TRIGONOMETRY (from Gr. rpiywvov, a triangle, /ATpov, measure), the branch of mathematics which is concerned with the measurement of plane and spherical triangles, that is, with the determination of three of the parts of such triangles when the numerical values of the other three parts are given. Since any plane triangle can be divided into rightangled triangles, the solution of all plane triangles can be reduced to that of rightangled triangles; moreover, according to the theory of similar triangles, the ratios between pairs of sides of a rightangled triangle depend only upon the magnitude of the acute angles of the triangle, and may therefore be regarded as functions of either of these angles. The primary object of trigonometry, therefore, requires a classification and numerical tabulation of these functions of an angular magnitude; the science is, however, now understood to include the complete investigation not only of such of the properties of these functions as are necessary for the theoretical and practical solution of triangles but also of all their analytical properties. It appears that the solution of spherical triangles is effected by means of the same functions as are required in the case of plane triangles. The trigonometrical functions are employed in many branches of mathematical and physical science not directly concerned with the measurement of angles, and hence arises the importance of analytical trigonometry. The solution of triangles of which the sides are geodesic lines on a spheroidal surface requires the introduction of other functions than those required for the solution of triangles on a plane or spherical surface, and therefore gives rise to a new branch of science,which is from analogy frequently called spheroidal trigonometry. Every new class of surfaces which may be considered would have in this extended sense a trigonometry of its own, which would consist in an investigation of the nature and properties of the functions necessary for the measurement of the sides and angles of triangles bounded by geodesics drawn on such surfaces.
HISTORY
Trigonometry, in its essential form of showing how to deduce the values of the angles and sides of a triangle when other angles and sides are given, is an invention of the Greeks. It found its origin in the computations demanded for the reduction of astronomical observations and in other problems connectedwith astronomical science; and since spherical triangles specially occur, it happened that spherical trigonometry was developed before the simpler plane trigonometry. Certain theorems were invented and utilized by Hipparchus, but material progress was not recorded until Ptolemy collated, amended and developed the work of his predecessors. In book xi. of the Almagest the principles of spherical trigonometry are stated in the form of a few simple and useful lemmas; plane trigonometry does not receive systematic treatment although several theorems and problems are stated incidentally. The solution of triangles necessitated the construction of tables of chords—the equivalent of our modern tables of sines; Ptolemy treats this subject in book i., stating several theorems relating to multiple angles, and by ingenious methods successfully deducing approximate results. He did not invent the idea of tables of chords, for, on the authority of Theon, the principle had been stated by Hipparchus (see PTOLEMY).
The Indians, who were much more apt calculators than the Greeks, availed themselves of the Greek geometry which came from Alexandria, and made it the basis of trigonometrical calculations. The principal improvement which they introduced consists in the formation of tables of halfchords or sines instead of chords. Like the Greeks, they divided the circumference of the circle into 36o degrees or 21,600 minutes, and they found the length in minutes of the arc which can be straightened out into the radius to be 3438. The value of the ratio of the circumference of the circle to the diameter used to make this determination is 62832:20000, or lr=3.1416, which value was given by the astronomer Aryabhata (476550) in a work called Aryabhaiya, written in verse, which was republished 1 in Sanskrit by Dr Kern at Leiden in 1874. The relations between the sines and cosines of the same and of complementary arcs were known, and the formula sin 2a=1/11719(3438cosa)} was applied to the determination of the sine of a half angle when the sine and cosine of the whole angle were known. In the SuryaSiddhanta, an astronomical treatise which has been translated by Ebenezer Bourgess in vol. vi. of the Journal of the American Oriental Society (New Haven, 186o), the sines of angles at an interval of 3° 45' up to 90° are given; these were probably obtained from the sines of 6o° and 45° by continual application of the dimidiary formula given above and by the use of the complementary angle. The values sin 15°=890', sin 7° 30'=449', sin 3° 45'=225' were thus obtained. Now the angle 3° 45' is itself 225'; thus the arc and the sine of Die of the circumference were found to be the same, and consequently special importance was attached to this arc, which was called the right sine. From the tables of sines of angles at intervals of 3° 45' the law expressed by the equation
sin (n + 1.225') –sin (n.225') = sin (n.225')
—sin (n— I . 225') —sin (n 2255')
was discovered empirically, and used for the purpose of recalculation. Bhaskara (fl. 1150) used the method, to which we have now returned, of expressing sines and cosines as fractions of the radius; he obtained the more correct values sin 3° 45' = 100/1529, cos 3° 45' =466/467, and showed how to form a table, according to degrees, from the values sin 1° = 10/573, cos 10=6568/6569, which are much more accurate than Ptolemy's values. The Indians did not apply their trigonometrical knowledge to the solution of triangles; for astronomical purposes they solved rightangled plane and spherical triangles by geometry.
