# The trigonometric functions

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The first work on trigonometric functions related to chords of a circle. Given a circle of fixed radius, 60 units were often used in early calculations, then the problem was to find the length of the chord subtended by a given angle. For a circle of unit radius the length of the chord subtended by the angle *x* was 2sin (*x*/2). The first known table of chords was produced by the Greek mathematician Hipparchus in about 140 BC. Although these tables have not survived, it is claimed that twelve books of tables of chords were written by Hipparchus. This makes Hipparchus the founder of trigonometry.

The next Greek mathematician to produce a table of chords was Menelaus in about 100 AD. Menelaus worked in Rome producing six books of tables of chords which have been lost but his work on spherics has survived and is the earliest known work on spherical trigonometry. Menelaus proved a property of plane triangles and the corresponding spherical triangle property known the *regula sex quantitatum* Ⓣ.

Ptolemy was the next author of a book of chords, showing the same Babylonian influence as Hipparchus, dividing the circle into 360° and the diameter into 120 parts. The suggestion here is that he was following earlier practice when the approximation 3 for π was used. Ptolemy, together with the earlier writers, used a form of the relation sin^{2} *x* + cos^{2} *x* = 1, although of course they did not actually use sines and cosines but chords.

Similarly, in terms of chords rather than sin and cos, Ptolemy knew the formulas

*x*+

*y*) = sinx cos

*y*+ cosx sin

*y*

*a*/sin *A* = *b*/sin *B* = *c*/sin *C*.

The first actual appearance of the sine of an angle appears in the work of the Hindus. Aryabhata, in about 500, gave tables of half chords which now really are sine tables and used jya for our sin. This same table was reproduced in the work of Brahmagupta (in 628) and detailed method for constructing a table of sines for any angle were give by Bhaskara in 1150.

The Arabs worked with sines and cosines and by 980 Abu'l-Wafa knew that

*x*= 2 sin

*x*cos

*x*

*x*+

*y*) = sin

*x*cos

*y*+ cos

*x*sin

*y*with

*x*=

*y*.

The Hindu word *jya* for the sine was adopted by the Arabs who called the sine *jiba*, a meaningless word with the same sound as *jya*. Now *jiba* became *jaib* in later Arab writings and this word does have a meaning, namely a 'fold'. When European authors translated the Arabic mathematical works into Latin they translated jaib into the word sinus meaning fold in Latin. In particular Fibonacci's use of the term *sinus rectus arcus* soon encouraged the universal use of sine.

Chapters of Copernicus's book giving all the trigonometry relevant to astronomy was published in 1542 by Rheticus. Rheticus also produced substantial tables of sines and cosines which were published after his death. In 1533 Regiomontanus's work *De triangulis omnimodis* Ⓣ was published. This contained work on planar and spherical trigonometry originally done much earlier in about 1464. The book is particularly strong on the sine and its inverse.

The term sine certainly was not accepted straight away as the standard notation by all authors. In times when mathematical notation was in itself a new idea many used their own notation. Edmund Gunter was the first to use the abbreviation sin in 1624 in a drawing. The first use of sin in a book was in 1634 by the French mathematician Hérigone while Cavalieri used *Si* and Oughtred *S*.

It is perhaps surprising that the second most important trigonometrical function during the period we have discussed was the versed sine, a function now hardly used at all. The versine is related to the sine by the formula

*x*= 1 - cos

*x*.

The cosine follows a similar course of development in notation as the sine. Viète used the term sinus residuae for the cosine, Gunter (1620) suggested co-sinus. The notation Si.2 was used by Cavalieri, s co arc by Oughtred and S by Wallis.

Viète knew formulas for sin *nx* in terms of sin *x* and cos *x*. He gave explicitly the formulas (due to Pitiscus)

*x*= 3 cos

^{2}

*x*sin

*x*- sin

^{3}

*x*

cos 3*x* = cos ^{3}*x* - 3 sin ^{2}*x* cos *x*.

The first known tables of shadows were produced by the Arabs around 860 and used two measures translated into Latin as *umbra recta* and *umbra versa.* Viète used the terms amsinus and prosinus. The name tangent was first used by Thomas Fincke in 1583. The term cotangens was first used by Edmund Gunter in 1620.

Abbreviations for the tan and cot followed a similar development to those of the sin and cos. Cavalieri used Ta and Ta.2, Oughtred used *t* arc and *t* co arc while Wallis used *T* and *t*. The common abbreviation used today is tan by we write tan whereas the first occurrence of this abbreviation was used by Albert Girard in 1626, but tan was written over the angle

*A*

The secant and cosecant were not used by the early astronomers or surveyors. These came into their own when navigators around the 15^{th} Century started to prepare tables. Copernicus knew of the secant which he called the hypotenusa. Viète knew the results

*x*/sec

*x*= cot

*x*= 1/tan

*x*

1/cosec *x* = cos *x*/cot *x* = sin *x*.

*s*and

*σ*. Albert Girard used sec, written above the angle as he did for the tan.

The term 'trigonometry' first appears as the title of a book *Trigonometria* by B Pitiscus, published in 1595. Pitiscus also discovered the formulas for sin 2*x*, sin 3*x*, cos 2*x*, cos 3*x*.

The 18^{th} Century saw trigonometric functions of a complex variable being studied. Johann Bernoulli found the relation between sin^{-1}*z* and log *z* in 1702 while Cotes, in a work published in 1722 after his death, showed that

*ix*= log(cos

*x*+

*i*sin

*x*).

*x*+

*i*sin

*x*)

^{n}= cos

*nx*+

*i*sin

*nx*

*ix*) = cos

*x*+

*i*sin

*x*.

**Article by:** *J J O'Connor* and *E F Robertson*

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