60

x x x

2 sin ⁡ ( x / 2 ) 2\sin (x/2) 2 sin ( x / 2 )

140

100

Ⓣ ( Rule of six quantities. )

360

120

3

sin ⁡ 2 x + cos ⁡ 2 x = 1 \sin^{2} x + \cos^{2} x = 1 sin 2 x + cos 2 x = 1

sin ⁡ ( x + y ) = sin ⁡ x cos ⁡ y + cos ⁡ x sin ⁡ y \sin(x + y) = \sin x \cos y + \cos x \sin y sin ( x + y ) = sin x cos y + cos x sin y



a / sin ⁡ A = b / sin ⁡ B = c / sin ⁡ C a/\sin A = b/\sin B = c/\sin C a / sin A = b / sin B = c / sin C .

3

4

5

6

10

36

72

60

90

120

30

(30

1

)

1

60

0

31

25

0

0087268

6

7

0

0087265

500

(

628)

1150

980

sin ⁡ 2 x = 2 sin ⁡ x cos ⁡ x \sin 2x = 2 \sin x \cos x sin 2 x = 2 sin x cos x

sin ⁡ ( x + y ) = sin ⁡ x cos ⁡ y + cos ⁡ x sin ⁡ y \sin(x + y) = \sin x \cos y + \cos x \sin y sin ( x + y ) = sin x cos y + cos x sin y

x = y x = y x = y

1542

1533

Ⓣ ( The properties of triangles )

1464

1624

1634

versin x = 1 − cos ⁡ x x = 1 - \cos x x = 1 − cos x .

(

)

90

(1620)

2

sin ⁡ n x \sin nx sin n x

sin ⁡ x \sin x sin x

cos ⁡ x \cos x cos x

(

)

sin ⁡ 3 x = 3 cos ⁡ 2 x sin ⁡ x − sin ⁡ 3 x \sin 3x = 3 \cos ^{2}x \sin x - \sin ^{3} x sin 3 x = 3 cos 2 x sin x − sin 3 x



cos ⁡ 3 x = cos ⁡ 3 x − 3 sin ⁡ 2 x cos ⁡ x \cos 3x = \cos ^{3}x - 3 \sin ^{2}x \cos x cos 3 x = cos 3 x − 3 sin 2 x cos x .

860

1583

1620

2

t t t

t t t

T T T

t t t

1626

tan

A

1674

15

th

cosec x x x /sec x x x = cot x = 1 / tan ⁡ x x = 1/\tan x x = 1 / tan x



1 /cosec x x x = cos x x x /cot x = sin ⁡ x x = \sin x x = sin x .

2

s s s

1595

sin ⁡ 2 x , sin ⁡ 3 x , cos ⁡ 2 x , cos ⁡ 3 x \sin 2x, \sin 3x, \cos 2x, \cos 3x sin 2 x , sin 3 x , cos 2 x , cos 3 x

18

th

sin ⁡ − 1 z \sin^{-1}z sin − 1 z

log ⁡ z \log z lo g z

1702

1722

i x = log ⁡ ( cos ⁡ x + i sin ⁡ x ) ix = \log(\cos x + i \sin x ) i x = lo g ( cos x + i sin x ) .

( cos ⁡ x + i sin ⁡ x ) n = cos ⁡ n x + i sin ⁡ n x (\cos x + i \sin x )^{n} = \cos nx + i \sin nx ( cos x + i sin x ) n = cos n x + i sin n x

1722

1748

(

)

e x p ( i x ) = cos ⁡ x + i sin ⁡ x exp(ix) = \cos x + i \sin x e x p ( i x ) = cos x + i sin x .

The use of trigonometric functions arises from the early connection between mathematics and astronomy. Early work with spherical triangles was as important as plane triangles. The first work on trigonometric functions related to chords of a circle. Given a circle of fixed radius,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 anglewas. The first known table of chords was produced by the Greek mathematician Hipparchus in aboutBC. 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 aboutAD. 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 Ptolemy was the next author of a book of chords, showing the same Babylonian influence as Hipparchus , dividing the circle into° and the diameter intoparts. The suggestion here is that he was following earlier practice when the approximationfor π was used. Ptolemy , together with the earlier writers, used a form of the relation, 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 Ptolemy calculated chords by first inscribing regular polygons ofandsides in a circle. This allowed him to calculate the chord subtended by angles of°,°,°,° and°. He then found a method of finding the cord subtended by half the arc of a known chord and this, together with interpolation allowed him to calculate chords with a good degree of accuracy. Using these methods Ptolemy found that sin' = half ofwhich is the chord of° was, as a number to base". Converted to decimals this iswhich is correct todecimal places, the answer todecimal places being The first actual appearance of the sine of an angle appears in the work of the Hindus. Aryabhata , in about, 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 inand detailed method for constructing a table of sines for any angle were give by Bhaskara in The Arabs worked with sines and cosines and by Abu'l-Wafa knew thatalthough it could have easily have been deduced from Ptolemy 's formulawith The Hindu wordfor the sine was adopted by the Arabs who called the sine, a meaningless word with the same sound as. Nowbecamein 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 termsoon encouraged the universal use of sine. Chapters of Copernicus 's book giving all the trigonometry relevant to astronomy was published inby Rheticus Rheticus also produced substantial tables of sines and cosines which were published after his death. In Regiomontanus 's workwas published. This contained work on planar and spherical trigonometry originally done much earlier in about. 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 inin a drawing. The first use of sin in a book was inby the French mathematician Hérigone while Cavalieri usedand Oughtred 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 formulaIt is just the sine turnedversedthrough°. The cosine follows a similar course of development in notation as the sine. Viète used the term sinus residuae for the cosine, Gunter suggested co-sinus. The notation Si.was used by Cavalieri , s co arc by Oughtred and S by Wallis Viète knew formulas forin terms ofand. He gave explicitly the formulasdue to Pitiscus The tangent and cotangent came via a different route from the chord approach of the sine. These developed together and were not at first associated with angles. They became important for calculating heights from the length of the shadow that the object cast. The length of shadows was also of importance in the sundial. Thales used the lengths of shadows to calculate the heights of pyramids. The first known tables of shadows were produced by the Arabs aroundand used two measures translated into Latin asand Viète used the terms amsinus and prosinus. The name tangent was first used by Thomas Fincke in. The term cotangens was first used by Edmund Gunter in Abbreviations for the tan and cot followed a similar development to those of the sin and cos. Cavalieri used Ta and Ta. Oughtred usedarc andco arc while Wallis usedand. The common abbreviation used today is tan by we write tan whereas the first occurrence of this abbreviation was used by Albert Girard in, but tan was written over the angle cot was first used by Jonas Moore inThe secant and cosecant were not used by the early astronomers or surveyors. These came into their own when navigators around theCentury started to prepare tables. Copernicus knew of the secant which he called the hypotenusa. Viète knew the resultsThe abbreviations used by various authors were similar to the trigonometric functions already discussed. Cavalieri used Se and Se. Oughtred used se arc and sec co arc while Wallis usedand 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 bookby B Pitiscus , published in Pitiscus also discovered the formulas for TheCentury saw trigonometric functions of a complex variable being studied. Johann Bernoulli found the relation betweenandinwhile Cotes , in a work published inafter his death, showed that De Moivre published his famous theorem inwhile Euler , in, gave the formulaequivalent to that of Cotes The hyperbolic trigonometric functions were introduced by Lambert