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Trigonometry

TRIGONOMETRY (from Gr. rplyuvov, a triangle, \ikrpov, 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 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 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 geodesies 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 connected with 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 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 1 = 3-1416, which value was given by the astronomer Aryabhata (476-550) in a work called Aryabhaiya, written in verse, which was republished ' 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 Ja = V(i7i9(3438-cosa)| 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 Surya-Siddhanta, 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 I5 = 89o', sin 7 3o' = 449', sin 3 45' = 225' were thus obtained. Now the angle 3 45' is itself 225'; thus the arc and the sine of j*j 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 (78.225') = sin (71.225')

-sin (tt-i 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 I " = 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, 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. 929-930, and whose Tables were translated in the 12th century by Plato of Tivoli into Latin, under the title De scientia stellar-urn, 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 <j> from the equation sin <#>/cos <t> = 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 6 sin c cos A for a spherical triangle ABC. Abu'1-Wafa 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 1008, 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 1 See also vol. ii. of the Asiatic Researches (Calcutta).

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 formulae for right-angled 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 Miiller (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 Trigonomelria 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 90. His method of calculation depends upon the formulae which give sin no. and cos no. in terms of the sines and cosines of ( i)a and (n 2)0; 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 J<#>) 3 3(2 cos Jtf>)=2 cos <j> to solve the cubic x* 3a*x=a*b(a> $6); he obtained, however, only one root of the cubic. In 1593 Van Roomen proposed, as a problem for all mathematicians, to solve the equation 45? -3795:v 3 +95634y' i - +945y 41 -45> |43 +/ 5 = C. Vieta gave y = 2 sin jrf, where C=2 sin </>, 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 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 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- l6 34)t 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 Euler had 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 V i; he published in 1712 the general formula for tan IKJ> 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, etc., 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 O, and initially coincident with a fixed straight line OA , to revolve round 0, and finally to take up any position OB. We shall sup- ConP" pose that, when this revolv- otA " ing straight line is turning J. aay ,. 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. 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 O in either direction. Each time that the straight line makes a complete revolution round O 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 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 ... ...

angle is taken as the unit and is called a degree; the vj degree is divided into sixty equal parts called minutes; and the minute into sixty equal parts called seconds; *f '" . angles smaller than a second are usually measured as decimals of a second, the "thirds," "fourths," etc., not MagaH 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 IT. 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 thesame 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 ir, and, since the same angle is 180, we see that the number of degrees in an angle of circular measure B is obtained from the formula i8oX0/x. The value of the radian has been found to 41 places of decimals by Glaisher (Proc. London Math. Soc. vol. iv.) ; the value of I/IT, from which the unit can easily be calculated, is ;ivento 140 places of decimals in Crunerts Archiv (1841), vol. i. To 10 lecimal 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 size of Definition 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.

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 . . is convenient to have a simple mode of indicating in Port'ns of wmcn direction on the straight line the length AB has in Infinite been described; this may be done by supposing that a straight P om t moving in one specified direction is describing Line. a positive length, and whe'n 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 length LM, taken with the positive or negative sign ** according to the convention as stated above, is called Lin 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, 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. i. Draw BOB' at right angles to and equal to AA'.

We shall suppose that the direction from A' to A is the positive one for the straight line AOA', and that from metrical B ' to B for BOB'. Suppose OP of fixed length, equal Functions. to ^A, 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 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 a be the angle which RS makes with A 'A , the projections of RS on A' A , B'B are RS cos o and RS sin a 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 TT + O 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 is negative and its cosine positive. If o 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 2ir-t-o, 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 2nv+a is positive if a is between o and IT and negative if a is between IT and 2v, and the cosine of the same angle is positive if a is between o and |T or |T and 2* and negative if a is between |TT and iSx.

In fig. 2 the angle POA is o, the angle P"'OA is o, P'OA is T a, P"OA is T+O, FOB is |T a. By observing the signs of the projections we see that sin( a) = sin a, sin(T a)=sin a, sin (ir+o) = sin o, cos( o)=cos a, COS(T o) = cos a, COS(T+U) = coso, sin(|ir a) =cos a, cos(|ir a) =sin o. Also sin(|ir+a) =sin(ir |ir o) = sin(jT a) = cosa, COS(|T-|-O) = cos(ir |T a) = cos(|ir a) = sin a.

