The history of
trigonometry goes back to the earliest recorded mathematics in
Egypt and Babylon. The Babylonians established the measurement
of angles in degrees, minutes, and seconds. Not until the time
of the Greeks, however, did any considerable amount of trigonometry
exist. In the 2nd century BC the astronomer Hipparchus compiled
a trigonometric table for solving triangles. Starting with 71°
and going up to 180° by steps of 71°, the table gave
for each angle the length of the chord subtending that angle
in a circle of a fixed radius r. Such a table is equivalent to
a sine table. The value that Hipparchus used for r is not certain,
but 300 years later the astronomer Ptolemy used r = 60 because
the Hellenistic Greeks had adopted the Babylonian base-60 (sexagesimal)
- In his great astronomical
handbook, The Almagest, Ptolemy provided a table of chords for
steps of 1°, from 0° to 180°, that is accurate to
1/3600 of a unit. He also explained his method for constructing
his table of chords, and in the course of the book he gave many
examples of how to use the table to find unknown parts of triangles
from known parts. Ptolemy provided what is now known as Menelaus's
theorem for solving spherical triangles, as well, and for several
centuries his trigonometry was the primary introduction to the
subject for any astronomer. At perhaps the same time as Ptolemy,
however, Indian astronomers had developed a trigonometric system
based on the sine function rather than the chord function of
the Greeks. This sine function, unlike the modern one, was not
a ratio but simply the length of the side opposite the angle
in a right triangle of fixed hypotenuse. The Indians used various
values for the hypotenuse.
- Late in the 8th century,
Muslim astronomers inherited both the Greek and the Indian traditions,
but they seem to have preferred the sine function. By the end
of the 10th century they had completed the sine and the five
other functions and had discovered and proved several basic theorems
of trigonometry for both plane and spherical triangles. Several
mathematicians suggested using r = 1 instead of r = 60; this
exactly produces the modern values of the trigonometric functions.
The Muslims also introduced the polar triangle for spherical
triangles. All of these discoveries were applied both for astronomical
purposes and as an aid in astronomical time-keeping and in finding
the direction of Mecca for the five daily prayers required by
Muslim law. Muslim scientists also produced tables of great precision.
For example, their tables of the sine and tangent, constructed
for steps of 1/60 of a degree, were accurate for better than
one part in 700 million. Finally, the great astronomer Nasir
ad-Din at- Tusi wrote the Book of the Transversal Figure, which
was the first treatment of plane and spherical trigonometry as
independent mathematical Science.
- The Latin West became
acquainted with Muslim trigonometry through translations of Arabic
astronomy handbooks, beginning in the 12th century. The first
major Western work on the subject was written by the German astronomer
and mathematician Johann Müller, known as Regiomontanus.
In the next century the German astronomer Georges Joachim, known
as Rheticus introduced the modern conception of trigonometric
functions as ratios instead of as the lengths of certain lines.
The French mathematician François Viète introduced
the polar triangle into spherical trigonometry, and stated the
multiple-angle formulas for sin(nq) and cos(nq) in terms of the
powers of sin(q) and cos(q).
- Trigonometric calculations
were greatly aided by the Scottish mathematician John Napier,
who invented logarithms early in the 17th century. He also invented
some memory aids for ten laws for solving spherical triangles,
and some proportions (called Napier's analogies) for solving
oblique spherical triangles.
- Almost exactly one
half century after Napier's publication of his logarithms, Isaac
Newton invented the differential and integral calculus. One of
the foundations of this work was Newton's representation of many
functions as infinite series in the powers of x. Thus Newton
found the series sin(x) and similar series for cos(x) and tan(x).
With the invention of calculus, the trigonometric functions were
taken over into analysis, where they still play important roles
in both pure and applied mathematics.
- Finally, in the 18th
century the Swiss mathematician Leonhard Euler defined the trigonometric
functions in terms of complex numbers (see Number). This made
the whole subject of trigonometry just one of the many applications
of complex numbers, and showed that the basic laws of trigonometry
were simply consequences of the arithmetic of these numbers.