The Arabs were acquainted with Ptolemy's Almagest, and they probably learned from the Indians the use of the sine. The celebrated astronomer of Batnae, Albategnius (q.v.), who died in A.D. 929930, and whose Tables were translated in the 12th century by Plato of Tivoli into Latin, under the title De scientia stellarum, employed the sine regularly, and was fully conscious of the advantage of the sine over the chord; indeed, he remarks that the continual doubling is saved by the use of the former. He was the first to calculate sin from the equation sin cls/cos 4 =k, and he also made a table of the length of shadows of a vertical object of height 12 for altitudes 1°, 2°, . . of the sun; this is a sort of cotangent table. He was acquainted not only with the triangle formulae in the Almagest, but also with the formula cos a=cos b cos c + sin b sin c cos A for a spherical triangle ABC. Abu'lWafa of Bagdad (b. 940) was the first to introduce the tangent as an independent function: his " umbra " is the half of the tangent of the double arc, and the secant he defines as the " diameter umbrae." He employed the umbra to find the angle from a table and not merely as an abbreviation for sin/cos; this improvement was, however, afterwards forgotten, and the tangent was reinvented in the 15th century. Ibn Yunos of Cairo, who died in loo8, showed even more skill than Albategnius in the solution of problems in spherical trigonometry and gave improved approximate formulae for the calculation of sines. Among the West Arabs, Geber (q.v.), who lived
I See also vol. H. of the Asiatic Researches (Calcutta).
272
at Seville in the 11th century, wrote an astronomy in nine books, which was translated into Latin in the 12th century by Gerard of Cremona and was published in 1534. The first book contains a trigonometry which is a considerable improvement on that in the .4lmagest. He gave proofs of the formulae for rightangled spherical triangles, depending on a rule of four quantities, instead of Ptolemy's rule of six quantities. The formulae cos B=cos b sin A, cos c= cot A cot B, in a triangle of which C is a right angle had escaped the notice of Ptolemy and were given for the first time by Geber. Strangely enough, he made no progress in plane trigonometry. Arrachel, a Spanish Arab who lived in the 12th century, wrote a work of which we have an analysis by Purbach, in which, like the Indians, he made the sine and the arc for the value 3° 45' coincide.
Georg Purbach (1423—1461), professor of mathematics at Vienna, wrote a work entitled Tractatus super propositiones Ptolemaei de sinubus et chordis (Nuremberg, 1541). This treatise consists of a development of Arrachel's method of interpolation for the calculation of tables of sines, and was published by Regiomontanus at the end of one of his works. Johannes Muller (1436—1476), known as Regiomontanus, was a pupil of Purbach and taught astronomy at Padua; he wrote an exposition of the Almagest, and a more important work, De triangulis planis et sphericis cum tabulis sinuum, which was published in 1533, a later edition appearing in 1561. He reinvented the tangent and calculated a table of tangents for each degree, but did not make any practical applications of this table, and did not use formulae involving the tangent. His work was the first complete European treatise on trigonometry, and contains a number of interesting problems; but his methods were in some respects behind those of the Arabs. Copernicus (1473—1543) gave the first simple demonstration of the fundamental formula of spherical trigonometry ; the Trigonometria Copernici was published by Rheticus in 1542. George Joachim (1514—1576), known as Rheticus, wrote Opus palatinum de triangulis (see TABLES, MATHEMATICAL), which contains tables of sines, tangents and secants of arcs at intervals of 10" from o° to go°. His method of calculation depends upon the formulae which give sin na and cos na in terms of the sines and cosines of (n—1)a and (n—2)a; thus these formulae may be regarded as due to him. Rheticus found the formulae for the sines of the half and third of an angle in terms of the sine of the whole angle. In 1599 there appeared an important work by Bartholomew Pitiscus (1561—1613), entitled Trigonometriae seu De dimensione triangulorum; this contained several important theorems on the trigonometrical functions of two angles, some of which had been given before by Finck, Landsberg (or Lansberghe de Meuleblecke) and Adriaan van Roomen. Francois Viete or Vieta (1540—1603) employed the equation (2 cos 2¢)3—3(2 cos 2d)) =2 cos ¢ to solve the cubic x3—3a'x=a2b(a>lb); he obtained, however, only one root of the cubic. In 1593 Van Roomen proposed, as a problem for all mathematicians, to solve the equation
45Y 3795Y'+95634Y' — • • • +945Y4I45y"3+y"S=C.
