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.
