Encyclopædia Britannica, Ninth Edition/Trigonometry

by Ernest William Hobson

TRIGONOMETRY

TRIGONOMETRY is primarily the science 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 right-angled triangles, the solution of all plane triangles can be reduced to that of right-angled triangles; moreover, according to the theory of similar triangles, the ratios between pairs of sides of a right-angled 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.

Greek. An account of Greek trigonometry is given under Ptolemy (q.v.).

Indian. 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 half-chords or sines instead of chords. Like the Greeks, they divided the circumference of the circle into 360 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 $\pi =3{\mathord {\cdot }}1416$, which value was given by the astronomer Âryabhata (476-550; see Sanskrit vol. xxi. p. 294) in a work called Âryabhatiya, written in verse, which was republished^[1] in Sanskrit by Dr Kern at Leyden in 1874. The relations between the sines and cosines of the same and of complementary arcs were known, and the formula $\sin {\frac {1}{2}}\alpha ={\sqrt {1719(3438-\cos \alpha )}}$ 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 Sûrya-Siddhânta, an astronomical treatise which has been translated by Ebenezer Bourgess in vol. vi. of the Journal of the American Oriental Society (New Haven, 1860), the sines of angles at an interval of 3° 45' up to 90° are given; these were probably obtained from the sines of 60° and 45° by continual application of the dimidiary formula given above and by the use of the complementary angle. The values $\sin 15^{\circ }=890'$, $\sin 7^{\circ }30'=449'$, $\sin 3^{\circ }45'=225'$, were thus obtained. Now the angle 3° 45' is itself 225'; thus the arc and the sine of ¹⁄₉₆ 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

$$\sin({\overline {n+1}}.225')-\sin(n.225')=\sin(n.225')-\sin({\overline {n-1}}.225)-\sin {\frac {n.225'}{225}}$$

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 $\sin 3^{\circ }45'=100/1529$, $\cos 3^{\circ }45=466/467$, and showed how to form a table, according to degrees, from the values $\sin 1^{\circ }=10/573$, $\cos 1^{\circ }=6568/6569$, 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 right-angled plane and spherical triangles by geometry.

The Arabs were acquainted with Ptolemy's Almagest, Arabian. and they probably learned from the Indians the use of the sine. The celebrated astronomer of Batnæ, Abú 'Abdallah Mohammed b. Jābir al-Battání (Bategnius), who died in 929/930 A.D., 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 war fully conscious of the advantage of the sine over the chord; indeed, he remarks that the continued doubling is saved by the use of the former. He was the first to calculate $\sin \phi$ from the equation $\sin \phi /\cos \phi =k$, and he also made a table of the lengths 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 formulæ in the Almagest, but also with the formula $\cos a=\cos b\cos c+\sin b\sin c\cos A$ for a spherical triangle $ABC$. Abú 'l-Wafá of Baghdad (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 umbræ." 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 re-invented in the 15th century. Ibn Yúnos of Cairo, who died in 1008, showed even more skill than Al-Battání in the solution of problems in spherical trigonometry and gave improved approximate formulæ for the calculation of sines. Among the West Arabs, Abú Mohammed Jābir b. Aflah, known as Geber b. Aflaḥ, who lived 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 Almagest. He gave proofs of the formulæ for right-angled spherical triangles, depending on a rule of four quantities, instead of Ptolemy's rule of six quantities. The formulæ $\cos B=\cos b\sin A$, $\cos c=\cot A\cot B$, 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.

Modern.Purbach (1423-1461), professor of mathematics at Vienna, wrote a work entitled Tractatus super propositiones Ptolemæi 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 Müller (1436-1476), known as Regiomontanus (q.v.), 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 re-invented 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 formulæ 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 (q.v.), wrote Opus Palatinum de triangulis (see Tables, p. 9 above), which contains tables of sines, tangents, and secants of arcs at intervals of 10" from 0° to 90°. His method of calculation depends upon the formulæ which give $\sin n\alpha$and $\cos n\alpha$ in terms of the sines and cosines of $(n-1)\alpha$ and $(n-2)\alpha$; thus these formulæ may be regarded as due to him. Rheticus found the formulæ 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 Pitiscus (1561-1613), entitled Trigonometriæ 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, and Adriaan van Roomen. François Viète or Vieta (q.v.) (1540-1603) employed the equation $(2\cos {\tfrac {1}{2}}\phi )^{3}-3(2\cos {\tfrac {1}{2}}\phi )=2\cos \phi$ to solve the cubic $x^{3}-3a^{2}x=a^{2}b(a>{\tfrac {1}{2}}b)$; he obtained, however, only one root of the cubic. In 1593 Van Roomen proposed, as a problem for all mathematicians, to solve the equation

$45y-3795y^{3}+95634y^{5}-{...}+945y^{41}-45y^{43}+y^{45}=C.$

Viète gave $y=2\sin {\tfrac {1}{45}}\phi$, where $C=2\sin \phi$, as a solution, and also twenty-two of the other solutions, but he failed to obtain the negative roots. In his work Ad angulares sectiones Viète gave formulæ 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 Napier's invention of logarithms in 1614. Napier also simplified the solution of spherical triangles by his well-known analogies and by his rules for the solution of right-angled 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 (1590-1634), 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 Wallis, Gregory, Mercator, and afterwards by Newton and Leibnitz. In the Analysis per æquationes 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. 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 Euler had in 1744 employed it in a memoir in the Acta eruditorum. John Bernoulli was the first to obtain real results by the use of the symbol ${\sqrt {-1}}$; he published in 1712 the general formula for $\tan n\phi$ in terms of $\tan \phi$, 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.

The preceding sketch has been mainly drawn from the following sources:—Cantor, Gesch. d. Math.; Hankel, Gesch. d. Math.; Marie, Hist. des sc. math.; Suter, Gesch. d. Math.; Klügel, Math. Wörterbuch.

Plane Trigonometry.

Conception of angles of any magnitude. Imagine a straight line terminated at a fixed point O, and initially coincident with a fixed straight line OA, to revolve round O, and finally to take up any position OB. We shall suppose that, when this revolving straight line is turning in one direction, say that opposite to that in which the hands of a clock turn, it is describing a positive angle, and when it is turning in the other direction it is describing a negative angle.

Fig. 1.

Before finally taking up the position OB the straight line may have passed any number of times through the position OB, making any number of complete revolutions round in either direction. Each time that the straight line makes a complete revolution round 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 one-half 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 less than two right angles, but of any positive or negative magnitude, to be generated.

Numerical measurement of angular magnitudes. 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 angle is taken as the unit and is called a degree; the degree is divided into sixty equal parts called minutes; and the minute into sixty equal parts called seconds; angles smaller than a second are usually measured as 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 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—(1) 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 unit mentioned above 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 quantity incommensurable with unity, usually denoted by $\pi$. We shall indicate later on (p. 571 sq.) some of the methods which have been employed to approximate to the value of this quantity. Its value to 20 places is 3·14159265358979323846; its reciprocal to the same number of places is ·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 quantity if, and, since the same angle is 180°, we see that the number of degrees in an angle of circular measure $\theta$ is obtained from the formula $180\times \theta /\pi$. The value of the unit of circular measure has been found to 41 places of decimals by Glaisher (Proc. London Math. Soc., vol. iv.): the value of ${\frac {1}{\pi }}$, from which the unit can be easily calculated, is given to 140 places of decimals in Grunert's Archiv, vol. i., 1841. To 10 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, are ultimately in a ratio of equality.

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 is convenient to have a simple mode of indicating in which direction on the straight line the length AB has been described; this may be done by supposing that a point moving in one specified directionSign of portions of an infinite straight line is describing 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.

Projections of straight lines on each otherIf perpendiculars AL, BM be drawn from two points A, B on any straight line, not necessarily in the same plane with AB, the length LM, taken with the positive or negative sign according to the convention as stated above, is called the projection of AB on the given straight line; the 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 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.

Definition of trigonometrical functions.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'.

Fig. 2.

We shall suppose that the direction from A' to A is the positive one for the straight line AOA', and that from B' to B for BOB'. Suppose OP of fixed length, equal to OA, and let PM, PN be drawn perpendicular to A'A, B'B; 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 the sine the cosecant of the angle. These functions of an angle of magnitude an are denoted by $\sin \alpha$, $\cos \alpha$, $\tan \alpha$, $\cot \alpha$, $\sec \alpha$, ${\text{cosec }}\alpha$ respectively. If any straight line RS be drawn parallel to OP, 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 $\alpha$ be the angle which RS makes with A'A, the projections of RS on A'A, B'B are $RS\cos \alpha$ and $RS\sin \alpha$ respectively, where RS denotes the absolute length RS. It must be observed that the line SR is to be considered as parallel not to OP but to OP'', and therefore makes an angle \pi+ a with A 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 P''OA are negative; and the sine of the angle P'''OA are negative and its cosine positive. If $\alpha$ 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 $2n\pi +\alpha$, 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 $\alpha$. Hence the sine of any angle $2n\pi +\alpha$ is positive if $\alpha$ is between 0 and $\pi$ and negative if $\alpha$ is between $\pi$ and $2\pi$, and the cosine of the same angle is positive if $\alpha$ is between 0 and ${\frac {1}{2}}\pi$ or ${\frac {3}{2}}\pi$ and $2\pi$ and negative if $\alpha$ is between ${\frac {1}{2}}\pi$ and ${\frac {3}{2}}\pi$.

In fig. 2 if the angle POA is $\alpha$, the angle $P{'}{'}OA$ is $-\alpha$, P'OA is $\pi -\alpha$, P''OA is $\pi +\alpha$, POB is ${\frac {\pi }{2}}-\alpha$. By observing the signs of the projections we see that

$$\sin(-\alpha )=-\sin \alpha ,\sin(\pi -\alpha )=\sin \alpha ,\sin(\pi +\alpha )=-\sin \alpha$$,

$$\cos(-\alpha )=\cos \alpha ,\cos(\pi -\alpha )=-\cos \alpha ,cos(\pi +\alpha )=-\cos \alpha$$,

$$\sin({\frac {1}{2}}\pi -\alpha )=\cos \alpha ,\cos({\frac {1}{2}}\pi -\alpha )=\sin \alpha$$.

Also

$$\sin({\frac {1}{2}}\pi +\alpha )=\sin(\pi -{\frac {1}{2}}\pi -\alpha )=\sin({\overline {{\frac {1}{2}}-\alpha }})=\cos \alpha$$,

$$\cos({\frac {1}{2}}\pi +\alpha )=\cos(\pi -{\frac {1}{2}}\pi -\alpha )=-cos({\overline {{\frac {1}{2}}\pi -\alpha }})-\sin \alpha$$.