Vieta gave y=2 sin where C=2 sin 0, as a solution, and also twentytwo of the other solutions, but he failed to obtain the negative roots. In his work Ad angulares sectiones. Vieta gave formulae for the chords of multiples of a given arc in terms of the chord of the simple arc.
A new stage in the development of the science was commenced after John Napier's invention of logarithms in 1614. Napier also simplified the solution of spherical triangles by his wellknown analogies and by his rules for the solution of rightangled triangles. The first tables of logarithmic sines and tangents were constructed by Edmund Gunter (1581—1626), professor of astronomy at Gresham College, London; he was also the first to employ the expressions cosine, cotangent and cosecant for the sine, tangent and secant of the complement of an arc. A treatise by Albert Girard (159o—1634), published at the Hague in 1626, contains the theorems which give areas of spherical triangles and polygons, and applications of the properties of the supplementary triangles to the reduction of the number of different cases in the solution of spherical triangles. He used the notation sin, tan, sec for the sine, tangent and secant of an arc. In the second half of the 17th century the theory of infinite series was developed by John Wallis, Gregory, Mercator, and afterwards by Newton and Leibnitz. In the Analysis per aequationes numero terminorum infinitas, which was written before 1669, Newton gave the series for the arc in powers of its sine; from this he obtained the series for the sine and cosine in powers of the arc; but these series were given in such a form that the law of the formation of the coefficients was hidden. James Gregory discovered in 1670 the series for the arc in powers of the tangent and for the tangent and secant in powers of the arc. The first of these series was also discovered independently by Leibnitz in 1673, and published without proof in the Acta eruditorum for 1682. The series for the sine in powers of the arc he published in 1693; this he obtained by differentiation of a series with undetermined coefficients.
In the 18th century the science began to take a more analytical form; evidence of this is given in the works of Kresa in 1720 and Mayer in 1727. Friedrich Wilhelm v. Oppel's Analysis triangulorum (1746) was the first complete work on analytical trigonometry. None of these mathematicians used the notation sin, cos, tan, which is the more surprising in the case of Oppel, since Leonhard Eulerhad in 1744 employed it in a memoir in the Acta eruditorum. Jean Bernoulli was the first to obtain real results by the use of the symbol I/ 1; he published in 1712 the general formula for tan n¢ in terms of tan 4,, which he obtained by means of transformation of the arc into imaginary logarithms. The greatest advance was, however, made by Euler, who brought the science in all essential respects into the state in which it is at present. He introduced the present notation into general use, whereas until his time the trigonometrical functions had been, except by Girard, indicated by special letters, and had been regarded as certain straight lines the absolute lengths of which depended on the radius of the circle in which they were drawn. Euler's great improvement consisted in his regarding the sine, cosine, &c., as functions of the angle only, thereby giving to equations connecting these functions a purely analytical interpretation, instead of a geometrical one as heretofore. The exponential values of the sine and cosine, De Moivre's theorem, and a great number of other analytical properties of the trigonometrical functions, are due to Euler, most of whose writings are to be found in the Memoirs of the St Petersburg Academy.
Plane Trigonometry.
