History of Trigonometry

Classical TrigonometryModern Trigonometry

Modern Trigonometry

• From geometric to analytic trigonometry

•Application to science

From geometric to analytic trigonometry

• In the 16th century trigonometry began to change its character from a purely geometric discipline to an algebraic-analytic subject.

From geometric to analytic trigonometry

Two developments spurred this transformation:• the rise of symbolic algebra,

pioneered by the French mathematician François Viète (1540–1603)• the invention of analytic

geometry by two other Frenchmen, Pierre de Fermat and René Descartes.

From geometric to analytic trigonometry

• Viète showed that the solution of many algebraic equations could be expressed by the use of trigonometric expressions.• For example, the equation x3 = 1

has the three solutions:• x = 1,• cos 120° + i sin 120° = −1 + i√3/2,

and• cos 240° + i sin 240° = −1 − i√3/2.

From geometric to analytic trigonometry

• (Here i is the symbol for √(−1) , the “imaginary unit.”) That trigonometric expressions may appear in the solution of a purely algebraic equation was a novelty in Viète's time;• he used it to advantage in a

famous encounter between King Henry IV of France and Netherlands' ambassador to France.

From geometric to analytic trigonometry

• The latter spoke disdainfully of the poor quality of French mathematicians and challenged the king with a problem posed by Adriaen van Roomen, professor of mathematics and medicine at the University of Louvain (Belgium), to solve a certain algebraic equation of degree 45.

From geometric to analytic trigonometry

• The king summoned Viète, who immediately found one solution and on the following day came up with 22 more.• Viète was also the first to

legitimize the use of infinite processes in mathematics.• In 1593 he discovered the infinite

product,

From geometric to analytic trigonometry

• 2/π = √2/2 ∙ √((2 + √2)) /2 ∙ √((2 + 

√((2 + √2)) )) /2⋯,• which is regarded as one of the

most beautiful formulas in mathematics for its recursive pattern.• By computing more and more

terms, one can use this formula to approximate the value of π to any desired accuracy.

From geometric to analytic trigonometry

• In 1671 James Gregory (1638–75) found the power series (see the table) for the inverse tangent function (arc tan, or tan−1), from which he got, by letting x = 1, the formulaπ/4 = 1 − 1/3 + 1/5 − 1/7 + ⋯,• which demonstrated a

remarkable connection between π and the integers.

From geometric to analytic trigonometry

• Although the series converged too slowly for a practical computation of π (it would require 628 terms to obtain just two accurate decimal places).• This was soon followed by Isaac

Newton's (1642–1727) discovery of the power series for sine and cosine (see the table).

From geometric to analytic trigonometry

• Recent research, however, has brought to light that some of these formulas were already known, in verbal form, by the Indian astronomer Madhava (c. 1340–1425).

From geometric to analytic trigonometry

• The gradual unification of trigonometry and algebra—and in particular the use of complex numbers (numbers of the form x + iy, where x and y are real numbers and i = √(−1) ) in trigonometric expressions—was completed in the 18th century.

From geometric to analytic trigonometry

• In 1722 Abraham de Moivre (1667–1754) derived, in implicit form, the famous formula• (cos ø + i sin ø) n = cos nø + i

sin nø, • which allows one to find the nth

root of any complex number.

From geometric to analytic trigonometry

• It was the Swiss mathematician Leonhard Euler (1707–83), though, who fully incorporated complex numbers into trigonometry.

From geometric to analytic trigonometry

• Euler's formula eiø = cos ø + i sin ø, where e 2.71828 is the base of natural logarithms, appeared in 1748 in his great work Introductio in analysin infinitorum—although Roger Cotes already knew the formula in its inverse form øi = log (cos ø + i sin ø) in 1714.

From geometric to analytic trigonometry

• Substituting into this formula the value ø = π, one obtains eiπ = cos π + i sin π = −1 + 0i = −1 or equivalently, eiπ + 1 = 0.• This most intriguing of all

mathematical formulas contains the additive and multiplicative identities (0 and 1, respectively), the two irrational numbers that occur most frequently in the physical world (π and e),

From geometric to analytic trigonometry

• and the imaginary unit (i), and it also employs the basic operations of addition and exponentiation—hence its great aesthetic appeal. Finally, by combining his formula with its companion formulae−iø = cos (−ø) + i sin (−ø) = cos ø − i sin ø,

From geometric to analytic trigonometry

• Euler obtained the expressions• cos ø = eiø + e−iø/2 and

• sin ø = eiø − e−iø/2i, • which are the basis of modern

analytic trigonometry.