From these equations we have $\tan(-\alpha )=-\tan \alpha$, $\tan(\pi -\alpha )=-\tan \alpha$, $\tan(\pi +\alpha )=-\tan \alpha$, $\tan({\frac {1}{2}}\pi -\alpha )=\cot \alpha$, $\tan({\frac {1}{2}}\pi +\alpha )=-\cot \alpha$, 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 ($=\alpha$) are the two sets of angles bounded by OP, OA, and OP', OA; these angles are $2n\pi +\alpha$ and $2n\pi +{\overline {\pi -\alpha }}$, and are all included in the formula $r\pi +(-1)^{r}\alpha$, where r is any integer; this therefore is the formula for all angles having the same sine as $\alpha$. The only angles which have the same cosine as an are those bounded by OA, OP and OA, OP'', and these are all included in the formula $2n\pi \pm \alpha$. Similarly it can be shown that $n\pi +\alpha$ includes all the angles which have the same tangent as $\alpha$.

From the Pythagorean theorem, the sum of the squares of the projections of any straight line upon two straight lines at right tween angles to one another is equal to the square on the projected line, we get $\sin ^{2}\alpha +\cos ^{2}\alpha =1$, and from this by the help of the definitions of the other functions we deduce the relations $1+\tan ^{2}\alpha ={\text{sec}}^{2}\alpha$, $1+\cot ^{2}\alpha ={\text{cosec}}^{2}\alpha$. We have now six relations between the six 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, ${\frac {1}{2}}\pi$, $\pi$, ${\frac {3}{2}}\pi$, $2\pi$, and the signs of the functions of angles between these values; I denotes numerical increase and D numerical decrease.

Angle	$0$	$0\dots {\frac {1}{2}}\pi$	${\frac {1}{2}}\pi$	${\frac {1}{2}}\pi \dots \pi$	$\pi$	$\pi \dots {\frac {3}{2}}\pi$	${\frac {3}{2}}\pi$	${\frac {3}{2}}\pi \dots 2\pi$	$2\pi$
Sine	0	+I	1	+D	0	-I	-1	-D	0
Cosine	1	+D	0	-I	-1	-D	0	+I	1
Tangent	0	+I	$\pm \infty$	-D	0	+I	$\pm \infty$	-D	0
Cotangent	$\pm \infty$	+D	0	-I	$\pm \infty$	+D	0	-I	$\pm \infty$
Secant	1	+I	$\pm \infty$	-D	-1	-I	$\pm \infty$	+D	1
Cosecant	$\pm \infty$	+D	1	+I	$\pm \infty$	-D	-1	-I	$\pm \infty$

The correctness of the table may be verified from the figure by considering the magnitudes of the projections of OP for different positions.

Values of trigonometrical functions for some angles. The following table shows the sine and cosine of some angles for which the values of the functions may be obtained geometrically:—

		sine	cosine
${\frac {\pi }{12}}$	15°	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	75°	${\frac {5}{12}}\pi$
${\frac {\pi }{10}}$	18°	${\frac {{\sqrt {5}}-1}{4}}$	${\frac {\sqrt {10+2{\sqrt {5}}}}{4}}$	72°	${\frac {2}{5}}\pi$
${\frac {\pi }{6}}$	80°	${\frac {1}{2}}$	${\frac {\sqrt {3}}{2}}$	60°	${\frac {1}{3}}\pi$
${\frac {\pi }{5}}$	36°	${\frac {\sqrt {10-2{\sqrt {5}}}}{4}}$	${\frac {{\sqrt {5}}+1}{4}}$	54	${\frac {3}{10}}\pi$
${\frac {\pi }{4}}$	45°	${\frac {1}{\sqrt {2}}}$	${\frac {1}{\sqrt {2}}}$	45°	${\frac {1}{4}}\pi$
		cosine	sine

These are obtained as follows. (1) ${\frac {\pi }{4}}$. The sine and cosine of this angle are equal to one another, since $\sin {\frac {\pi }{4}}=\cos({\frac {\pi }{2}}-{\frac {\pi }{4}})$, and since the sum of the squares of the sine and cosine is unity each is ${\frac {1}{\sqrt {2}}}$. (2) ${\frac {\pi }{6}}$ and ${\frac {\pi }{3}}$. Consider an equilateral triangle; the projection of one side on another is obviously half a side; hence the cosine of an angle of the triangle is ${\frac {1}{2}}$ or $\cos {\frac {\pi }{3}}={\frac {1}{2}}$, and from this the sine is found. (3) ${\frac {\pi }{10}},{\frac {\pi }{5}},{\frac {2\pi }{5}},{\frac {3\pi }{10}}$. In the triangle constructed in Euc. iv., each angle at the base is ${\frac {2\pi }{5}}$, and the vertical angle is ${\frac {\pi }{5}}$/. If a be a side and b be the base, we have by the construction $a(a-b)=b^{2}$; hence $2b=a({\sqrt {5}}-1)$; the sine of ${\frac {\pi }{10}}$ is ${\frac {b}{2a}}$ or ${\frac {{\sqrt {5}}-1}{4}}$, and $\cos {\frac {\pi }{5}}$ is ${\frac {a}{2b}}={\frac {{\sqrt {5}}+1}{4}}$. (4) ${\frac {\pi }{12}},{\frac {5\pi }{12}}$. Consider a right-angled triangle, having an angle ${\frac {\pi }{6}}$. Bisect this angle, then the opposite side is cut by the bisector in the ratio of ${\sqrt {3}}$ to 2; hence the length of the smaller segment is to that of the whole in the ratio of ${\sqrt {3}}$ to ${\sqrt {3}}+2$, therefore $\tan {\frac {1}{12}}\pi ={\frac {\sqrt {3}}{{\sqrt {3}}+2}}\tan {\frac {1}{6}}\pi$ or $\tan {\frac {1}{12}}\pi =2-{\sqrt {3}}$, and from this we can obtain $\sin {\frac {1}{12}}\pi$ and $\cos {\frac {1}{12}}\pi$.

Formulæ for some and cosine of sum and difference of two angles. Draw a straight line OD making any angle A with a fixed straight line OA, and draw OF making an angle B with OD, this angle being measured positively in the same direction as A ; draw FE a perpendicular on DO (produced if necessary). The projection of OF on OA is the sum of the projections of OE and EF on OA. Now OE is the projection of OF on DO, and is therefore equal to OF cos B, and EF is the projection of OF on a straight line making an angle + JT with OD, and is therefore equal to O^sin B ; hence OF cos (A +)= OEcos A+EFcos (ir+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 -) = cos A cos B + sin A sin B.

If we projected the sides of the triangle OEF on a straight line making an angle +^ir with OA we should obtain the formulas

= sin A cosJ5cos A sin B,

which are really contained in the cosine formula, since we may put frr - B for B. The formulæ

/^_LT> tan^4tan.B ,/>i DN cot ^4 ta.n(A) = - l^tan^tan^ ~ cot^?cot^

are immediately deducible from the above formulas. The equations

sin C+ sin D=2 sin (G+D) cosfc (G- D), sin C f -sin2)=2sini(C -Z>)cos|(C +Z)),\\ cosD + cos C=2cosl(G+D)cosl(C-D), cos D- cos G=2sin^(C+D)sin^(G-D),

may be obtained directly by the method of projections. Take two equal straight lines OG, OD, making angles G, D with OA, and draw OE perpendicular to CD. The angle which OE makes

with OA is ^(G+D) and that which DC makes the angle COE is (G-D). The sum of the projections of OD and DE on OA is equal to that of OE, and the sum of the projections of OD and DE is equal to that of OC ; hence the sum of the projections of 00 and OD is twice that of OE, or cos G + cos D = 2 cos ^(G+D) cos (C-D). The difference of the projections of OD and OG Fi S- 4 - on OA is equal to that of ED, hence we have the formula cos D - cos G=2 sin %(G+D) sin ^(C- D). The other two formulas will be obtained by projecting on a straight line inclined at an angle + ir to OA.

Sum of series of cosines in arithmetical progression. As another example of the use of projections, we will find the sum of the series cos a + cos (o + /3) + cos (a + 2/3) + . . . + cos (a + ?i-l |8). series of Suppose an unclosed polygon each angle of which is TT - /3 to be in- scribed in a circle, and let A lt A 2 , A 3 , . . ., A n be n + 1 consecutive angular points ; let D be the diameter of the circle ; and suppose a straight line drawn making an angle a with AA lt then a + /3, progres- a + 2/3, . . . are the angles it makes with A 1 A 2 , A 2 A 3 , . . . ; we have by s i u - projections (1 /3 a -) j-*C j = ^^ 1 (cos o + cos o + /3 + . .". -f cos a + n - 1 ft), J^lgO AA 2 2 hence the sum of the series of cosines is cos ( a H ^ -

Formulæ for some and cosine of sum of angles By a double application of the addition formulas we may obtain the formulas

sin (A l + A 2 + A 3 ) = sin A cos A% cos A 3 + cos A l sin A 2 cos A 3 + cos A l cos y* 2 sin ^ 3 - sin A l sin ^ 2 sin A 3 ; cos (-<4 1 + A 2 + A 3 ) = cos ^j cos A 2 cos ^4 3 cos A sin ^4 2 sin A 3 - sin ^4 X cos A 2 sin ^ 3 - sin A l sin ^ 2 cos ^ 3 .

We can by induction extend these formulas to the case of n angles. Assume sin (A 1 +A 2 + ... + A n ) = S 1 -S 3 + S 5 - .. . cos (A-i + A 2 + . . . + A n ) = SQ S 2 + 84 ... where S r denotes the sum of the products of the sines of? of the angles and the cosines of the remaining n-r angles ; then we have

sin (A 1 + A 2 + ... + A n + A n+1 ) = cos A n+l (S l S 3 + S 6 . . . ) + sin A n+ i(S -S 2 + S t - ...).

