Trigonometry Essay, Research Paper

Trigonometry

Trigonometry uses the fact that ratios of pairs of sides of triangles are

functions of the angles. The basis for mensuration of triangles is the right-

angled triangle. The term trigonometry means literally the measurement of

triangles. Trigonometry is a branch of mathematics that developed from simple

measurements.

A theorem is the most important result in all of elementary mathematics. It was

the motivation for a wealth of advanced mathematics, such as Fermat’s Last

Theorem and the theory of Hilbert space. The Pythagorean Theorem asserts that

for a right triangle, the square of the hypotenuse is equal to the sum of the

squares of the other two sides. There are many ways to prove the Pythagorean

Theorem. A particularly simple one is the scaling relationship for areas of

similar figures.

Did Pythagoras derive the Pythagorean Theorem or did he piece it together by

studying ancient cultures; Egypt, Mesopotamia, India and China? What did these

ancient cultures know about the theorem? Where was the theorem used in their

societies? In “Geometry and Algebra in Ancient Civilizations”, the author

discusses who originally derived the Pythagorean Theorem. He quotes Proclos, a

commentator of Euclid’s elements, “if we listen to those who wish to recount the

ancient history we may find some who refer this theorem to Pythagoras, and say

that he sacrificed an ox in honor of his discovery”. If this statement is

considered as a statement of fact, it is extremely improbable, for Pythagoras

was opposed to the sacrifice of animals, especially cattle. If the saying is

considered as just a legend, it is easy to explain how such a legend might have

come into existence. Perhaps the original form of the legend said something

like he who discovered the famous figure sacrificed a bull in honor of his

discovery.

Van der Waerden goes on to comment that he believes the original discoverer was

a priest, before the time of Babylonian texts, who was allowed to sacrifice

animals and also was a mathematician. This question can never be answered, but

evidence that societies used the theorem before the time of Pythagoras can be

found.

The Theorem is useful in everyday life. For example, at a certain time of day,

the sun’s rays cast a three foot shadow off a four foot flag pole. Knowing

these two lengths, and the fact that the pole forms a ninety degree angle with

the ground, the distance from the end of the shadow to the top of the pole can

be found without measuring. The first step is to substitute the given data

into the actual formula. Now you can find from the length of the third side,

which is five feet. Trigonometry is basicly the study of the relationship

between the sides and the angles of right triangles. Knowing how to use these

relationships and ratios, is absolutly necessary for almost everthing. It

might not seem like it, but trigonometry is used almost everywhere.

Another example of the importance of the theorem is the world orb symbol, which

depicts engineering studies. Although there are many parts to this symbol, the

Pythagorean theorem is appropriately at the center, since much of engineering,

mensuration, logarithms etc., are based on trigonometric functions.