I. Imagine a straight line terminated at a fixed point 0, and initially coincident with a fixed straight line OA, to revolve round 0, and finally to take up any Conception
position OB. We 'shall sup ~B pose that, when this revolv o'Angles
ing straight line is turning y
in one direction, say that magnitude. opposite to that in which the hands o a clock turn, it is describing a positive angle, and when it is turning in the other direction it is describing a negative angle. Before finally taking up the position 0.$ the straight line may have passed any number of times through the position OB, making any number of complete revolutions round 0 in either direction. Each time that the straight
line makes a complete revolution round 0 we consider it to have described four right angles, taken with the positive or negative sign, according to the direction ,in which it has revolved; thus, when it stops in the position OB, it may have revolved through any one of an infinite number of positive or negative angles any two of which differ from one another by a positive or negative multiple of four right angles, and all of which have the same bounding lines OA and OB. If OB' is the final position of the revolving line, the smallest positive angle which can have been described is that described by the revolving line making more than onehalf and less than the whole of a complete revolution, so that in this case we have a positive angle greater than two and less than four right angles. We have thus shown how we may conceive an angle not restricted to be less than two right angles, but of any positive or negative magnitude, to be generated.
2. Two systems of numerical measurement of angular magnitudes are in ordinary use. For practical measurements the sexagesimal system is the one employed: the ninetieth part of a right Nnmeruai
angle is taken as the unit and is called a degree; the measure' eri
degree is divided into sixty equal parts called minutes; meaand the minute into sixty equal parts called seconds; An Iar angles smaller than a second are usually measured as Magnitudes. decimals of a second, the " thirds, fourths, &c., not
being in ordinary use. In the common notation an angle, for example, of 120 degrees, 17 minutes and 14.36 seconds is written 120° 17' 14.36". The decimal system measurement of angles has never come into ordinary use. In analytical trigonometry the circular measure of an angle is employed. In this system the unit angle or radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. The constancy of this angle follows from the geometrical propositions—(L) the circumferences of different circles vary as their radii; (2) in the same circle angles at the centre are proportional to the arcs which subtend them. It thus follows that the radian is an angle independent of the particular circle used in defining it. The constant ratio of the circumference of a circle to its diameter is a number incommensurable with unity, usually denoted by lr. We shall indicate later on some of the methods which have been employed to approximate to the value of this number. Its value to 20 places is 3.14159265358979323846; its reciprocal to the same number of places is 0.31830988618379067153. In circular measure every angle is measured by the ratio which it bears to the unit angle. Two right angles are measured by the number and, since the same angle is 18o°, we see that the number of degrees in an angle of circular measure B is obtained from the formula 18oXO/ir. The value of the radian has been found to 41 places of decimals by Glaisher (Prot. London Math. Soc. vol. iv.) ; the value of from which the unit can easily be calculated, is given to 140 places of decimals in Grunerts Archiv (1841), vol. i. To lo decimal places the value of the unit angle is 57° 17' 44.8062470964". The unit of circular measure is too large to be convenient for practical purposes, but its use introduces a simplification into the series in analytical trigonometry, owing to the fact that the sine of
an angle and the angle itself in this measure, when the magnitude of the angle is indefinitely diminished, arc ultimately in a ratio of equality.
3. If a point moves from a position A to another position B on a straight line, it has described a length AB of the straight line. It Sign of is convenient to have a simple mode of indicating in portions of which direction on the straight line the length AB has an infinite been described; this may be done by supposing that a
Straight point moving in one specified direction is describing
Line. a positive length, and when moving in the opposite
direction a negative length. Thus, if a point moving from A to B is moving in' the positive direction, we consider the length AB as positive; and, since a point moving from B to A is moving in the negative direction, we consider the length BA as negative. Hence any portion of an infinite straight line is considered to be positive or negative according to the direction in which we suppose this portion to be described by a moving point; which direction is the positive one is, of course, a matter of convention.
If perpendiculars AL, BM be drawn from two points, A, B on any straight line, not necessarily in the same plane with AB, the projections length LM, taken with the positive or negative sign of straight according to the convention as stated above, is called Lines on the projection of AB on the given straight line; the each other. projection of BA being ML has the opposite sign to the
projection of AB. If two points A, B be joined by a number of lines in any manner, the algebraical sum of the projections of all these lines is LM—that is, the same as the projection of AB. Hence the sum of the projections of all the sides, taken in order, of any closed polygon, not necessarily plane, on any straight line, is zero. This principle of projections we shall apply below to obtain some of the most important propositions in trigonometry.