The right-hand side of this equation may be written (S 1 cos A n+1 + S sin A n+l ) - (S 3 cos A n+l + S 2 sin A n+1 ) + ..., or S 1 -S 3 + ... where S r denotes the quantity which corresponds for n + l angles to S r for n angles ; similarly we may proceed with the cosine for mula. The theorems are true for n=2 and n=3 ; thus they are true generally. The formulas cos 2 A = cos- A - sin 2 A = 2 cos z A -1 = 1-2 sin 2 ^, sin 2A = 2 sin A cos A, tan 2A = - , sin 5 A = 3 sin A - 4 sin 3 A, cos 2 A = 4 cos 3 ^ - 3 cos A, sin nA = , . . . 1 ^ sm A n(n- s n ~ 3 A sin 3 A + . .. Formulas for mul tiple and sub-mul tiple angles. + (-1) r n(n-l)...(n-2r) 2r + l coa n ~ 2r ~ l A cos nA = cos nA cos"~ 2 ^ sin? A + ...

may all be deduced from the addition formula?, by making the angles all equal. From the last two formulas we obtain by division tannA ntan ^- 3 tan A

In the particular case of n= 3 we have tan ZA = -i _ o t a~s The values of $\sin {\frac {1}{2}}A$, $\cos {\frac {1}{2}}A$, $\tan {\frac {1}{2}}A$ are given in terms of $\cos A$ by the formulae

$$\sin {\frac {1}{2}}A=(-1)^{p}({\frac {1-\cos A}{2}})^{\frac {1}{2}},\cos {\frac {1}{2}}A=(-1)^{q}({\frac {1+\cos A}{2}})^{\frac {1}{2}},\tan {\frac {1}{2}}A=(-1)^{r}({\frac {1-\cos A}{1+\cos A}})^{\frac {1}{2}},$$

where '$p$ is the integral part of ${\frac {A}{2\pi }}$, $q$ the integral part of ${\frac {A}{2\pi }}+{\frac {1}{2}}$, and $r$ the integral part of ${\frac {A}{\pi }}$.

$\sin {\frac {1}{2}}A$, $\cos {\frac {1}{2}}A$ are given in terms of $\sin A$ by the formulae

$$2\sin {\frac {1}{2}}A=(-1)^{p'}(1+\sin A)^{\frac {1}{2}}+(-1)^{q'}(1-\sin A)^{\frac {1}{2}}$$ $$2\cos {\frac {1}{2}}A=(-1)^{p'}(1+\sin A)^{\frac {1}{2}}-(-1)^{q'}(1-\sin A)^{\frac {1}{2}}$$

where $p'$ is the integral part of ${\frac {A}{2\pi }}+{\frac {1}{4}}$ and $q'$ the integral part of ${\frac {A}{2\pi }}-{\frac {1}{4}}.$

Properties of triangles. In any plane triangle ABC we will denote the lengths of the ties of sides BC, CA, AB by a, b, c respectively, and the angles BAC, ABC, ACB by A, B, C respectively. The fact that the projections of b and c on a straight line perpendicular to the side a are equal to one another is expressed by the equation $b\sin C=c\sin B$; this equation and the one obtained by projecting c and a on a straight line perpendicular to a may be written ${\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}$. The equation $a=b\cos C+c\cos B$ expresses the fact that the side a is equal to the sum of the projections of the sides b and c on itself ; thus we obtain the equations

$$\left.{\begin{aligned}a&=b\cos C+c\cos B\\b&=c\cos A+a\cos C\\c&=a\cos B+b\cos A\\\end{aligned}}\right\rbrace .$$

If we multiply the first of these equations by $-a$, the second by $b$, and the third by $c$, and add the resulting equations, we obtain the formula $b^{2}+c^{2}-a^{2}=2bc\cos A$ or $\cos A={\frac {b^{2}+c^{2}-a^{2}}{2bc}}$, which gives the cosine of an angle in terms of the sides. From this expression for $\cos A$ the formulæ $\sin {\frac {1}{2}}A=\left\{{\frac {(s-b)(s-c)}{bc}}\right\}^{\frac {1}{2}}$, $\cos {\frac {1}{2}}A=\left\{{\frac {s(s-a)}{bc}}\right\}^{\frac {1}{2}}$, $\tan {\frac {1}{2}}A=\left\{{\frac {(s-b)(s-c)}{s(s-a)}}\right\}^{\frac {1}{2}}$, $\sin A={\frac {2}{bc}}\{s(s-a)(s-b)(s-c)\}^{\frac {1}{2}}$ where $s$ denotes ${\frac {1}{2}}(a+b+c)$, can be deduced by means of the dimidiary formula. From any general relation between the sides and angles of a triangle other relations may be deduced by various methods of transformation, of which we give two examples.

(α) In any general relation between the sines and cosines of the angles A, B, C of a triangle we may substitute $pA+qB+rC$, $rA+pB+qC$, $qA+rB+pC$ for A, B, C respectively, where p, q, r are any quantities such that $p+q+r+1$ is a positive or negative multiple of 6, provided that we change the signs of all the sines. Suppose $p+q+r+1=6n$, then the sum of the three angles $2n\pi -(pA+qB+rC)$, $2n\pi -(rA+pB+qC)$, $2n\pi -(qA+rB+pC)$ is $\pi$; and, since the given relation follows from the condition $A+B+C=\pi$, we may substitute for A, B, C respectively any angles of which the sum is $\pi$; thus the transformation is admissible.

(β) It may easily be shown that the sides and angles of the triangle formed by joining the feet of the perpendiculars from the angular points A, B, C on the opposite sides of the triangle ABC are respectively $a\cos A$, $b\cos B$, $c\cos C$, $\pi -2A$, $\pi -2B$, $\pi -2C$; we may therefore substitute these expressions for a, b, c, A, B, C respectively in any general formula. By drawing the perpendiculars of this second triangle and joining their feet as before, we obtain a triangle of which the sides are $a\cos A\cos 2A$, $b\cos B\cos 2B$, $c\cos C\cos 2C$ and the angles are $4A-\pi$, $4B-\pi$, $4C-\pi$; we may therefore substitute these expressions for the sides and angles of the original triangle; for example, we obtain thus the formula $$\cos 4A={\frac {a^{2}\cos ^{2}A\cos ^{2}2A-b^{2}\cos ^{2}B\cos ^{2}2B-c^{2}\cos ^{2}C\cos ^{2}2C}{2bc\cos B\cos C\cos 2B\cos 2C}}$$

Solution of triangles.This transformation obviously admits of further extension.

(1) The three sides of a triangle ABC being given, the angles can be determined by the formula and two corresponding formulae for the other angles.

(2) The two sides a, b and the included angle C being given, the angles A, B can be determined from the formulae L tan (A -B)= log (a - b) - log (a+ b) +L cot C, and the side c is then obtained from the formula logc= loga + isin(7-isin^.

(3) The two sides a, b and the angle A being given, the value of sin B may be found by means of the formula L sin B=L sin A + log b - log a ; this gives two supplementary values of the angle B, if b sin A<a. If bsinA>a there is no solution, and if isin^=a there is one solution. In the case b sin A < a, both values of B give solutions provided b>a, but the acute value only of B is admissible if b<a. The other side c can then be determined as in case (2).

(4) If two angles A, B and a side a are given, the angle C is de termined from the formula C=ir- A -B and the side b from the formula log b = log a + L sin B - L sin A.

The area of a triangle is half the product of a side into the per- Areas pendicular from the opposite angle on that side ; thus we obtain of tri- the expressions bcsmA, {s(s-a)(s-b)(s-c)}l for the area of a angles triangle. A large collection of formulae for the area of a triangle and are given in the Annals of Mathematics for 1885 by M. Baker. quadri-

Let a, b, c, d denote the lengths of the sides AB, BC, CD, DA laterals. respectively of any plane quadrilateral and A + C=2a; we may obtain an expression for the area S of the quadrilateral in terms of the sides and the angle a. We have 2S=adsinA + bcsin(2a-A) and $(a? + d?-b 2 -c-) = ad cos A -be cos (2a- A) ; hence 4S 2 + 1( 2 + d 2 - & 2 - c 2 ) 2 = a 2 cF + & 2 c 2 - 2abcd cos 2a. If 2s=a + b + c + d, the value of S may be written in the form S= {s(s - a)(s - b)(s - c)(s - d) - abed cos 2 a} *.

Let R denote the radius of the circumscribed circle, r of the in- Radii of scribed, and r lt r 2 , r 3 of the escribed circles of a triangle ABC ; the circum- values of these radii are given by the following formulae. Radii of circumscribed, inscribed, and escribed circles of a triangle.

_ _ a be _ a ~ = n r=- = (s-a) tan A = s Of sin B sinj C, = - = s tan A = iB sinj A cosj B cos J C. s a

Spherical Trigonometry.

We shall throughout assume such elementary propositions in spherical geometry as are required for the purpose of the investiga tion of formulae given below.

A spherical triangle is the portion of the surface of a sphere Defini- bounded by three arcs of great circles of the sphere. If BC, CA, tion of AB denote these arcs, the circular measure of the angles subtended spherical by these arcs respectively at the centre of the sphere are the sides triangle. a, b, c of the spherical triangle ABC ; and, if the portions of planes passing through these area and the centre of the sphere be drawn, the angles between the portions of planes intersecting at A, B, C respectively are the angles A, B, C of the spherical triangle. It is not necessary to consider triangles in which a side is greater than IT, since we may replace such a side by the remaining arc of the great circle to which it belongs. Since two great circles intersect each Asso- other in two points, there are eight triangles of which the sides are ciated arcs of the same three great circles. If we consider one of these triangles, triangles ABC as the fundamental one, then one of the others is equal in all respects to ABC, and the remaining six have each one side equal to, or common with, a side of the triangle ABC, the opposite angle equal to the corresponding angle of ABC, and the other sides and angles supplementary to the corresponding sides and angles of ABC. These triangles may be called the associated Transfor- triangles of the fundamental one ABC. It follows that from any mation. general formula containing the sides and angles of a spherical triangle we may obtain other formulae by replacing two sides and the two angles opposite to them by their supplements, the remain ing side and the remaining angle being unaltered, for such formulae are obtained by applying the given formulae to the associated triangles.

If A ,S, (7 are those poles of the arcs BC, CA, ^^respectively which lie upon the same sides of them as the opposite angles A, B, C, then the triangle A B C is called the polar triangle of the triangle ABC. The sides of the polar triangle are v - A, ir-B, ir-C, and the angles IT -a, w-b, TT-C. Hence from any general formula connecting the sides and angles of a spherical triangle we may obtain another formula by changing each side into the supplement of the opposite angle and each angle into the supplement of the opposite side.