Modern Trigonometry

• From geometric to analytic trigonometry

•Application to science

Application to science• While these developments

shifted trigonometry away from its original connection to triangles, the practical aspects of the subject were not neglected.

Application to science• The 17th and 18th centuries

saw the invention of numerous mechanical devices—from accurate clocks and navigational tools to musical instruments of superior quality and greater tonal range—all of which required at least some knowledge of trigonometry.

Application to science• A notable application was the science

of artillery—and in the 18th century it was a science.• Galileo Galilei (1564–1642) discovered

that any motion—such as that of a projectile under the force of gravity—can be resolved into two components,• one horizontal and the other vertical,

and that these components can be treated independently of one another.

Application to science• This discovery led scientists to the

formula for the range of a cannonball when its muzzle velocity v0 (the speed at which it leaves the cannon) and the angle of elevation A of the cannon are given.• The theoretical range, in the absence of

air resistance, is given byR = v0

2 sin2A/g,

• where g is the acceleration due to gravity (about 9.81 metres/second2).

Application to science• This formula shows that, for a given

muzzle velocity, the range depends solely on A; it reaches its maximum value when A = 45° and falls off symmetrically on either side of 45°. (See the animation.)• These facts, of course, had been

known empirically for many years, but their theoretical explanation was a novelty in Galileo's time.

Application to science• Another practical aspect of

trigonometry that received a great deal of attention during this time period was surveying.• The method of triangulation was first

suggested in 1533 by the Dutch mathematician Gemma Frisius (1508–55): one chooses a base line of known length, and from its endpoints the angles of sight to a remote object are measured.

Application to science• The distance to the object from

either endpoint can then be calculated by using elementary trigonometry.• The process is then repeated

with the new distances as base lines, until the entire area to be surveyed is covered by a network of triangles.

Application to science• The method was first carried out

on a large scale by another Dutchman, Willebrord van Roijen Snell (1581–1626), who surveyed a stretch of 130 km (80 miles) in Holland, using 33 triangles.

Application to science• The French government, under

the leadership of the astronomer Jean Picard (1620–82), undertook to triangulate the entire country, a task that was to take over a century and involve four generations of the Cassini family (Gian, Jacques, César-François, and Dominique) of astronomers.

Application to science• The British undertook an even

more ambitious task—the survey of the entire subcontinent of India.• Known as the Great Trigonometric

Survey, it lasted from 1800 to 1913 and culminated with the discovery of the tallest mountain on Earth—Peak XV, or Mount Everest.

Application to science• Concurrent with these

developments, 18th-century scientists also turned their attention to aspects of the trigonometric functions that arose from their periodicity.

Application to science• If the cosine and sine functions

are defined as the projections on the x- and y-axes, respectively, of a point moving on a unit circle (a circle with its centre at the origin and a radius of 1), then these functions will repeat their values every 360°, or 2π radians.

Application to science• Hence the importance of the sine

and cosine functions in describing periodic phenomena—the vibrations of a violin string, the oscillations of a clock pendulum, or the propagation of electromagnetic waves.

Application to science• These investigations reached a

climax when Joseph Fourier (1768–1830) discovered that almost any periodic function can be expressed as an infinite sum of sine and cosine functions, whose periods are integral divisors of the period of the original function.

Application to science• For example, the “sawtooth”

function, as shown in the animation, can be written as2(sin x − sin 2x/2 + sin 3x/3 − ⋯);

• as successive terms in the series are added, an ever-better approximation to the sawtooth function results.

Application to science• These trigonometric or Fourier series

have found numerous applications in almost every branch of science, from optics and acoustics to radio transmission and earthquake analysis.• Their extension to nonperiodic

functions played a key role in the development of quantum mechanics in the early years of the 20th century.

Application to science• Trigonometry, by and large,

matured with Fourier's theorem; further developments (e.g., generalization of Fourier series to other orthogonal, but nonperiodic, functions) are well beyond the scope of this encyclopedia article.