4. Let us now return to the conception of the generation of an angle as in fig. 1. Draw BOB' at right angles to and equal to AA'.
We shall suppose that the direction from A' to A is the
Definition
of Mono, positive one for the straight line AOA', and that from metrical B' to B for BOB'. Suppose OP of fixed length, equal Functions. to OA, and let PM, PN be drawn perpendicular to
A'A, B'B respectively ; then OM and ON, taken with their proper signs, are the projections of OP on A'A and B'B. The ratio of the projection of OP on B'B to the absolute length of OP is dependent only on the magnitude of the angle POA, and is called the sine of that angle; the ratio of the projection of OP on A'A to the length OP is called the cosine of the angle POA. The ratio of the sine of an angle to its cosine is called the tangent of the angle, and that of the cosine to the sine the cotangent of the angle; the reciprocal of the cosine is called the secant, and that of sine the cosecant of the angle. These functions of an angle of magnitude a are denoted by sin a, cos a, tan a, cot a, sec a, cosec a respectively. If any straight line RS be drawn parallel to OF, the projection of RS on either of the straight lines A'A, B'B can be easily seen to bear to RS the same ratios which the corresponding projections of OP bear to OP; thus, if a be the angle which RS makes with A'A, the projections of RS on A'A, B'B are RS cos a and RS sin a respectively, where RS denotes the absolute, length K.S. It must he observed that the line SR is to be considered as parallel not to OP but to OP", and therefore makes an angle Zr+a with .4'A ; this is consistent with the fact that the projections of SR are of opposite sign to those of RS. By observing the signs of the projections of OP for the positions P, P', P", P"' of P we see that the sine and cosine of the angle POA are both positive; the sine of the angle P'OA is positive and its cosine is negative; both the sine and the cosine of the angle PTA are negative; and the sine of the angle P"'OA is negative and its cosine positive. If a be the numerical value of the smallest angle of which OP and OA are boundaries, we see that, since these straight lines also bound all the angles 2nr+a, where n is any positive or negative integer, the sines and cosines of all these angles are the same as the sine and cosine of a. Hence the sine of any angle 2n7r+a is positive if a is between o and 7r and negative if a is between r and 27r, and the cosine of the same angle is positive if a is between o and 17r Or IT and 27r and negative if a is between Zr and 17r.
In fig. 2 the angle P0A is a, the angle P"'OA is —a, P'OA is 7r—a, P"O A is 7r+a, POB is 17r—a. By observing the signs of the projections we see that
sin(—a) _ —sin a, sin(Zra) asin a, sin (a+a) —sin a, cos(— a) =cog Cl, COS(Tr—a) = —COS a, cos(7r+a) = cos a, sin(4Tr—a) =cos a, cos(17r—a) =sin a.
Also sin(;ir+a) =sin(Zr—17r—a) = sin(za — a) = cos a, cos(7r+a) = —cos(1r—y1r—a) = —cos(27r—e) = —sin a.
From these equations we have tan(—a) = —tan a, tan(Zr—a) _ —tan a,tan (a+a)= —tan a, tan(ir—a) =cot a, tan (jn+a) _ —cot a, with corresponding equations for the cotangent.
The only angles for which the projection of OP on B'B is the same as for the given angle POA( a) are the two sets of angles bounded by OP, OA and OP', OA; these angles are 2n,r+a and 2n7r+(lr—a), and are all included in the formula r7r+(—1)'a, where r is any integer.; this therefore is the formula for all angles having the same sine as a. The only angles which have the sane cosine as a are those bounded by OA, OP and OA, OP", and these are all included in the formula 2117r =a. Similarly it can be shown that nir+a includes all the angles which have the same tangent as a.
From the Pythagorean theorem, the sum of the squares of the projections of any straight line upon two straight lines at right angles to one another is equal to the square on the Relations projected line, we get sin2a+costa=l, and from this between by the help of the definitions of the other functions we Trigonodeduce the relations i + tan2a = sec2a, I + cot2a = metrical cosec2a. We have now six relations between the six Functions. functions; !these enable us to express any five of these
functions in terms of the sixth. The following table shows the values of the trigonometrical functions of the angles 0, ir, ,r, ',7r, z7r, and the signs of the functions of angles between these values; I denotes numerical increase and D numerical decrease:
Angle . 0 0...2ir . it 17r...7r 7r 7r...o7r :; 7r ;; 7r...27r 27r
Sine . o +I I +D o —I 1 —D o
Cosine I +D o —I —I —D o +I I
Tangent . o +I too —D o +I =oo —D o
Cotangent too +D o —I too +D o —I too
Secant i +I too —D — i —I too +D I
Cosecant . too +D I +I =oo —D —I —I too
The correctness of the table may be verified from the figure by considering the magnitudes of the projections of OP for different positions.