Let O be the centre of the sphere on which is the spherical triangle ABC. Draw AL per pendicular to OC and AM perpendicular to o* the plane OBC. Then the projection of OA on OB is the sum of the projections of OL, LM, MA on the same straight Jane. Since AM has no projection on any straight line in the plane OBC, this gives angles. OA cos c = OL cos a + LM sin a. Now OL = OA cos b, LM= AL cos C= OA sin b cos C ; therefore cos c cos a cos b + sin a sin b cos C. We may obtain similar formulae by interchanging the letters a, b, c, thus cos a = cos b cos c + sin 6 sin c cos A ^j cos& = cosc cosa + sinc sina cos 2? j- (1). cos c = cos a cos b + sin a sin b cos C ) Fig. 5. These formulæ (1) may be regarded as the fundamental equations connecting the sides and angles of a spherical triangle; all the other relations which we shall give below may be deduced analytically from them; we shall, however, in most cases give independent proofs. By using the polar triangle transformation we have the formulæ

$$\left.{\begin{aligned}\cos A&=-\cos B\cos C+\sin B\sin C\cos a\\\cos B&=-\cos C\cos A+\sin C\sin A\cos b\\\cos C&=-\cos A\cos B+\sin A\sin C\cos b\\\end{aligned}}\right\rbrace .$$

In the figure we have $AM=AL\sin C=r\sin b\sin C$, where r denotes the radius of the sphere. By drawing a perpendicular from A on OB, we may in a similar manner show that $AM=r\sin c\sin B$, therefore

$$\sin B\sin c=\sin C\sin b.$$

By interchanging the sides we have the equation ${\frac {\sin A}{\sin a}}={\frac {\sin B}{\sin b}}={\frac {\sin C}{\sin c}}$ we shall find below a symmetrical form for k.

If we eliminate $\cos b$ between the first two formulae of (1) we have

$$\cos a\sin ^{2}c=\sin b\sin c\cos A+\sin c\cos c\sin a\cos B;$$

therefore

$${\begin{aligned}\cot a\sin c&=\cos A+\cos c\cos B\\&=\sin B\cot A+\cos c\cos B.\end{aligned}}$$

We thus have the six equations

$$\left.{\begin{aligned}\cot a\sin b&=\cot A\sin C+\cos b\cos C\\\cot b\sin a&=\cot B\sin C+\cos a\cos C\\\cot b\sin c&=\cot B\sin A+\cos c\cos A\\\cot c\sin b&=\cot C\sin A+\cos b\cos A\\\cot c\sin a&=\cot C\sin B+\cos a\cos B\\\cot a\sin c&=\cot A\sin B+\cos c\cos B\\\end{aligned}}\right\rbrace .$$

When $C={\frac {\pi }{2}}$formula (1) gives

$$\cos c=\cos a\cos b,$$

and (3) gives

$$\left.{\begin{aligned}\sin b=\sin B\sin c\\\sin a=\sin A\sin c\\\end{aligned}}\right\rbrace ;$$

from (4) we get

$$\left.{\begin{aligned}\tan a&=\tan A\sin b&=\tan c\cos B\\\tan b&=\tan B\sin a&=\tan c\cos A\\\end{aligned}}\right\rbrace .$$

The formulae

$$\cos c=\cot A\cot B$$

and $$\left.{\begin{aligned}\cos A&=\cos a\sin B\\\cos B&=\cos b\sin A\end{aligned}}\right\rbrace .$$

follow at once from (α), (β), (γ). These are the formulas which are used for the solution of right-angled triangles. Napier gave mnemonical rules for remembering them.

The following proposition follows easily from the theorem in equation (3) : If AD, BE, CFare three arcs drawn through A, B, V to meet the opposite sides in D, E, F respectively, and if these arcs pass through a point, the segments of the sides satisfy the relation sin BD sin CE sin AF= sin CD sin AEsin BF; and conversely if this relation is satisfied the arcs pass through a point. From this theorem it follows that the three perpendiculars from the angles on the opposite sides, the three bisectors of the angles, and the three arcs from the angles to the middle points of the opposite sides, each pass through a point.

Formulae for sine and cosine of half-angles; If D be the point of intersection of the three bisectors drawn perpendicular to BC, it may be shown that BE=$(a + c-b) and CE=(a + b-c), and that the angles BDE, ADC are supplementary. We have sine sin ADB sin b sin ADC , . A sinBD s[n A sinCD s ^A 2 2 _siuBDsinCD sinCDEsinBDE n ._ . . 2 BDE=smBE a + b-c sin b sin c ( . a+c-b . a+6-c i . (5). A 1 sm 9 " 9 fore -in ) 2 f sin o sin c

Apply this formula to the associated triangle o

• " - B, C are the angles and ir - a, ir-b, c are the s

a ; n b+c a n . a + b + c 

which IT -A, des ; we obtain i (6). tlifi formula <>n s A < - z 2 / sin & sin c By division we have ( . a + c-b . a + b-c^ I ...a>, tan^J 2 2 | ^ J . b+c-a . a+b+c b . a+b-cl ( 2 Sm 2 J and by multiplication sinA- 2 i in a + 6 + C flin 6 + C ~%iT, c + a - sinftsiacl 811 2 8m 2 2 sm 2 j-

os b cos c} i-

"sin&sinc 1 Cos2a cos s 6-cos z c + 2cosa(

Hence the quantity k in (3) is sina sin 6 sine* 1 -coa a-cos 6-cosc + 2 cosacosb cose}* (8).

Of half-sides.Apply the polar triangle transformation to the formulaj (5), (6), (7), (8) and we obtain ( A + O-B A+B-Cl " cos cos sin B sin C B + C-A A + B + C - cos cos - sin B sin C J B+C-A A+B+C - cos 2 cos A+C-B A+B-C cos cos 2 ..(10); .(11).

If j. - 1 sinAsiiiBsinC we have kk = -2cos^4cos J BcosC ! ^, (12).

Delambre's formulæLet E be the middle point of AB ; draw ED at right angles to AB to meet AC in D ; then DE bisects ,? the angle ADB. Let CF bisect the angle /]r DCB and draw FG perpendicular to BC, then =90 -5. 3Q

From the triangle CFG we have cos = cos (7<? sin JF 1 ^, and from the triangle FEE cos JEFB = cos EB sin .F&E. Now Fig. 7. the angles CFG, EFB are each supplementary to the angle DFB, therefore a-b C . A + B c cos - cos 2= sm cos .(13).

Also sin CG = sin CF sin CFG and sin EB = sin BF sin EFB ; .a-b C A- B . c therefore sm jr cos - = sm - sm- (14). 22 22

Apply the formulae (13), (14) to the associated triangle of which a, IT - b, rr - c, A, IT - B, tr - (7 are the sides and angles, we then have . a+b . C A-B . c sin sin = cos - sin -. Si 22 a + b . C cos -- sin = A + B c - cos .(15), .(16).

The four formulae (13), (14), (15), (16) were first given by Delambre in the Connaissance des Temps for 1808. Formulae equivalent to these were given by Mollweide in Zach's Monatliche Correspondenz for November 1808. They were also given by Gauss (Theoria motus, 1809), and are usually called after him.

Napier's analogies. From the same figure we have tan FG = tan FCG sin CG = tan FBG sin BG ; therefore .0 . a-b , A-B . cot -sin -g- =tan ^ sm a-b tan A-B cot 2 (17).

Apply this formula to the associated triangle (IT - a, b, tr-c, ir-A, B, v-C], and we have a + b S -2- C tan a-b A+B cos -r- cos .(18).

If we apply these formulae (17), (18) to the polar triangle, we have . A-B , sin - a- b _ 2 c tan-g - sm , 1f ., (19 ; A-B . COS jr a + b 2 tan~2~- = .(20). cos- 2 - The formulae (17), (18), (19), (20) are called Napier s "Analogies" ; they were given in the Mirif. logar. canonis descriptio. Schmeisser's formulæ. If we use the values of $\sin {\frac {a}{2}}$, $\sin {\frac {b}{2}}$, $\sin {\frac {c}{2}}$, $\cos {\frac {a}{2}}$, $\cos {\frac {b}{2}}$, $\cos {\frac {c}{2}}$, given by (9), (10) and the analogous formulae obtained by interchanging the letters, we obtain by multiplication

$$\left.{\begin{aligned}\sin {\frac {a}{2}}\cos {\frac {b}{2}}\sin C&=\sin {\frac {c}{2}}\cos {\frac {B+C-A}{2}}\\\cos {\frac {a}{2}}\cos {\frac {b}{2}}\sin C&=\cos {\frac {c}{2}}\cos {\frac {B+C-A}{2}}\\\sin {\frac {a}{2}}\sin {\frac {b}{2}}\sin C&=\cos {\frac {c}{2}}\cos {\frac {B+C-A}{2}}\\\end{aligned}}\right\rbrace .$$

These formulae were given by Schmeisser in Crelle's Journ., vol. x.

Caguoli's formulæ. The relation $\sin b\sin c+\cos b\cos c\cos A=\sin B\sin C-\cos B\cos C\cos a$ was given by Cagnoli in his Trigonometry (1786), and was rediscovered by Cayley (Phil. Mag., 1859). It follows from (1), (2), and (3) thus: the right-hand side of the equation equals $\sin B\sin C+\cos a(\cos A-\sin B\sin C\cos a)=\sin B\sin C\sin ^{2}a+\cos a\cos A$, and this is equal to $\sin b\sin c+\cos A(\cos a-\sin b\sin c\cos A)$ or $\sin b\sin c+\cos b\cos c\cos A.$

Solution of triangles; The formulæ we have given are sufficient to determine three parts of a triangle when the other three parts are given; moreover such formulæ may always be chosen as are adapted to logarithmic calculation. The solutions will be unique except in the two cases (1) where two sides and the angle opposite one of them are the given parts, and (2) where two angles and the side opposite one of them are given.

