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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 instrumen**t 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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# 2º

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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# 3º

→ 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. :-(

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**** =** a^{2} + b^{2} = c^{2 } without fractions.

A well-known example is 3, 4, 5 because 3^{2} + 4^{2} = 5^{2 }

Another example is 6, 8, 10 because 6^{2} + 8^{2} = 10^{2}

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 a ^{2} + b^{2} = c^{2 }**

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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