From these equations we have tan( a) = tan a, tan(7r o) = tan a, tan(+a) = tan o, tan(Jr a) =cot o, tan (Jir+o) = 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 27nr+a and 2nir+(ir o), and are all included in the formula fjr+( i) r a, where r is any integer; this therefore is the formula for all angles having the same sine as o. The only angles which have the same cosine as o are those bounded by OA, OP and OA, OP", and these are all included in the formula 2nira. Similarly it can be shown that nir+o 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 . .. projected line, we get sin 2 a+cos 2 a = i, and from this ?' ; by the help of the definitions of the other functions we Trironodeduce the relations i + tan 2 o = sec 2 o, I + cot 2 o = metrical cosec 2 a. We have now six relations between the six Fmnctloas. 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 o, |T, *, ITT, 2ir, and the signs of the functions of angles between these values ; / denotes numerical increase and D numerical decrease :

Angle . .

O...|?r I*..* 7r...|ir |T...2T Sine . Cosine Tangent . Cotangent Secant Cosecant .

o i 00 +D +1 +D +1 +D i o 00 00 -D -I -D +1 00 00 +1 +D -I -D 00 OO -D +1 -D -I +D -I O I O 8 i 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 :

sine cosine V6 V2 V6+V2 Vs-i VIO+2V5 1zz T "6 2 "3* IT 1 Vio 2VJ5 ft- ~V2 TT cosine sine These are obtained as follows, (i) \ir. The sine and cosine of this angle are equal to one another, since sin Values of JT=COS (|ir Jir) ; and since the sum of the squares Trlgonoof the sine and cosine is unity each is l/V 2. (2) |ir and Jir. n ^ etrk ^ 1 Consider an equilateral triangle; the projection of one l ~ uact side on another is obviously half a side; hence the cosine lrson ' e of an angle of the triangle is | or cos ir = |, and from "X es - this the sine is found. (3) T/IO, jr/5, 2ir/5, 3Yi- In the triangle constructed in Euc. iv. 10 each angle at the base is fjr, and the vertical angle is ir. If a be a side and b the base, we have by the construction a(a b)=V; hence 2& = a(V5 i); the sine of ir/io is b/2a or i(V5 i), and cos JTT is a/2b J (V5 + l). (4) j'j.x, yjjir. Consider a right-angled triangle, having an angle JTT. Bisect this angle, then the opposite side is cut by the bisector in the ratio of V3 to 2; hence the length of the smaller segment is to that of the whole in the ratio of V3 to V3+2, therefore tan jVH V3/(V3+2)j tan Jir or tan i 1 5 ir = 2-V3. and from this we can obtain sin -j^ir and cos j'jTr.

5. 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 p la angle being measured posi- S!ng lively in the same direction _ , as A; draw FE a perpen- < s ' ne dicular on DO (produced i{ necessary). The projection , of OF on OA is the sum of T Aa s' es - the projections of OE and EF on OA . P Now OE is the projection of OF on DO, and is therefore equal to OF cos B, and EF is the projection of OF ere ce of on a straight line making an angle + Jjr with OD, and is therefore equal to OF sin B ; hence OF cos (A +B) =OE cos A +EF cos (\*+A)

= OF (cos A cos B sin A sin B), or cos (/I +5) =cos A cos 5 sin A sin 5.

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 (.4 B) =cos A cos B+sin 4 sin B.

If we projected the sides of the triangle OEF on a straight line making an angle +Jir with OA we should obtain the formulae sin (A B) =sin A cos Bcos A sin B, which are really contained in the cosine formula, since we may put Jjr-BforB. The formulae //( tan (A tan B , co t cot<4cotB=Fi are immediately deducible from the above formulae. The equations sin C+sin D = 2 sin \(C+D) cos %(C-D), sin C sin D=2 sin |(C D) cos J(C+.D), cosD+cos C = 2 cos %(C-\-D) cos J(C D), cos > cos C = 2 sin KC+D) sin |(C P), 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 %(C+D) and that which DC makes is J(ir-t-C+.D) ; the angle COE is j(C 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 OC and CE is equal to that of OE; hence the sum of the projections of OC and OD is twice that of OE, or cos C+cos D=2 cos |(C+>) cos %(C-D). The difference of the A projections of OD and OC an 04 p IG . is equal to twice that of ED, hence we have the formula cos D cos C = 2 sin J(C+D) sin \(C D). The other two formulae will te obtained by projecting on a straight line inclined at an angle + iir to OA.