Ambiguous cases. Suppose a, b, A are the given parts. We determine B from the formula $\sin B={\frac {\sin b}{\sin a}}\sin A$; this gives two supplementary values of B, one acute and the other obtuse. Then C and c are determined from the equations $\tan {\frac {C}{2}}={\frac {\sin {\frac {a-b}{2}}}{\sin {\frac {a+b}{2}}}}\cot {\frac {A-B}{2}}$, $\tan {\frac {c}{2}}={\frac {\sin {\frac {A+B}{2}}}{\sin {\frac {A-B}{2}}}}\tan {\frac {a-b}{2}}$. Now $\tan {\frac {C}{2}}$, $\tan {\frac {c}{2}}$ must both be positive; hence $A-B$ and $a-b$ must have the same sign. We shall distinguish three cases. First, suppose $\sin b<\sin a$; then we have $\sin B<\sin A$. Hence A lies between the two values of B, and therefore only one of these values is admissible, the acute or the obtuse value according as a is greater or less than b; there is therefore in this case always one solution. Secondly, if $\sin b>\sin a$, there is no solution when $\sin b\sin A>\sin a$; but if $\sin b\sin A<\sin a$ there are two values of B both greater or both less than A. If a is acute, $a-b$, and therefore $A-B$, is negative; hence there are two solutions if A is acute and none if A is obtuse. These two solutions fall together if $\sin b\sin A=\sin a$. If a is obtuse there is no solution unless A is obtuse, and in that case there are two, which coincide as before if $\sin b\sin A=\sin a$. Hence in this case there are two solutions if $\sin b\sin A\leq \sin a$ and the two parts A, a are both acute or both obtuse, these being coincident in case $\sin b\sin A=\sin a$; and there is no solution if one of the two A, a is acute and the other obtuse, or if $\sin b\sin A>\sin a$. Thirdly, if $\sin b=\sin a$ then $B=A$ or $\pi -A$. If a is acute, $a-b$ is zero or negative, hence $A-B$ is zero or negative; thus there is no solution unless A is acute, and then there is one. Similarly, if a is obtuse, A must be so too in order that there may be a solution. If $a=b={\frac {\pi }{2}}$, there is no solution unless $A={\frac {\pi }{2}}$, and then there are an infinite number of solutions, since the values of C and c become indeterminate.

The other case of ambiguity may be discussed in a similar manner, or the different cases may be deduced from the above by the use of the polar triangle transformation. The method of classification according to the three cases $\sin b{\underset {<}{\overset {>}{=}}}\sin a$ was given by Professor Lloyd Tanner (Messenger of Math., vol. xiv. ).

Radii of circles related to triangles. If r is the angular radius of the small circle inscribed in the triangle ABC, we have at once tan r=tan-2-sin (s-a), where triangles. 2s=a + b + c ; from this we can derive the formulas tanr=i cosecs= sec sec ^-sec^ =sin sin sin sec (21), 2t 2&gt; t Z B SI Z where n, N denote the expressions !sin s sin (* - a) sin (s - b) sin (s - c} &, { - cos Scos (S- A) cos (S- B) cos (S- 0)} J. The escribed circles are the small circles inscribed in three of the associated triangles ; thus, applying the above formulae to the triangle (a, v - b, ir - c, A, IT - B, w - C), we have for r v the radius of the escribed circle opposite to the angle A, the following formulae A . . . N A B C tan r^ = tan - 9 - sm s= n cosec (s - a) = ^-sec -^- cosec -^- cosec ^-

• *

The pole of the circle circumscribing a triangle is that of the circle inscribed in the polar triangle, and the radii of the two circles are complementary ; hence, if R be the radius of the circum scribed circle of the triangle, and R v R& R^ the radii of the circles circumscribing the associated triangles, we have by writing - - R 2 for r, - - RI for r v ir-a for A, &c., in the above formulae cot R= cot cos (S- A) = - cosec - cosec ^ cosec |= - JVsec S = sin A cos g cos = cosec = a v n a b c cot BI-- cot ^ cos S= ~ cosec ^ sec ~ sec ^ = a . . . b . c a = sm .4 sin - sm - cosec ^. . . (23) ; (S-A) .(24). The following relations follow from the formulae just given : 2 tan R = cot r t + cot r z + cot r 3 - cot r, 2 tan .Kj = cot r + cot r 2 + cot r s - cot r lt tan r tan r x tan r 2 tan r 3 = n 2 , sin 2 s=cotrtanrj tan r 2 tan 7*3, sin* (s - a) = tan r cot r t tan r 2 tan r 3.

Formulæ for spherical excess. If E=A+B + C-ir, it may be shown that E multiplied by the square of the radius is the area of the triangle. We give some of the more important expressions for the quantity E, which is called the spherical excess.

A + B cos 2 We have - -^ . O sin a + b A + B a-b c cos and (J cos . C SU1 2 and a-b 5 ~2~ hence therefore Similarly cos -= c " a + b COS 2~ COS ~2~ . G . sm - + sm COS jj + COS II a + b tan -7 4 tan 2 -^ E , n C-E . tan tan- - = tan ^ tan s-b i 22 ^ f E (, s. s-a. s-b. s-c~ I ,.... therefore tan - = -j tan - tan -- tan -^- tan - - Y (2o). This formula was given by L Huillier. a + b Also C E C . E cos s a-b C E . cos - cos -fsin- E .(26). whence, solving for cos , we get E l + cosa + cosi + cosc cos - = - 2 a b c 4 COS jr COS x COS 5 2&gt; 2t

This formula was given by Euler (Nova acta, vol. x.). If we find siii from this formula, we obtain after reduction I . E n sm -r = ^ , A _ M/ V If 2 cos- cos - cos - m jm- m a formula given by Lescell (Acta Petrop., 1782).

From the equations (21), (22), (23), (24) we obtain the following formulae for the spherical excess : sin 2 -= = tan R cot RI cot R. 2 cot R 3 _ 4(cot rj + cot r 2 + cot ?,,) (cot r - cot r x + cot r + cot r 3 ) (cot r + cot r x - cot r 2 + cot r 3 ) (cot r + cot T-J + cot r a - cot r 3 ) The formula (26) may be expressed geometrically. Let M, NM the middle points of the sides AB, AC. Then we find cos MN 1 + cos a + cos b + cos c ~~b ^ 4 cos - cos 5 2, 2t hence cos - = cos MN sec . A geometrical construction has been given for E by Gudermann (in Crelle s Journ., vi. and viii. ). It has been shown by Cornelius Keogh that the volume of the parallelepiped of which the radii of the sphere passing through the middle points of the sides of the triangle are edges is sin - .

Properties of spherical quadrilateral inscribed in small circle. Let ABCD be a spherical quadrilateral inscribed in a small circle ; let a, b, c, d denote the sides AS, BC, CD, DA respectively, and x, y the diagonals AC, SD. It can easily be shown by joining the angular points of the quadrilateral to the pole of the circle that A + C=B + D. If we use the last expression in (23) for the radii of the circles circumscribing the triangles BAD, BCD, we have whence .ad ?/.& C y sin A cos x cos ~ cosec 5 = sin C cos ^ cos = cosec ^ SB * 9 ^ 82 2 sin (7 b c cos -cos g a d cos ^ cos x. This is the proposition corresponding to the relation A + C=irfor a plane quadrilateral. Also we obtain in a similar manner the theorem sin B cos - sin yf cos analogous to the theorem for a plane quadrilateral, that the diagonals are proportional to the sines of the angles opposite to them. Also the chords AB, BC, CD, DA are equal to 2 sin , 2 sin |, 2 sin 1 2 sin f m m W m respectively, and the plane quadrilateral formed by these chords is inscribed in the same circle as the spherical quadrilateral ; hence by Ptolemy s theorem for a plane quadrilateral we obtain the analogous theorem for a spherical one . x . y . a . c sm 2 Sm 2 =Sm 2 S1U 2

It has been shown by Remy (in Crete s Journ., vol. iii.) that for any quadrilateral, if z be the spherical distance between the middle points of the diagonals, cos a + cos b + cos c + cos d = 4 cos Jar cos y cos z. This theorem is analogous to the theorem for any plane quadri lateral, that the sum of the squares of the sides is equal to the sum of the squares of the diagonals, together with twice the square on the straight line joining the middle points of the diagonals. A theorem for a right-angled spherical triangle, analogous to the Pythagorean theorem, has been given by Gudermaun (in Crelle s Journ., vol. xlii.).

Analytical Trigonometry.

Periodicity of functions. Analytical trigonometry is that branch of mathematical analysis in which the analytical properties of the trigonometrical functions , are investigated. These functions derive their importance in ana lysis from the fact that they are the simplest singly periodic functions, and are therefore adapted to the representation of undu lating magnitude. The sine, cosine, secant, and cosecant have the single real period 2ir ; i.e., each is unaltered in value by the addi tion of 2w to the variable. The tangent and cotangent have the period ir. The sine, tangent, cosecant, and cotangent belong to the class of odd functions ; that is, they change sign when the sign of the variable is changed. The cosine and secant are even func tions, since they remain unaltered when the sign of the variable is reversed.

Connexion with theory of complex quantities. The theory of the trigonometrical functions is intimately con nected with that of complex quantities, that is, of quantities of the form x + iy (t = V - 1 ). Suppose we multiply together, by the rules of ordinary algebra, two such quantities, we have . . We observe that the real part and the real factor of the imaginary part of the expression on the right-hand side of this equation are similar in form to the expressions which occur in the addition formulae for the cosine and sine of the sum of two angles ; in fact, if we put a^riCOUtfj, Vl = ri mn0 lt X 2 =r 2 cose 2&gt; y^r^sinO^ the above equation becomes r^cos 0j + 1 sin X ) x r 2 (cos 2 + 1 sin 2 ) = r^cos 6 l + 2 + 1 sin 0j + 2 ). We may now, in accordance with the usual mode of representing complex quantities, give a geometrical interpretation of the meaning of this equation. Let P a be the point whose coordinates referred to rectangular axes Ox, Oy are a^, Vl ; then the point P 1 is employed to represent the quantity x 1 + iy 1 . In this mode of representation real quantities are measured along the axis of x and imaginary ones along the axis of y, additions being performed according to the parallelogram law. The points A,A 1 represent the magnitudes 1, the points a, ^ the magnitudes i. Let P 2 represent the expression x. 2 + ty. 2 and P the expression (x 1 + iy l )(x 2 + iy 2 ). The quantities ^Vfn^VAj are the polar coordinates of P and P 2 respectively referred to as origin and Ox as initial line ; the above equation shows that 7y- 2 and 0) + 0., are the polar coordinates of P ; hence A

OP l : .OP ;t :OP and the angle P0P 2 is equal to the angle

P]OA. Thus we have the following geometrical construc tion for the determination of the point P. On OP 2 draw a triangle similar to the triangle OA P l so that the sides OP 2 , OP are homologous to the sides OA, OP, and so that the angle POP 2 is positive ; then the vertex P represents the product of the expressions represented by Pj.Pa. If x 2 + iy 2 were to be divided V by x 1 + iy l , the triangle OPP Z would be drawn on the negative side of P 2 , similar to the triangle OA P l and having the sides OP, OP 2 homologous to OA, OP, and P would represent the quotient.

