# Earliest Uses of Symbols for Trigonometric and Hyperbolic Functions

Sine. In 1583, Thomas Fincke (or Finck) used sin. (with a period) in Book 14 of his Geometria rotundi. Cajori writes that "perhaps the first use of abbreviations for the trigonometric lines goes back to ... Finck" (Cajori vol. 2, page 150).

In 1624, Edmund Gunter (1581-1626) used sin (without a period) in a drawing representing Gunter's scale (Cajori vol. 2, page 156). However, the symbol does not appear in Gunter's work published the same year.

In 1626, Girard designated the sine of A by A, and the cosine of A by a (Smith vol. 2, page 618).

In a trigonometry published by Richard Norwood in London in 1631, the author states that "in these examples s stands for sine: t for tangent: sc for sine complement: tc for tangent complement: sc for sine complement: tc for tangent complement : sec for secant" (Smith vol. 2, page 618).

In 1632, William Oughtred (1574-1660) used sin (without a period) in Addition vnto the Vse of the Instrvment called the Circles of Proportion (Cajori vol. 1, page 193, and vol. 2, page 158).

According to Smith (vol. 2, page 618), "the first writer to use the symbol sin for sine in a book seems to have been the French mathematician Hérigone (1634)." However, the use by Oughtred would seem to predate that of Hérigone.

Cosine. In his Geometria rotundi (1588) Thomas Fincke used sin. com. for the cosine.

In 1674, Sir Jonas Moore (1617-1679) used Cos. in Mathematical Compendium (Cajori vol. 2, page 163).

Samuel Jeake (1623-1690) used cos. in Arithmetick, published in 1696 (Cajori vol. 2, page 163).

The earliest use of cos Cajori shows apparently is by Leonhard Euler in 1729 in Commentarii Academiae Scient. Petropollitanae, ad annum 1729 (Cajori vol. 2, page 166).

Ball and Asimov say cos was first used by William Oughtred (1574-1660). Ball gives the date 1657; Asimov gives 1631. However, Cajori indicates Oughtred used only sco for cosine (Cajori vol. 1, page 193) and Cajori reports Glaisher says Oughtred did not even use the word cosine.

Tangent. In 1583, Thomas Fincke (1561-1656) used tan. in Book 14 of his Geometria rotundi (Cajori vol. 2, page 150). Fincke also used tang.

In 1632, William Oughtred (1574-1660) used tan in The Circles of Proportion (Cajori vol. 1, page 193).

Secant. In 1583, Thomas Fincke (1561-1656) used sec. in Book 14 of his Geometria rotundi (Cajori vol. 2, page 150).

In 1632, William Oughtred (1574-1660) used sec in The Circles of Proportion (Cajori vol. 1, page 193).

Cosecant. In his Geometria rotundi (1588), Thomas Fincke used sec. com. for the cosecant (Cajori vol. 2, page 150).

Samuel Jeake in his Arithmetick (1696) used cosec. for cosecant (Cajori vol. 2, page 163).

Simon Klügel in Analytische Trigonometrie (1770) used cosec for cosecant.

It appears that the earliest use Cajori shows for csc is in 1881 in Treatise on Trigonometry, by Oliver, Wait, and Jones (Cajori vol. 2, page 171).

Cotangent. In his Geometria rotundi (1588), Thomas Fincke used tan. com. for the cotangent (Cajori vol. 2, page 150).

In 1674, Sir Jonas Moore (1617-1679) used Cot. in Mathematical Compendium (Cajori vol. 2, page 163).

Samuel Jeake in his Arithmetick (1696) used cot. for cotangent (Cajori vol. 2, page 163).

A. G. Kästner in Anfangsgründe der Arithmetik, Geometrie ... Trigonometrie (1758) used cot for cotangent (Cajori vol. 2, page 166).

Cis notation. Irving Stringham used cis β for cos β + i sin β in 1893 in Uniplanar Algebra (Cajori vol. 2, page 133).

Inverse trigonometric function symbols. Except where noted otherwise, the following citations are from Cajori vol. 2, page 175-76.

Daniel Bernoulli was the first to use symbols for the inverse trigonometric functions. In 1729 he used A S. for arcsine in Comment. acad. sc. Petrop., Vol. II.

