics, astronomy, computer science.

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Neat, but completely unnecessary

The Pythagorean Triple = a2 + b2 = c2   without fractions.

A well-known example is 3, 4, 5 because 32 + 42 =  52

Another example is  6, 8, 10 because 62 + 82 = 102

They can be collected and represented in a pretty graph.

by Jrkenti at http://en.wikipedia.org/wiki/File:Pythagorean_Triples_from_Grapher.png   under CC Attribution-Share Alike 3.0 Unported license.

Owl photo under CC Attribution-Share Alike 3.0 Unported license at http://commons.wikimedia.org/wiki/File:Athene_noctua_%28portrait%29.jpg published  by Trebol-a..
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Also unnecessary

Proof that  a2 + b2 = c

Pythagoras showed that  a²+ b² = c²
The proofs are complicated and are not needed to understand how trigonometry works.

It is really a great example of how some things have to be taken on trust…. even in mathematics :-D.

There are some 370 proofs.

:-D

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Emergency help:

SOHCAHTOA = ?

It is a word made up to remember  the trigonometry definitions

SOH means Sine equals Opposite over Hypotenuse.

CAH means Cosine equals Adjacent over Hypotenuse.

TOA means Tangent equals Opposite over Adjacent.

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Pythagoras: Music and Space

“Pythagoras observed that when the blacksmith struck his anvil, different notes were produced according to the weight of the hammer.”

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