As another example of the use of projections, we will find the sum of the series cos o+cos (<z+/3)+cos (a+2/3)+ . . . +cos (a+n-i(3).

Suppose an unclosed polygon each angle of which s = ft to be inscribed in a circle, and let A, 4,, At, . . ., A n be n + l consecutive angular points; iriunnuui.ai P be the diameter of the circle; and suppose a Progression. stra 'S nt ' me drawn making an angle a with AAi, then ' o+/3, a+2/3, . . . are the angles it makes with A\ AI, At, AS, . . .; we have by projections A A n cos (a+j 1/3) =AAi [cos a+ cos a+/3+... + cosa+(n 1)/3), also AAi = T> sin /9, AA n = D sin Jn/3; hence the sum of the series of cosines is cos (a+^ i /S) sin JjS cosec J/3.

By a double application of the addition formulae we may obtain the formulae sin (Ai+At+A 3 )=sin AI cos At cos 4 3 +cos AI sin 4 2 cos 4 3 +cos A\ cos 42 sin A 3 sin 4i sin 42 sin 4 3 ; cos (4i+4 2 +4 3 ) =cos 4i cos 42 cos 4 3 cos AI sin At sin At sin 4i cos At sin 4 3 sin AI sin 4j cos 4 3 . We can by induction extend these formulae to the case of n angles. Assume sin (4!+4 2 + . . . +4 n ) =Si-S 3 +S 6 - . . .

Series of Cosines la ' et Formulae for Sine and Cosine of Sum of Angles.

where S r denotes the sum of the products of the sines of r of the angles and the cosines of the remaining n r angles; then we have sin (A,+A t + . . . +A n +A^ =cos X+i(5,-S,+5,- . . .)

+sin A^StSt+St . . .). I he right-hand side of this equation may be written (5i cos /l^n-l-5o sin 4^i) (5 3 cosy4^n-f5 2 sin^4 + i)+ . . ., or S'i-5',+ . . .

where S' r denotes the quantity which corresponds for n+i angles to 5, for n angles ; similarly we may proceed with the cosine formula. The theorems are true for n = 2 and n = 3; thus they are true generally. The formulae Formulae cos 2 A - cos 2 A sin* A = 2 cos 2 4 1 = 12 sin 2 A , for Multiple aodSub- sin 2A = 2 sin A cos A, Multiple Angles.

tan zA = 2 * a A < I -tan 2 A sin $A =3 sin A 4 sin 3 A, cos j,A =4 cos 3 A 3 cos A, sin nA =n cos-i A sin A -" (n ~' ) | (n ~ 2) cos"-' A sin 8 A + . . .

A sin- A, cos nA =cosM n ^ ^ cos"" 2 A sin 2 A + . . .

may all be deduced from the addition formulae by making the angles all equal. From the last two formulae we obtain by division n tan A - = 3 ta " A ~ In the particular case of n = 3 we have tan 3^ = a " .

I 3 l-3.Il A The values of sin \A, cos J^4, tan \A are given in terms of cos A by the formulae where #> is the integral part of A/2ir, gthe integral part of A/2ir+$, and r the integral part of A ITT.

Sin j.4, cos \A are given"in terms of sin A by the formulae 2 sin M = (-i)*'(l+sin4)* + (-l)'(l-sin4)i, 2 cos \A =(-i)*'(i +sin 4)*- (-i)'(i -sin 4)*, where ' is the integral part of A/2ir+l and q' the integral part of A/ar-i.

6. In any plane triangle ABC we will denote the lengths of the sides BC, CA, AB by a, b, c respectively, and the angles BAG, ABC, ACB by A, B, C respectively. The fact that the projections of b and c on a straight line perpendicular to the Properties side a are equal to one another is expressed by the equa- of ' r ' aa g les - tion 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 a/sin A=b/sin B = 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 a = b cos C+c cos Bl b = c cos A -j-o cos C p c = o cos B+6cos 4J 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 6 2 +c 2 -a 2 = 2etc cos A or cos A =(i 2 +c 2 -o 2 )/2etc, which gives the cosine of an angle in terms of the sides. From this expression for cos A the formulae (s-b)(s-c) - -- tan \A where s denotes 5(a+&+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.

o. 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+l is a positive or negative multiple of 6, provided that we change the signs of all the sines. Suppose p-{-q-\-r-\-l =6n, then the sum of the three angles 2nir-(pA+qB+rC),2m r -(rA+pB+qC),2mr-(qA+rB+pC)\sir; and, since the given relation follows from the condition A+B + C = TT, we may substitute for A, B, C respectively any angles of which the sum is IT; thus the transformation is admissible.

