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...

...Real Life Applications of Trigonometry
A. Importance/Application of Trigonometry in General
Fields that use trigonometry or trigonometric functions include astronomy (especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.
B. PythagoreanTheoremPythagoreanTheorem can be use in the field of cartography, mapping out distances between two distinct points or locations.
C. Special Right Triangle Theorem
Main use for the Special triangle theorems is on the field of architecture or engineering as well as geophysics.
D. Distance and Midpoint Formula
One of the main uses of the distance and midpoint theorem is on the field of optometry. It involves the calculation of the lenses...

...In mathematics, the Pythagoreantheorem — or Pythagoras' theorem — is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
Thetheorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:[1]
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
The Pythagoreantheorem is named after the Greek mathematician Pythagoras (ca. 570 BC—ca. 495 BC), who by tradition is credited with its discovery and proof,[2][3] although it is often argued that knowledge of the theorem predates him. There is evidence that Babylonian mathematicians understood the formula, although there is little surviving evidence that they used it in a mathematical framework.[4][5]
The theorem has numerous proofs, possibly the most of any mathematical theorem. These are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including...

...The PythagoreanTheorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many contributions to mathematics although some of them may have actually been the work of his students.
ThePythagoreanTheorem is Pythagoras' most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery that the square root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers. They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that the square root of 2 is irrational. It is even said that the man who divulged the secret was drowned at sea.
The PythagoreanTheorem is a statement about triangles containing a right angle. The PythagoreanTheorem states that:
"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."
Figure 1
According to...

...Teaching trigonometry using Empirical Modelling
0303417
Abstract
The trigonometric functions sin(x), cos(x) and tan(x) are relationships that exist between the angles
and length of sides in a right-angled triangle. In Empirical Modelling terms, the angles in a triangle
and the length of the sides are observables, and the functions that connect them are the definitions.
These well-defined geometric relationships can be useful when teaching GCSE-level students about
the functions, as they provide a way to visualise what can be thought of as fairly abstract functions.
This paper looks at how different learning styles apply to Empirical Modelling, and presents a practical example of their use in a model to teach trigonometry.
1 Introduction
The trigonometric functions sin(x), cos(x) and tan(x)
are relationships that exist between the angles and
length of sides in a right-angled triangle. In Empirical Modelling terms, the angles in a triangle and the
length of the sides are observables, and the functions
that connect them are the definitions. These welldefined geometric relationships can be useful when
teaching GCSE-level students about the functions,
as they provide a way to visualise what can be
thought of as fairly abstract functions. Rather than
teaching students by showing them diagrams in an
instructive way (already a good way of doing it), a
constructive approach may allow students to gain a
better understanding...

...PYTHAGOREANTHEOREM
More than 4000 years ago, the Babyloneans and the Chinese already knew that a triangle with the sides of 3, 4 and 5 must be a right triangle. They used this knowledge to construct right angles. By dividing a string into twelve equal pieces and then laying it into a triangle so that one side is three, the second side four and the last side five sections long, they could easily construct a right angle.
A Greek scholar named Pythagoras, who lived around 500 BC, was also fascinated by triangles with these special side ratios. He studied them a bit closer and found that the two shorter sides of the triangles squared and then added together, equal exactly the square of the longest side. And he proved that this doesn't only work for the special triangles, but for any right triangle. Today we would write it somehow like this: a2 + b2= c2. In the time of Pythagoras they didn't use letters yet to replace variables. (They weren't introduced until the 16th century by Vieta.) Instead they wrote down everything in words, like this: if you have a right triangle, the squares of the two sides adjacent to the right angle will always be equal to the square of the longest side.
We can't be sure if Pythagoras really was the first person to have found this relationship between the sides of right triangles, since no texts written by him were found. In fact, we can't even prove the guy lived. But the theorem a2 + b2= c2 got his...

...In mathematics, the PythagoreanTheorem — or Pythagoras' theorem — is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
Thetheorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:
Where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
The PythagoreanTheorem is named after the Greek mathematician Pythagoras (ca. 570 BC—ca. 495 BC), who by tradition is credited with its discovery and proof.
The converse of the theorem is also true:
For any three positive numbers a, b, and c such that a2 + b2 = c2, there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b.
An alternative statement is:
For any triangle with sides a, b, c, if a2 + b2 = c2, then the angle between a and b measures 90°.
This converse also appears in Euclid's Elements (Book I, Proposition 48):
"If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of...

... When using the PythagoreanTheorem, the hypotenuse or its length is often labeled with a lower case c. The legs (or their lengths) are often labeled a and b.
Either of the legs can be considered a base and the other leg would be considered the height (or altitude), because the right angle automatically makes them perpendicular. If the lengths of both the legs are known, then by setting one of these sides as the base ( b ) and the other as the height ( h ), the area of the right triangle is very easy to calculate using this formula:
(1/2)
This is intuitively logical because another congruent right triangle can be placed against it so that the hypotenuses are the same line segment, forming a rectangle with sides having length b and width h. The area of the rectangle is b × h, so either one of the congruent right triangles forming it has an area equal to half of that rectangle.
Right triangles can be neither equilateral, acute, nor obtuse triangles. Isosceles right triangles have two 45° angles as well as the 90° angle. All isosceles right triangles are similar since corresponding angles in isosceles right triangles are equal. If another triangle can be divided into two right triangles, then the area of the triangle may be able to be determined from the sum of the two constituent right triangles. Also the Pythagoreantheorem can be used for non right triangles. a2+b2=c2-2c
The side lengths of a...

...
PythagoreanTheorem
Pythagoras was born in Samos, Greece around 570 BCE. From there he emigrated to Croton, Italy where most of his most important ideas and theories would develop. Early on, Pythagoras founded a society of disciples where he introduced the idea eternal recurrence into Greek thought, and it was Pythagoras’ ambition to reveal the basis of divine order. This is how Pythagoras came to mathematics, and he saw math as a purifier of the soul, and saw numbers in everything. He was convinced that the divine principles of the universe can be expressed in terms of relationships of numbers.
Over 4000 years ago, the Babylonians and the Chinese already knew that a triangle with the sides of 3, 4, and 5 must be a right triangle. They used this knowledge to construct right angles. Pythagoras studied them a bit closer and found that the two shorter sides of the triangles squared and then added together, equal exactly the square of the longest side. And he proved that this doesn't only work for the special triangles, but for any right triangle. This can be written in the form a^2 + b^2 = c^2, and today this is what is known as the PythagoreanTheorem.
The PythagoreanTheorem was one of the first times in human history that people could calculate a length or distance using only outside information. The train of thought used by Pythagoras was the first time the idea of a unset...