*Last revision: Sept. 12, 2007*

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. Herschell in 1813, and at an earlier date in
Germany by an analyst named Burmann. See Herschell, *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 sinIn 1922 in^{-1}yis generally used; on the continent of Europe the symbol arc sinyis the one that is met. The symbol sin^{-1}yis read "the inverse sine ofy," "the antisine ofy," or "the angle whose sine isy." The symbol arc sinyis read "the arc whose sine isy," or "the angle whose sine isy."

**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 cs^{2} (for cosine) and
s^{2} (for sine) (Cajori vol. 2, page 179).

**Degrees, minutes, seconds.** See the *geometry* page.

**Radians.** G. B. Halstead 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).