De Moivre's theorem. If we extend the above to n complex quantities by continual repeti tion of a similar operation, we ^ have (cos X + 1 sin 0j) (cos 2 + 1 sin 2 ) . . . (cos B + 1 siu 0,,) 2 + ... +0 n ) + 1 sin(0 1 + 2 + ... If 0j 2 ... =0 n = 0j, this equation becomes (cos + tsin 0)" = cos nO + 1 sin ?i0 ; this shows that cos + i sin0 is a value of (cos n6 + 1 f / f i sin ?i0) n. If now we change into -, we see that cos- + t siu - is a i n n n value of (cos + 1 sin 0) ; raising each of these quantities to any positive integral power m, cos_ + ts i n is one value of (cos m n n + tsin0)". Also (TO. . / m, )0 + tsinl --0 )= n) n / m . TO COS 0+i sin n hence the expression of the left-hand side is one value of

m

x , . sin 0)" 1 1 or of (cos + isin0)~n~. We have thus De Moivre s theorem that cos + 1 sin k6 is always one value of (cos + 1 sin 0)*, where k is any real quantity.

The n roots of a complex quantity. The principal object of De Moivre s theorem is to enable us to The n c j 11 ,, , , roots of a nnd all the values ot an expression of the form (a + ib) n , where m com piex and n are positive integers prime to each other. If a=rcos0, auan titv m m * " 6 = rsin0, we require the values of r" (cos + t sin 0)". One value is immediately furnished by the theorem ; but we observe that, sincn the expression cos + tsin is unaltered by adding any multiple of o . a 1 n i. e -/ m.0 + 2s7T . m.0 + 2s7rY 2irto0, the -thpowerofr"! cos htsm hs a + ib, m n n / if s is any integer ; hence this expression is one of the values re quired. Suppose that for two values s 1 and s 2 of s the values of this expression are the same ; then we must have M 1?r - m + 2 V ; n n a multiple of 2ir or 5 X - s 2 must be a multiple of n. Therefore, if we give s the values 0, 1, 2, ... n - 1 successively, we shall get n differ ent values of (a + tb) n , and these will be repeated if we give s other m values; hence all the values of (a + ib)" are obtained by giving s the values 0, 1, 2, . . . n-l in the expression r"(c&lt; + t sin Y wherer=(a 2 +& 2 )iand = arc tan -

We now return to the geometrical representation of the complex quantities. If the points B lt B& B s&gt; ... B n repre sent the expression * + iy, (x + a/) 2 , (x + lyf, . . . (x + iy) respectively, the triangles OAB V OB^, . . . OB n _ l B n are all similar. Let (x + iy) n = a + ib, then the con-" verse problem of finding the nth root of a + ib is equivalent to the geometrical problem of describ ing such a series of triangles that OA is the first side of the first triangle and OB n the second side of the ftth. Now it is obvious that this geometrical problem has more solutions than one, since any number of com-, plete revolutions round may be made in travelling from B l to B n . The first solution is that in which the vertical angle of each triangle

is ${\frac {1}{n}}B_{n}OA$; the second is that in which each is ${\frac {1}{n}}(B_{n}OA+2\pi )$, in this case one complete revolution being made round; the third has ${\frac {1}{n}}(B_{n}OA+4\pi )$ for the vertical angle of each triangle ; and so on. There are n sets of triangles which satisfy the required conditions.

For simplicity we will take the case of the determination of the values of $(\cos \theta +i\sin \theta )^{\frac {1}{3}}$. Suppose B to represent the expression $\cos \theta +i\sin \theta$. If the angle $AOP_{1}$ is ${\frac {1}{3}}\theta$, $P_{1}$ represents the root $\cos {\frac {\theta }{3}}+i\sin {\frac {\theta }{3}}$; the angle AOB is filled up by the angles of the three similar triangles $AOP_{1},P_{1}Op_{1},p_{1}OB$. Also, if $P_{2}$, $P_{3}$ be such that the angles $P_{1}OP_{2}$, $P_{1}OP_{3}$ are ${\frac {2\pi }{3}}$, ${\frac {4\pi }{3}}$ respectively, the two sets of triangles $AOP_{2}$, $P_{2}Op_{3}$, $p_{3}OB$ and $AOP_{3}$, $P_{3}Op_{2}$, $p_{2}OB$ satisfy the conditions of similarity and of having OA, OB for the bounding sides; thus $P_{2}$, $P_{3}$ represent the roots $\cos {\frac {\theta +2\pi }{3}}+i\sin {\frac {\theta +2\pi }{3}}$, $\cos {\frac {\theta +4\pi }{3}}+i\sin {\frac {\theta +4\pi }{3}}$ respectively. If B coincides with A, the problem is reduced to that of finding the three cube roots of unity. One will be represented by A and the others by the two angular points of an equilateral triangle, with A as one angular point, inscribed in the circle.

The nth roots of unity. The problem of determining the values of the nth roots of unity is equivalent to the geometrical problem of inscribing a regular polygon of n sides in a circle. Gauss has shown in his Disquisitiones arithmeticæ that this can always be done by the compass and ruler only when n is a prime of the form $2^{p}+1$. The determina tion of the nth root of any complex quantity requires in addition, for its geometrical solution, the division of an angle into n equal parts.

Factorizations. We are now in a position to factorize an expression of the form $x^{n}-(a+ib)$. Using the values which we have obtained above for ib) n , we have s=0 If& = 0, a=l, this becomes + 2S7T cos H 0J-2S7T n ...(1). s=i-l x n -l = P s=0 X-( !"~V 2STT, . 2STT [ x cos i sin =l V 5t n ) n r2 ~ (a?-2xcos^ + l eveu)(2). 5= o / 9cir aj-l = (a?-l)P (s-2xcoa- --+11 (ra odd) ...(3). s=l V / If in (1) we put a= - 1, 6 = 0, and therefore = 7r, we have

• = n - l r 2s + ITT

x n + 1 = P a: - cos 1 sn s =o L n = P x 2 -2xcos hi I (n even) (4). s=0 I l J s = Also x- n - 2x"y n cos n6+y- n fa? - 2x cos %2l + 1~| ( B odd) (5). L n J = (x n - y n cos n6 + 1 sin 7i D =7l ~Y = P [x- s=o V - y" cos nd - 1 sin ?i0) o S = n-l = P s=0 + 2*77-, . ycos - rttrin 5"! - + y 2 n J (6).

Airy and Adams have given proofs of this theorem which do not involve the use of the symbol i (see Camb. Phil. Trans., vol. xi.). A large number of interesting theorems may be derived from De Moivre's theorem and the factorizations which we have deduced from it; we shall notice one of them.

Example of De Moivre's theorem. In equation (6) put $y={\frac {1}{x}}$, take logarithms, and then differentiate each side with respect to x, and we get

$${\frac {2n(x^{2n-1}-x^{-2n-1})}{x^{2n}-2\cos n\theta +x^{-2n}}}=\sum _{s=0}^{s=n-1}{\frac {2(x-x^{-3})}{x^{2}-2\cos \theta +{\frac {2s\pi }{n}}+x^{-2}}}.$$

Put $x^{2}={\frac {a}{b}}$, then we have the expression

$${\frac {n(a^{2n}-b^{2n})}{(a^{2}-b^{2})(a^{2n}-2a^{n}b^{n}\cos n\theta +b^{2n})}}$$

for the sum of the series

$$\sum _{s=0}^{s=n-1}{\frac {1}{a^{2}-2ab\cos \theta +{\frac {2s\pi }{n}}+b^{2}}}.$$

Complex quantities as exponentials. We shall now consider what meaning can be assigned to the symbol $e^{x+iy}$. The quantity e is defined as the limit of $(1+{\frac {1}{n}})^{n}$ where n is a positive quantity, and is increased indefinitely; then, for a real value of x, $e^{x}$ is the limit of $(1+{\frac {1}{n}})^{nx}$ or $(1+{\frac {x}{m}})^{\frac {1}{m}}$ where $m=nx$, when m is increased indefinitely. We may define $e^{x+iy}$ as the limit of $(1+{\frac {x+iy}{m}})^{\frac {1}{m}}$ when m is increased indefinitely. To determine the value of this limit put $1+{\frac {x}{m}}=r\cos \theta ,{\frac {y}{m}}=r\sin \theta$; then e x+ly is the limit of r m (cos70 + t sin7?z0), and r m is equal to S9^. ^4-2 / 9v - 1 + f + 1 ( 2 or ultimately to ( 1 + ) 2 which has f for m m 2 J J n/ its limiting value. Also is arc tan in the limit ; x + m x + m hence 7n0 is ultimately equal to y, and thus the equation e x+l y=e c (cosy + i sin?/) follows from our definition. It may be shown at once that e*+ i y x e*i+ t 2 i= e x + x i+ l &lt;y+yi an d, if we suppose that a x+iy denotes e^ +l ^ loga , we may show that complex expon ents defined thus obey the same laws as real ones.

Exponential values of sine and cosine. When the exponent is entirely imaginary we have, in accordance with the above definition,

$e^{iy}=\cos y+i\sin y$ and $e^{-iy}=\cos y-i\sin y$; we thus obtain the exponential values of the sine and cosine—

$$\sin y={\frac {1}{2i}}(e^{iy}-e^{-iy}),\cos y={\frac {1}{2}}(e^{iy}+e^{-iy}).$$

Expansions of sines and cosines and their powers. If we give imaginary or complex values to the variables in algebraical expansions we obtain analogous trigonometrical theorems; it is, however, necessary to consider the convergency of the series so obtained in order to determine within what limits the values of the variables must lie. If we expand e?U and e~ y by putting iy ifl i/ 3 and their and-ty in the series l + y+ jS+s o~ 5+ we obtain the series sin y = y - ^~- + .-^- - ^=- I I

These series are convergent for all finite values of y. They may also be got from the expressions which we have obtained for the cosine and sine of a multiple of an angle in terms of the cosine and sine of the angle, and would thus be made to rest upon a basis independent of the symbol ı.

Expansion of powers of sines and cosines in series of sines and cosines of multiple arc. Consider the binomial theorem /_ i T. -_n i n n ~*-i~n-iT.i , 7^(7^-l)... (n-r+l) n _ rbr ^ E Putting a=e ıθ , b=e -ıθ , we obtain (2 cos 0)" = 2 cos 710 + 2 cos 71 - 20 + j~ ^ 2 cos 71 - 40 + ... 71(71 - 1) ... (n - r - cos(n - 2r)0 + . . .