0. 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, ir 2A, v2B, ir 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 aA, b cos B cos 2B, c cos C cos 2C and the angles are &,A ir, 4B IT, 4.C IT; we may therefore substitute these expressions for the sides and angles of the original triangle; for example, we obtain thus the formula A _a 2 cos 2 A cos 2 2A -6 2 cos 2 B cos 2 2B-c 2 cos 2 C cos 2 2C 2bc cos B cos C cos 2B cos 2C This transformation obviously admits of further exten- sion. Solution of (i) The three sides of a triangle ABC being given, Triangles. the angles can be determined by the formula L tan \A =io+J1og (s 5)+| log (s c) } log i i log (so) 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 and Escribed Circles of a Triangle.

L tan JG4-B)=log ( o _)_l O g (a+b) +L cot JC, and the side c is then obtained from the formula log c=log a+Z, sin CL sin A.

(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=Z, sin A +log b log a; this gives two supplementary values of the angle B, if b sin A < a. 'If b sin A > o there is no solution, and if b sin A= a there is one solution. In the case b sin A < a, both values of B give solutions provided 6 > o, but the acute value only of B is admissible if b < a. The other side c can be then determined as in case (2).

(4) If two angles A, B and a side a are given, the angle C is determined from the formula C = ir A B and the side b from the formula log 6= log a+Z, sin BL sin 4.

The area of a triangle is half the product of Areas o a gjjg ; nto tne perpendicular from the opposite . . angle on that side; thus we obtain the expressions laterals T % sin A > !*(*-<*) (s-b)(s-c)}\ for the area of a triangle. A large collection of formulae for the area of a triangle are given in the Annals of Mathematics for 1885 by M. Baker.

Let a, 6, c, d denote the lengths of the sides. AB, BC, CD, DA respectively of any plane quadrilateral and A-}-C = 2a; we may obtain an expression for the area 5 of the quadrilateral in terms of the sides and the angle a.

We have 2S = ad sin A+bc sin(2a A)

and J(a 2 +<P-6 2 -c 2 )=o<2 cos A-bc cos (?.aA); hence 4^ 2 + ?(a 2 +^ 2 b 2 c z ) 2== <i 2 d 2 -\-b 2 c 2 - L 2abcd cos 2a. If 2S = a + b+c +d, the value of 5 may be written in the form S=ls(s-a)(s-b')(s-c)(s-d)-abcdcos 2 a}L Let R denote the radius of the circumscribed circle, r of the inscribed, and ri, r 2 , r 3 of the escribed circles of a triangle Radii of Clr- ABC; the values of these radii are given by the followcumscrlbed, ; n g formulae : Inscribed R = abc ^ s = a/2 s ; n A ^ r = S/s = (sa)tan %A =4^? sin \A sin JB sin JC, ri=S/(s a) =s tan \A =\R sin \A cos jB cos JC.

Spherical Trigonometry. 7. We shall throughout assume such elementary propositions in spherical geometry as are required for the purpose of the investigation of formulae given below.

A spherical triangle is the portion of the surface of a Sphere bounded by three arcs of great circles of the Sphere. If BC, CA, AB denote these arcs, the circular measure of the ofSohertcal an g' es subtended by these arcs respectively at the Trlansfle centre of the Sphere are the sides a, b, c of the spherical triangle ABC; and, if the portions of planes passing through these arcs 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 Associated wmcn '* belongs. Since two great circles intersect Triangles. eacn otrl er in two points, there are eight triangles of which the sides are arcs of the same three great circles. If we consider one of these triangles .4BC as the fundamental one, then one of the others is equal in all respects to ^4BC, 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 ^4BC. These triangles may be called the associated triangles of the fundamental one ABC. It follows that from any 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 remaining side and the remaining angle being unaltered, for such formulae are obtained by applying the given formulae to the associated triangles.

If A', B', C' are those poles of the arcs BC, CA, AB 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 j4BC. The sides of the polar triangle are TT A, r B, ir C, and the angles v a, itb, TTC. 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 jangle and each angle into the supplement of the opposite side.

8. Let O be the centre of the Sphere on which is the spherical triangle ^4BC. Draw ^4Z. perpendicular to OC and AM perpendicular to the plane FIG. 5.

