# HEXAGON: An Aid in Remembering the Eight Fundamental Trigonometric Identities

Updated on April 13, 2011

BRIEF HISTORY

Trigonometry is one of the oldest branches of mathematics, not by name, perhaps, but in its application to the measurement of distances, heights, depths, areas, and volumes. In the early centuries, the seasons of the year and the hours of the day are indicated by shadows cast by an upright pole, tree or rock. A tree or other upright objects which casts a shadow for this purpose was called by the Greeks a gnomon (which means inspector) and the shadow was called an umbra, Latin for shadow. When the umbra is longest at  noon, thus it is called the winter solstice; whereas when shortest, it is called the summer solstice, usually occurring during the month of June.

Other notable evidences of the use of the triangle in measurement are found in sundials used to measure or indicate the time of the day. These sundials were commonly used a few centuries ago and some are still being used in some oriental countries.

As means of measurement, trigonometry seems to have developed very early in Asia, specifically in India, Iraq, Egypt, and China. Some traces appear about 4, 000 years ago, generally in the surveying of land, the planning of irrigation ditches and the determination of the height of edifices (as in the case of the Pyramid at Khafre, Egypt).

As a science, it is to the Greeks, at one time the most scholarly of the ancient peoples of the Mediterranean region that trigonometry owes its origin.

Hipparchus (about 140 B.C.) and Ptolemy about three centuries later studied not only plane but also spherical trigonometry. They knew some of our formulas including sin2a + cos2a = 1, although of course, not using the same symbols, notations and representations.

The trigonometry which is being used now dates from the seventeenth century, stemming immediately after the Rebirth or Renaissance Period, when it was greatly improved particularly in relation to the symbols in Italy, Germany,  England, and adjacent countries.

DEFINITION

Etymologically speaking, trigonometry comes from the Greek words:

“tri” meaning “three”

gonia meaning “angle”

metronmeaning “a measure”

Therefore, Trigonometry is the measurement of three angles or triangles.

TRIGONOMETRY involves functions of angles in a triangle, called trigonometric functions, which depend on ratios of the sides of the triangle. Trigonometry is widely utilized in the physical sciences, particularly in navigation, astronomy, surveying and mechanics. Without climbing a mountain top, its altitude can be measured through trigonometry. Without crossing a river, its breadth can be obtained through trigonometry. Trigonometric functions are also in other fields, such as analysis of financial markets, probability theory, statistics, biology, medical imaging, computer graphics and crystallography.

From six equilateral triangles, what polygon can be formed such that one of its side is equal to that of the triangle?

HEXAGON

Now, learn how to use the hexagon in remembering trigonometric identities...and prove these identities.

57

5

7