When n is odd the last term is 2 cos e ... n(n-).. . and when n is even it is J - , , - , I In If we put a=e e , b= -e~^, we obtain the formula n (-if (2 sin 0)" = 2 cos ?i0 - 2n cos(n - 2)0 + - ( ^ ~^2 cos(n - 4)0 - ... when 7i is even, and n-l (-1) 2 (2 sin 0)"= 2 sin 710 -?i. 2 sin (TI -2)0 + 71-1 1.2 -2sin(?i-4)0... when ?i is odd. These formulae enable us to express any positive integral power of the sine or cosine in terms of sines or cosines of multiples of the argument. There are corresponding formulse when 71 is not a positive integer. 570 Expan- Considerthe identity og(l- sion of Expand both sides of this equation in powers of a 1 , and equate the sines and coefficients of x n , we then get cosines of . . , n(n - 3) multiple n n - n -*- arcsiu TRIGONOMETRY = sin0 [ANALYTICAL. , ,, powers of + ( - 1 ) sines and . (p + q) n r p r q r + ... cosines of If we write this series in the reverse order, we have when n is even, and &lt; n-5 n-l

-i) 2

when vi is odd. If in these three formulae we put jj=e tfl , q=e~ l , we obtain the following series for cos n8 : 2 cos nd = (2 cos 0)" - ?i(2 cos g)n-2 + ^ TO ~ 3 cos 0)- 4 - ... - when 71 is any positive integer ; 7i 2 , 7i 2 (7t 2 -2 2 ) ? i a (?i 2 -2 2 )(i 2 -4 2 ) .- ( - 1 ) 2 cos n0 - 1 - ps- cos 2 + i-r-j - cos 4 -- * - j-^ - cos 6 I * * o + ... + (-l)22- 1 cos0 when n is an even positive integer ; ~ n(t 2 - 1 2 ) 7 t (7i 3 - I 2 )(n 2 - 3 2 ) (-1) 2 cos0=7icos?i0- - 3 - - (8) ...+(-1) 2 2- 1 cos"0 ..................... (9) when ?i is odd. If in the same three formulae we put p = e t6 , q -e~ i8 , we obtain the following four formulae : - 1) 2 2 cos 7i0 = (2 sin 0)" - w(2 sin 0)- 2 2 sin 0)"- 4 - ... (-1) 2 2 sin 7i0 = the same series (n odd) n 2 . , n 2 (7i 2 -2 2 ) . . i 2 (7i 2 -2 2 )(n 2 -4 2 ) . cos 710 = 1 - nr- siu 2 + y-j - - y sm 4 -- S - r - sin 6 !__ I * | o + ... + 2"- 1 sin"0 (71 even) ............... (12) ; , 7l(7l 2 -! 2 ) . , (t 2 -l 2 )(7l 2 -3 2 ) . sm 710 = n sm - -^-5 - sin 3 + - r-^ - ? sm 5 - ... LJL LA n-l + (-1) 2 2"- 1 sin" ( odd) ............... (13). P~1 Next consider the identity - - - - - _ l-px l-qx l-(p + q)x+pqa? Expand both sides of this equation in powers of x, and equate the coefficients of x"~ l , then we obtain the equation If, as before, we write this in the reverse order, we have the series IJL when n is even, and n - 1 _. n - 1 when ?i is odd. If we put^=e 9 , q = e~ l9 , we obtain the formulae i-r-l)(?t - r-2). . . (71 - 2?-), where n is any positive integer ; (-1) siu7i0=sin0i 71COS0- ) 7( M 2 -2 2 ) Jl 2 -4 2 ) y COS 3 0+ - r^ - - ; COS S - ... + (-l) 2 (2COS0)"- 1 (71 even) ............ (15); + (-l) 2 (2 cos 0)"- 1 (TI odd) (16). If we put in the same three formulae ^=6^, q= - e ~ l6 , we obtain the series n-2 (-1) , , ., + (-1) ~ "1 ... J . ?teven)(17) ; (-1) 2 cos 710 = the same series (n odd) ...... (18); a /.i /, ( 2 - 2-) . , a 7i(7i 2 -2-)(t--4 2 ) . ... sin H0 = cos } )i sin - v , . / sm 3 + 3 - X sm 5 + I o I 4 - ... + (-1)2 (2 sin 0)~ l }(TI even) ............ (19); + (2 sin 0)"- 1 } (71 odd; ............... (20). We have thus obtained formulae for cos ?i0 and sin 710 both in ascending and in descending powers of cos 6 and sin 6. Viete ob tained formula? for chords of multiple arcs in powers of chords of the simple or complementary arcs equivalent to the formulae (13) and (19) above. These are contained in his work Theoremata ad angulares scctiones. James Bernoulli found formulae equivalent to (12) and (13) (Mem. de I Academic des Sciences, 1702), and trans formed these series into a form equivalent to (10) and (11). John Bernoulli published in the Acta eruditorum for 1701, among other formula already found by Viete, one equivalent to (17). These formulas have been extended to cases in which n is fractional, nega tive, or irrational ; sec a paper by D. F. Gregory in Camb. Math. Journ., vol. iv., in which the series for cos 7i0, sin0 in ascending powers of cos and sin are extended to the case of a fractional value of n. These series have been considered by Euler in a memoir in the Nova acia, vol. ix., by Lagrange in his Calcul d?s fonctioiis (1806), and by Poinsot in Rccherchcs sur Vatuilyse des sec tions angulaircs (1825). The general definition of Napierian logarithms is that, if 6*+^ Theory = a + ib, then x + iy= log (a + ib). Now we know that e x+t y= e*cos y y= ic c siny; hence e? cos y=a, e* sin y=b, or e x =(a? arc tan - mir, where m is an integer. If b=Q, then m must be even or odd according as a is positive or negative ; hence log,, (a + i&)=log e ( 2 +& 2 )i+t (arc tan -2ra7r)

rithnis - or log e (a + ib) = log e (a 2 + 6 2 )* + 1 (arc tan - 2?i + ir), (L according as a is positive or negative. Thus the logarithm of any complex or real quantity is a multiple-valued function, the differ ence between successive values being 27rt ; in particular, the most general form of the logarithm of a real positive quantity is obtained by adding positive or negative multiples of 27n to the arithmetical logarithm. On this subject, see De Morgan s Trigonometry and Double Algebra, chap, iv., and a paper by Prof. Cayley in vol. ii. of Proc. London Math. Soc. We may suppose the exponential values of the sine and cosine Hyper- extended to the case of complex arguments ; thus we accept bolic e i(x+iy) + e -i(x+iy) e t(x+iy)_ e -i(x+ty) trigono- and-- as the definitions of the me try. 2 2t functions cos (x + iy), sin (x + iy) respectively. If x=Q, we have cosiy= and sin iy=.~($-e ?/ ). The quantities s are called the hyperbolic cosine and sine of y and arc written cosh y, sinh y ; thus cosh j/ = cos ly, sinh y= i sin ty. The functions cosh y, sinh y are connected with the rectangular hyperbola in a manner analogous to that in which the cosine and sine are ANALYTICAL.] TRIGONOMETRY 571 connected with the circle. We may easily show from the definitions that cos 2 (a; + iy) + sin*(x + iy) = 1, cosh 2 y - sinh - y 1 ; cos(ar + iy) = cos x cosh y - 1 sin x sinh y, sin(x + iy) = sin x cosh y + 1 cos x sinh y, cosh(a + |3) = cosh a cosh /3 + sinh a sinh /3, sinh(a + /3) = sinh a cosh - cosh a sinh /3. These formulae are the basis of a complete hyperbolic trigonometry. The connexion of these functions with the hyperbola was first pointed out by Lambert. Expan- If we equate the coefficients of 71 on both sides of equation (13), sionofan we set . . , . . * a , _ . ... . 1 aaro 1 . 3 sm 5 1.3.5 sm 7 an g e m 6 = sm6 + - 5- + ;r-7 iT~ + o t ~ ^~ + (21); powers of 2 3 2.4 5

its sine. Q must Ue ^tween the va i ues ?T

2.4.6 7 This equation may also be written in the form when x lies between 1. By equating the coefficients of n 2 on both sides of equation (12) we get 2sin 4 2.4 sin0 2.4.6sin 8 which may also be written in the form / arcsma . =a j! ,2 ** 2^4 x 6 2_ L 4J5a 32 3.53 3.5.74 when x is between 1. Differentiating this equation with regard to x, we get arc sin a; 2 , 2.4 2.4.6 , if we put arc sin x arc tan y, this equation becomes &gt;, l 2 1 + s. ! Gregory s I series. Series for calcula tion of TT. (23). This equation was given with two proofs by Euler in the Nova acta We have o^S-j = x+ ^- + -=- + -= + ... ; put iy for x, the left side then becomes {log (1 + iy) - log (1 - iy)} or larctanyimTT ; I/O yd |/7 The series is convergent if y lies between 1 ; if we suppose arc tan y restricted to values between -, we have 7/"^ V^ arc tan y=y- "t + ^ - (24), which is Gregory s series. Various series derived from (24) have been employed to calculate the value of TT. At the end of the 17th century ir was calculated to 72 places of decimals by Abraham Sharp, by means of the series obtained by putting arc tan y=, V=j= in (24). The cal culation is to be found in Sherwin s Mathematical Tables (1742). About the same time Machin employed the series obtained from the equation 4 arc tan ^ - arc tan HOQ = I to ca l cu l ate T to 100 de cimal places. Long afterwards Euler employed the series obtained from 7 = arc tan- + arctan-, which, however, gives less rapidly con verging series (Introd., Anal, infin., vol. L). Lagny employed the 1 7T formula arc tan /== ~ to calculate ?r to 127 places ; the result was communicated to the Paris Academy in 1719. Vega calculated IT to 140 decimal places by means of the series obtained from the equation - = 5 arc tan = + 2 arc tan . The formula ^ = arc tan - + 1 4 j / 10 4 2 arc tan = + arc tan - was used by Base to calculate IT to 200 decimal 11 places. Rutherford used the equation TT= 4 arc tan = - arc tan =- + 1 5 70 If in (23) we put y-- and ^, we have o / 56 a rapidly convergent series for ir which was first given by Hutton in Phil. Trans, for 1776, and afterwards by Euler in Nova acta for 1793. Euler gives an equation deduced in the same manner from t q the identity ir = 20 arc tan = + 8 arc tan ^. The calculation of TT has been carried out to 707 places of decimals ; see Proc. Roy. Soc., xxi. and xxii. ; also SQUARING THE CIRCLE (vol. xxii. p. 435 sq.). We shall now obtain expressions for sin x and cos x as infinite Factoriz- products of rational factors. We have sina;=2sin s sin =2 3 sin 7 sin & A 444 4 proceeding continually in this way with each factor, we obtain . , . x . x + ir . a: + 2jr x + n-lir sma;=2 n - 1 sin - sm sin ... sin ^ n n n n where n is any positive integral power of 2. Now ation of sine and cosine. . x + rv . sm sm and -nr x + rir . rir-x . nr . .x =sm sm =sm 2 sin 2 -, I n n n n . x + &nir x sm = = cos -. 7i n Hence the above may be written sina:=2 n - 1 sin -(sin 2 - - sin 2 - Vsin 2 _ sin 2? . .