In 1736 Leonhard Euler (1707-1783) used A t for arctangent in Mechanica sive motus scientia. Later in the same publication he used simply A

In 1737 Euler used A sin for arcsine in Commentarii academiae Petropolitanae ad annum 1737, Vol. IX.

In 1769 Antoine-Nicolas Caritat Marquis de Condorcet (1743-1794) used arc (sin. = x) (Katz, page 42).

In 1772 Carl Scherffer used arc. tang. in Institutionum analyticarum pars secunda.

In 1772 Joseph Louis Lagrange (1736-1813) used arc. sin in Nouveaux memoires de l'academie r. d. sciences et belles-lettres.

According to Cajori (vol. 2, page 176) the inverse trigonometric function notation utilizing the exponent -1 was introduced by John Frederick William Herschel in 1813 in the Philosophical Transactions of London. A full-page footnote explained his choice of notation for the inverse trigonometric functions, such as cos.-1 e, which he used in the body of the article (Cajori vol. 2, page 176).

However, according to Differential and Integral Calculus (1908) by Daniel A. Murray, "this notation was explained in England first by J. F. W. Herschel in 1813, and at an earlier date in Germany by an analyst named Burmann. See Herschel, A Collection of Examples of the Application of the Calculus of Finite Differences (Cambridge, 1820), page 5, note."

According to Cajori, in France, Jules Houël (1823-1886) used arcsin.

In 1914, Plane Trigonometry by George Wentworth and David Eugene Smith has:
In American and English books the symbol $\sin^{-1} y$ is generally used; on the continent of Europe the symbol $arc \sin y$ is the one that is met. The symbol $\sin^{-1} y$ is read "the inverse sine of y," "the antisine of y," or "the angle whose sine is y." The symbol arc sin y is read "the arc whose sine is y," or "the angle whose sine is y."
In 1922 in Introduction to the Calculus, William F. Osgood wrote, "The usual notation on the Continent for sin-1, x, tan-1 x, etc., is arc sin x, arc tan x, etc. It is clumsy, and is followed for a purely academic reason; namely, that sin-1 x might be misunderstood as meaning the minus first power of sin x. It is seldom that one has occasion to write the reciprocal of sin x in terms of a negative exponent. When one wishes to do so, all ambiguity can be avoided by writing (sin x)-1."

Powers of trigonometric functions. The practice of placing the exponent beside the symbol for the trigonometric function to indicate, for example, the square of the sine of x, was used by William Jones in 1710. He wrote cs2 (for cosine) and s2 (for sine) (Cajori vol. 2, page 179).

Degrees, minutes, seconds. See the geometry page.

Radians. G. B. Halsted in Mensuration (1881) suggested using the Greek letter rho to indicate radians.

G. N. Bauer and W. E. Brooke in Plane and Spherical Trigonometry (1907) suggested the use of r (the lower case r in a raised position) to indicate radians.

A. G. Hall and F. G. Frink in Plane Trigonometry (1909) suggested the use of R (the capital R in a raised position).

P. R. Rider and A. Davis in Plane Trigonometry (1923) suggested the use of (r) (the lower case r in parentheses) to indicate radians.

Hyperbolic functions. Vincenzo Riccati (1707-1775), who introduced the hyperbolic functions, used Sh. and Ch. for hyperbolic sine and cosine (Cajori vol. 2, page 172). He used Sc. and Cc. for the circular functions.

Johann Heinrich Lambert (1728-1777) further developed the theory of hyperbolic functions in Histoire de l'académie Royale des sciences et des belles-lettres de Berlin, vol. XXIV, p. 327 (1768). According to Cajori (vol. 2, page 172), Lambert used sin h and cos h.

According to Scott (page 190), Lambert began using sinh and cosh in 1771.

According to Webster's New International Dictionary, 2nd. ed., sinh is an abbreviation for sinus hyperbolus.

In 1902, George M. Minchin proposed using hysin, hycos, hytan, etc.: "If the prefix hy were put to each of the trigonometric functions, all the names would be pronounceable and not too long." The proposal appeared in Nature, vol. 65 (April 10, 1902).