OBC. Then the projection of OA on OB is the sum of the projections of OL, LM, MA on the same straight line. Since AM has no projection on any straight line in the p aa j a . plane OBC, this gives angles. mental OA cos c = OL cos a-\-LM sin a. Equations Now OL = OA cos b, LM = AL cos C = OA sin b cos C; between therefore cos c = cos a cos 6+sin a sin b cos C. Sides and We may obtain similar formulae by interchanging the Angles. letters a, b, c, thus cos a=cos 6 cos c+sin b sin c cos A 1 cos 6 = cos c cos <z+sin c sin a cos B C (i)

cos c =cos a cos 6-j-sin a sin b cos C } These formulae (i) 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 formulae cos A = cos B cos C+sin B sin C cos a )

cos B = cos C cos A +sin C sin A cos 6 > (2)

cos C= cos A cos B+sin A sin B cos c )

In the figures 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 6.

By interchanging the sides we have the equation sin A sin B sin C = K sin a sin b sin c (3) If we eliminate cos b we shall find below a symmetrical form for k. between the first two formulae of (i) we have cos a sin 2 c = sin b sin c cos A +sin c cos c sin a cos B; therefore cot a sin c = (sin b/sin a) cos A +cos c cos B = sin B cot A +cos c cos B. We thus have the six equations cot a sin 6 = cot A sin C+cos 6 cos C cot b sin o = cot B sin C+cos a cos C cot b sin c cot B sin A +cos c cos cot c sin b = cot C sin A +cos b cos cot c sin a=cot C sin B+cos a cos cot a sin c=cot A sin B+cos c cos When C Jir formula (i) gives cos c = cos a cos b sin b sin B sin c ) sin a=sin A sin c \ tan o = tan A sin 6 = tan c cos B) tan b = tan B sin a = tan c cos A ( cos c = cot A cot B and (3) gives from (4) we get The formulae and cos A =cos A sin B I cos B=cos b sin A I (a) (ft)

(T) W (r)

follow at once from (a), (0), (7). These are the formulae 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, CF are three arcs drawn through A, B, C 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 AE sin 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.

9. If D be the point of intersection of the three Formulee bisectors of the angles A, B, C, and if DE be drawn for Sine perpendicular to BC, it may be shown that BE and Cosine = i(a + c-Z>) and C = i(o + 6-c), and that of Half the angles BDE, ADC are supplementary. We have Angles.

sin c sin ADB sin 6 sin ADC .< c -51/1 also 5TT = = nr- ' ^n = m' therefore sin 2 \A sin BD sin %A sin CD sin %A sin BD sin CD sin CDE sin BDE -- : r :

sin o sin c sin i(a+c-6), and sin T> . . But sin ... , therefore CD sin C>E = sin C = sin |(a+6-c); l(g+c b) sin %(q+b c) ) j , , sin & sin " - f (5)

Apply this formula to the associated triangle of which ir A, ir B, C are the angles and v a, v b, c are the sides; we obtain c-a)sin|(a+6+c)) i sin b sin c )

the formula cos :

. A (sir m-= j- la to t igles an A _ ( sin '2~ I (7)

By division we have i sin J(a+c b) sin J(a+& c) ) J ! sin i(b-^ca) sin J(a+6+e) ) and by multiplication sin A = 2Jsin (a+6+c) sin J(6+c a)sin J(c+a 6) sin J(a+6 c)[J sin b sin c = |l cos 2 a cos 2 6 cos'c +2 cos a cos jcoscjj sin b sin c. Hence the quantity k in (3) is (i cos 2 o cos 2 6 cos 2 c+2 cosacos b cos cjS/sina sin 6 sin c. (8) Of Half- Apply the polar triangle transformation to the formulae sides. (5), (6), (7) (8) and we obtain a. (cos^A+C-B) cos JQ4+B-C)i ( sin B sin C )

cos J(B+C .4) cos JM+B + C ) J .

(10)

cos- = tan a ( 2~ ( -cos sin B sin C )

-A) cos |Q4 +B + CM (n) cos J(/l+C'-.0) cos $(A+H-L > ^ ' If k' = {i cos'X cos 2 B cos'C 2 cos A cosBcosC)J/siny4 sinBsinC, we have. kk' = I (12)

10. Let be the middle point of AB ; draw ED at right angles to p AB to meet AC in D; then DE bisects the angle A DB. Let CF bisect the angle DCB and draw FG perpendicular to BC, then Delambre's Formulae.

