Trigonometry is the branch of mathematics concerned with certain functions of angles and their applications to geometry. It developed from the study of right-angles triangles. From the Babylonian civilisation until the time of Descartes, simple trigonometry had been used in surveying, astronomy and navigation. Both astronomers and sailors, scanning the heavens and the seas, often needed to calculate distances not directly measurable. They applied certain basic rules about the relationship between the sides and angles of triangles.

The Egyptians used these relationships in land surveying and when building the pyramids. Babylonian astronomers related angles to arcs of circles and to the legths of the chords subtending the arcs. The Greeks developed trigonometry into an ordered science by analysing the arcs of circles. The first person known to have used trigonometric ratios was the Greek astronomer and mathematician Hipparchus. He used them in around 140 BC to find the straight-line distances across the curved ‘heavens’, and constructed tables of chords at half-degree intervals for all central angles from 0 to 180^o, calculated to three sexagesimal places.

These trigonometric tables (corresponding to sine, cosine and tangent) were later improved and extended by other mathematicians. But for a long time they were used only by astronomers, sailors and cartographers. Then the remarkable French mathematician Francis Vieta (who preceeded Descartes by half a century) made an amazing observation. He discovered that a trigonometric ratio could be used to solve an algebraic equation. In effect, a series of numbers in a table could represent successive values taken by an unknown. The statement ‘sine of angle x is y’ could also be written as an equation ‘y = sin x’. This new way of looking at trigonometric ratios broadened the scope of trigonometry. Then Descartes developed his plotting techniques. An equation like ‘y = sin x’ could be plotted point by point to create a curve. This curve is an endless wave, and is the exact graphic equivalent of the ebb and flow of electric current in an ordinary AC power cable.

Origin of ‘Sine’

The Hindu mathematician and astronomer Aryabhata the Elder called the sine ratio ardha-jya (‘half-chord’), later abreviated to jya (‘chord’). Arab translators turned this phonetically into jiba — a meaningless word in Arabic — which they wrote as jb, according to Arabic practice of omitting the vowels in writing.

Later Western scholars, with no knowledge of the Sanskrit origin, assumed jb to be an abbreviation of jaib, the Arabic word for ‘cove’, ‘bay’, ‘bulge’, ‘bosom’. Gerard of Cremona, who translated the Arabic Almagest in the late twelfth century, substituted for jaib its Latin equivalent sinus, from which our word ‘sine’ derives.