The following table shows the sine and cosine of some angles for which the values of the functions may be obtained geometrically:
I' 15° sine cosine 75° r
'sine v6+J2 712
JJ2
4 4
18° 'J5—I IO+2J5 72° 2 a
10 4 4 5
° 6o°
6 30 2 23 37r
36° JIO—2s/ 5 J5+1 54° Zr
5 4 4 10
I 1
45° 45° 7r
4 cosine sine 4
are obtained as follows. (I) fir. The sine and
cosine of this angle are equal to one another, since sin values of ir=cos (sr—',ir); and since the sum of the squares Trigonoof the sine and cosine is unity each is 1/J 2. (2) *7r and 37r. metrical Consider an equilateral triangle; the projection of one u functions side on another is obviously half a side; hence the cosine of an angle of the triangle is 2 or cos 37r =i, and from Angles. this the sine is found. (3) 7r/I0, 45, 277/5, 37r/10. In the triangle constructed in Euc. iv. 10 each angle at the base is ?7r, and the vertical angle is i ir. If a be a side and b the base, we have by the construction a(a—b) =b"; hence 2b=a (115 —I); the sine of Zr/io is b/2a or 4 (J5—I), and cos IT is a/2b=& (J5+1). (4) i 27r, P .27r. Consider a rightangled triangle, having an angle s7r. Bisect this angle, then the opposite side is cut by the bisector in the ratio of 113 to 2; hence the length of the smaller segment is to that of the whole in the ratio of 1,13 to J3+2, therefore tan l?27r={113/(J3+2)} tan 67r or tan 1s7r=2'3, and from this we can obtain sin i 7r and cos ir.
5. Draw a straight line OD making any angle A with a fixed straight line OA, and draw OF making D
an angle B with OD, this Formulae angle being measured Pose for Sine and tively in the same direction cosine of as A ; draw FE a perpen Sum and dicular on DO (produced if Difference of necessary). The projection Tic o Angles. of OF on 0A is the sum of
the projections of OE and EF on OA. F
Now OE is the projection of OF on DO, FIG. 3.
and is therefore equal to OF cos B, and EF is the projection of OF
B
274
on a straight line making an angle + Iv with OD, and is therefore equal to OF sin B; hence
OF cos (A +B) =OE cos A+EF cos (27r+A)
=OF (cos A cos B—sin A. sin B),
or cos (A+B) =cos A cos B—sin A sin B.
The angles A, B are absolutely unrestricted in magnitude, and thus
this formula is perfectly general. We may change the sign of B, thus
cos (A —B) =cos A cos (—B) —sin A sin (—B),
or cos (A —B) =cos A cos B+sin A sin B.
If we projected the sides of the triangle OEF on a straight line making an angle+j;7r with OA we should obtain the formulae
sin (A rsB) =sin A cos B *cos A sin B,
which are really contained in the cosine formula, since we may put ,7r—B for B. The formulae
tan A t tan B cot (A tB) = cot A cot B;=
tan (A tB) = i
tan A tan B' cot B * cot A
are immediately deducible from the above formulae. The equations sin C+sin D=2 sin I(C+D) cos 2(C—D),
sin C—sin D=2 sin 1(C—D) cos'2(C+D),
cos D+cos C=2 Cos I(C+D) cos 2(C—D),
cos D—cos C=2 sin I(C+D) sin 1(C—D),
may be obtained directly by the method of projections. Take two equal straight lines OC, OD, making angles C, D, with OA, and draw OE perpendicular to CD. The angle which OE makes
with OA is 1(C+D) and that which
DC makes is I(3r+C+D); the angle
End of Article: TRIGONOMETRY (from Gr. rpiywvov, a triangle, /ATpov, measure) 

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