n Ti/V n nj 

(lew a; x sin 2 sin 2 - I cos -. ?i n/ n where k=

- 1. Let x be indefinitely small, then we have 

hence sm a: = . 2"" . * . ,27r ,/br 1 = - sm- - sin 2 ... sm- ; n n n n sin 2 x/n in a! =nsin^ C os^l- s 4H^Vl- si t a;/ rV./l- n n sin 2 7r/7i/ siu-2ir/nj sl We may write this . x xf si sma;=?isin- cos-( 1 r n 7i si where R denotes the product / sin 2 ? / (l- =-}(!-

sin 2 / V 

N n / x and m is any fixed integer independent of n. It is necessary, when we make n infinite, to determine the limiting value of the quantity R ; then, since the limit of ^^ is s ~, and that of sin THTT/Ti . n sin - cos - mr y n &gt; U1 "ty. we have sin a; Now R is less than unity, since sin - is less than sin n s n

also by an elementary algebraical proposition R is greater

than 1 -sin 2 |(cosec 2 1 ^~ + ... +cosec 2 ^ and cosec ^&lt;^,if &lt; ; R is therefore greater than orthan 4 Im 7W + 1 m+l 7/1 + 2 By? __ k-1 k or than 1 - . Hence R=l - , where is some proper fraction whence When TO is indefinitely increased this becomes The expression for cos x in factors may be found in a similar manner by means of the equation cos x = 2 sin ^ cos ~ , or may be deduced thus 4 sin 2a; 2 sin x Pil- n=+&lt;a 2a; .(26). n=-&lt;n 2?i + l?r&gt; If we change x into tx, we have the formulae for sinh x, cosh x as infinite products v=xP (l+_ ) C osha;=P (T+ |. 7i=0 n-ir*J n=Q 271 + 1 -IT-/ In the formula for sin a; as an infinite product put x^, we then get 1 =

if we stop after 2?t factors in the numer

- . ator and denominator, we obtain the approximate equation j.ri j^.^-iji (2re+1) ~2 2 2 .4 2 .6 2 ... (2ra) 2 -^ n + i &gt; 572 TRIGONOMETRY [ANALYTICAL. or 2 " 7 % = V7i7r, where n is a large integer. This ex- 1 . O 9 * 6*ft&gt; * pression was obtained in a quite different manner by Wallis (Arith- metica infinitorum, vol. i. of Opp.). Series for We have ^ + ^ p f 1 + cot,cosec, sin (x + y) __ V tan, and sinx sec. or cos y + sin y cot x Equating the coefficients of the first power of y on both sides we obtain the series = x + xT^r x^rx + 27r x-2,r From this we may deduce a corresponding series for cosec x, for, since cosec x = cot~ -cot x, we obtain 1111 11 1 cosec x = x x + ir x-ir x+2ir x-2ir x + 3ir x-3ir By resolving ^ z) into factors we should obtain in a similar cos a; manner the series 22222 tanx= -- ^ r-^- + . "ir-2a; and thence 37r-2a: +...(29), .. ....(30). These four formulae may also be derived from the product formulae for sin x and cos x by taking logarithms and then differentiating. COS X Glaisher has proved them by resolving the expressions for and . ... as products into partial fractions (see Quart. Journ. smx Math., vol. xvii.). The series for cot a; may also be obtained by a continued use of the equation cotx = ^l cot ^ + cot g J (see a paper by Dr Schrbter in Schlbmilch s Zeitschrift, vol. xiii. ). Series for Various series for IT may be derived from the series (27), (28), (29), TT derived (30), and from the series obtained by differentiating them one or more from times. For example, in the formula (27) and (28), by putting series for ^T t cot and n cosec. irf, 1 1 1 1 TT = n tan -11 -- ? + ; - - ;- + n = ... r , n I Ti-1 + l 2n-l 271+1 ; . 7T/, 1 1 1 1 1 m 7U 1+ 77^1~7 l +T~27i-l + 277+T i&gt; J ; if we put n = 3, these become i_i_i i i 2 4 5 + 7 + 8"V By differentiating (27) we get coseca;=- z=, and we get ir z = (x- irf (x + 2 (x-2ir) 2 These series, among others, were given by Glaisher (Quart. Journ. Math., vol. xii.). Sums of We certain series. we differentiate these formulae after taking logarithms, we obtain the series 1 1 1 , . .. (a: 2 / y? 1 +&gt;). cosh iry= P[ 1 + I ; if n*/ 2n + l 2 / . rt, 2a: CO 2a^~ gtanh ^= These series were given by Kummer (in Crelle s Journ., vol. xvii.). The sum of the more general series + . . . , has been found by Glaisher (Proc. Land, Math. Soc., vol. vii.). Certain If in the series (12) and (13) we put n = 2x, 0=^, we get series for sine and cosine. S T~ ""| x* a?(x&gt;-l*) 15 &gt;...}. These series were given by Schellbach (in Crelle s Journ., vol. xlviii. ). TT 2a/ If in the same series (12), (13) we put = o&gt; n= we get cosx=l- . - 4V 2 ) 4x 2 (4a! 2 -2V 2 ) 1.2.3.4JT* 1.2.3.4.5.67T 6 ! -7r 2 )(4x 2 -3V 2 ) + ..., Sin X- - - _ _ 5- -r 1 o Q 4 c^ 7T J. A IT lJbt) 1 Vva We have of course assumed the legitimacy of the substitutions made. These last series have been discussed by M. David (Bull. Soc. Math, de France, vol. xi.) and Glaisher (Mess, of Math., vol. vii.). If U m denotes the sum of the series =r^,+^+ 5+ &gt; V m that Sums of powers of the series ^ + q^+ ?^+ - &gt; an( i W m that of the series - ^ of reci " + = sr-+ ..., we obtain by taking logarithms in the formulie . u ~ 5" 7 m ral uum- (25) and (26) log (x cosec x) = Lers. and differentiating these series we get (32). In (31) x must lie between TT and in (32) between ir. Write equation (30) in the form -,. ,. (27l+l)7T secx=S(-l) n - and expand each term of this series in powers ,of a; 2 , then we get secx= -- -j ? I / h (33), where x must lie between JTT. By comparing the series (31), (32), (33) with the expansions of cot x, tan x, sec x obtained otherwise, we can calculate the values of Uz, U 4 . . . Vz, V t .. . and W v W z When U n has been found, V n may be obtained from the formula For Lord Brounker s series of TT, see SQUARING THE CIRCLE (vol. Cou- xxii. p. 435). It can be got at once by putting = 1, b=3, tinued & 2 factors . . ,, , , ,, 111 1 0=5. ... in Liners theorem = rH --- .... = i -- 1 .... t a b c a+ b-a+ c-b + for TT. Sylvester gave (Phil. Mag., 1869) the continued fraction TT_ J_ 1.22.33.4 2~ + 1+ 1+ 1+ 1+ " which is equivalent to Wallis s formula for IT. This fraction was originally given by Euler (Comm. Acad. Petropol. , vol. xi. ) ; it is also given by Stern (in Crelle s Journ., vol. x.). It may be shown by means of a transformation of the series for Con- , sin x , , , x cos x and that tan x= x 2 a? y? m.- T- i tinned This may be also fractions 1- 3- 5- 7- " easily shown as follows. Let j/=cos /x, and let y , y" . . . denote . the differential coefficients of y with regard to x, then by forming caTfunc- these we can show that 4xy"+2y + y=Q, and thence by Leibnitz s . theorem we have Therefore .= -2 -f^= " " ^ y y/y y( n + l ) hence -2VxcotVx= -2- -6- -10- -14- " Replacing Vx by x we have tan x= = ^ - Euler gave the continued fraction _n tan x (ri* - 1 ) tan 2 x (7i 2 - 4) tan 2 x (n? - 9) tan 2 x Lclll ?ZC , J this was published in Mem. de I Acad. de St Petersb., vol. vi. Glaisher has remarked (Mess, of Math., vol. iv. ) that this may be derived by forming the differential equation where ?/=cos(rearc cosx), then replacing a; by cos x, and proceeding as in the former case. If we put 7i = 0, this becomes ^_tan x tan 2 x 4 tan 2 x 9 tan 2 x whence we have arctanx- 1+ ^ ^ 7+ ... It is possible to make the investigation of the properties of the simple circular functions rest on a purely analytical basis. The sine Purely malyti^al treat of x would be defined as a function such that, if x= I j===. then . J ^ l ~f y= sin x; the quantity -would be defined to be the complete integral f vf=? We * UM to haV6 i -= ff ^change J *9 the variable in the integral to z, where?/ 2 + 2 2 =l, we then have - - x = I i and z must be defined as the cosine of a;, and is

The integral may also be obtained in the form

thus equal to sin (s-*)i satisfying the equation sin 2 a; + cos 2 a3=l. Next consider the differential equation This is equivalent to hence the integral is 2/Vl - s 2 + zVl - 2/ 2 =a constant. The constant will be equal to the value u of y when 2=0; whence 2/ Vl - z 2 + z Vl - y- = u. 2 - Vl - j/ 2 Vl - z a = dr /" (fit o vf^; Let . _ . __ Vl-2 2 we have o + /3 = 7, and sin 7 = sin a cos + cos a sin /3, cos 7 = cos a cos /3 - sin a sin $ the addition theorems. By means of the addition theorems and the values sin |=1, cos |= Owe can prove that sin | + a; ) = cos x, cos ( o + x ) = ~ s i n ^ 5 and thence by another use of the addition theorems that sin (IT + x) = - sin x cos (v + x) = - cos x, from which the periodicity of the functions sin x, cos x follows. We have also I -/==,= - t log* J VI -y* whence log,, ( Vl - y 2 + ty) + log e (Vl-z 2 + tz) = a constant. Therefore (Vl-y 3 + cy)(Vl -z a +iz) = vT^ + m, since =?/ when 2=0; whence we have the equation (cos a + 1 sin a)(cos p + 1 sin |3) = cos (a + /3) + 1 sin (a + /3), from which De Moivre's theorem follows. (e. w. h.)

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1. See also vol. ii. of the Asiatic Researches (Calcutta).