# Trigonometry - Historic Development Of Trigonometry, Angles, Triangles And Their Properties, Right Triangles And Trigonometric Functions

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Trigonometry is a branch of applied **mathematics** concerned with the relationship between angles and their sides and the calculations based on them. First developed as a branch of **geometry** focusing on triangles during the third century B.C., trigonometry was used extensively for astronomical measurements. The major trigonometric functions, including sine, cosine, and tangent, were first defined as ratios of sides in a right triangle. Since trigonometric functions are intrinsically related, they can be used to determine the dimensions of any triangle given limited information. In the eighteenth century, the definitions of trigonometric functions were broadened by being defined as points on a unit **circle**. This allowed the development of graphs of functions related to the angles they represent which were periodic. Today, using the periodic nature of trigonometric functions, mathematicians and scientists have developed mathematical models to predict many natural periodic phenomena.

## Additional Topics

Trigonometry was initially considered a field of the science of astronomy. It was later established as a separate branch of mathematics—largely through the work of the mathematicians Johann Bernoulli (1667-1748) and Leonhard Euler (1707-1783). …

Central to the study of trigonometry is the concept of an angle. An angle is defined as a geometric figure created by two lines drawn from the same point, known as the vertex. The lines are called the sides of an angle and their length is one defining characteristic of an angle. Another characteristic of an angle is its measurement or magnitude, which is determined by the amount of rotation, aroun…

The principles of trigonometry were originally developed around the relationship between the sides of a triangle and its angles. The idea was that the unknown length of a side or size of an angle could be determined if the length or magnitude of some of the other sides or angles were known. Recall that a triangle is a geometric figure made up of three sides and three angles, whose sum is equal to …

The triangles used in the previous example were right triangles. During the development of trigonometry, the parts of a right triangle were given certain names. The longest side of the triangle, which is directly across from the right angle, is known as the hypotenuse. The sides that form the right angle, denoted by a box in the diagram, are the legs of the triangle. For either acute angle in the …

One immediate application for trigonometric functions is the simple determination of the dimensions of a right triangle, also known as the solution of a triangle, when only a few are known. For example, if the sides of a right triangle are known, then the magnitude of both acute angles can be found. Suppose we have a right triangle whose sides are 2 in (5 cm) and 4.7 in (12 cm), and whose hypotenu…

In addition to the reciprocal relationships of certain trigonometric functions, two other types of relationships exist. These relationships, known as trigonometric identities, include cofunctional relationships and Pythagorean relationships. Cofunctional relationships relate functions by their complementary angles. Pythagorean relationships relate functions by application of the Pythagorean theore…

For hundreds of years, trigonometry was only considered useful for determining sides and angles of a triangle. However, when mathematicians developed more general definitions for sine, cosine and tangent, trigonometry became much more important in mathematics and science alike. The general definitions for the trigonometric functions were developed by considering these values as points on a unit ci…

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