AFCG=90-JC.

From the triangle CFG we have cos CFG = cos CG sin FCG, and B from the triangle FEB cos FB = cos B sin FB. Now the angles CFG, EFB are each supplementary to the angle DFB, therefore jC = sinJ(.<4+B)cos2 l c. (13)

Also sin CG = sin CFsin CFG and sin B = sin BF sin EFB; therefore sinj(a 6)cosJC = sinJ(.<4 B)sinjc. (14)

Apply the formulae (13), (14) to the associated triangle of which a, TT b, TC, A, IT B, ic C are the sides and angles, we then have B)sinlc (15)

cosjC. (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 1 808. They were also given by Gauss ( Theoria motus, 1809), and are usually called after him.

II. From the same figure we have Napier 1 * tan FG = tan FCG sin CG = tan FBG sin BG; Analogies, therefore cotJCsinJ(a 6)tanJ(.4 B)sinJ(a+&), MA r>\ sin i(a 6) ,_ . .

or tanJl4-B)= sin |^ +6) cotJC. <'?)

Apply this formulae to the associated triangle (T a, b, rc, vA, B, TT C), and we have If we apply these formulae (17), (18) to the polar triangle, we have ,. sin \(A B) .

n Jc (19)

n Jc. (20)

The formulae (17), (18), (19), (20) are called Napier's " Analogies "; they were given in the Mirif. logar.*canonis descriptio.

12. If we use the values of sin Ja, sin Jft, sin Jc, cos Ja, cos J6, cos Jc, given by (o), (10) and the analogous formulae obtained by interchanging the letters we obtain by multiplication ... . . .. c _^ cos JacosJ6sinC=cosJccosJ(.(4+B C) V . sin Jasin Jisin C = cos JccosJ(.4+B + C) )

These formulae were given by Schmiesser in Crelle's Journ., vol. x. 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 (i), (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 6 cos c cos A.

13. The formulae we have given are sufficient to determine three parts of a triangle when the other three parts are given ; moreover such formulae may always be chosen as are adapted , to logarithmic calculation. The solutions will be unique zjr* " except in the two cases (i) where two sides and the angle '" aa x ies - opposite one of them are the given parts, and (2) where two angles and the side opposite one of them are given.

Suppose a, b, A are the given .parts. We determine B from the formula sin B = 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 cot ^ - B) > tan *- Now tan JC, tan Jc, must both be positive; hence A B and a 6 must have the same sign. We shall distinguish three cases. First, suppose sin 6<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 6>sin a, there is no solution when sin b sin A > sin a; but if sin 6 sin ^4<sin a there are two values of B, both greater or both less than A. If a is acute, ab, 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 <_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 6 sin A>sin a. Thirdly, if sin 6 = sin a then B=A or v = 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 = %ir, there is no solution unless .4 = Jx, 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 & sin a was given by Professor Lloyd Tanner (Messenger of Math., vol. xiv.).

14. If r is the angular radius of the small circle inscribed in the triangle ^4BC, we have at once tan r = tan \A sin (s a), where 2s = a+b+c; from this we can derive the formulae tan r = n cosec s = %N sec \A sec JB sec JC = Radii of sin a sin |B sin JC sec \A (21) Circles where n, N denote the expressions Related to [sin s sin (s-a) sin (s-b) sin (i-c))J, Triangles.

j cos 5 cos (SA) cos (5 B) cos (S C)|J.

The escribed circles are the small circles inscribed in three of the associated triangles; thus, applying the above formulae to the triangle (a, JT b, irc, A, vB, v Q, we have for r\, the radius of the escribed circle opposite to the angle A , the following formulae tan fi=tan \A sin s n cosec (s a) = %Nsec \A cosec JB cosec JC = sin a cos ^B cos jC sec J.4. (22)

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 circumscribed circle of the triangle, and 1?!, R 2 , R the radii of the circles circumscribing the associated triangles, we have by writing Jrr R for r, %-ir Ri for TI, va for A, etc., in the above formulae cot R = cot Jacos (S A) \n cosec Ja cosec J6 cosec $c=-N sec 5 = sin A cos J6 cos Jc cosec Ja (23)

cot /?i = cot Ja cos S = Jn cosec Ja sec J6 sec \c = N sec (SA)

= sin A sin Jft sin Jc cosec Ja. (24)

The following relations follow from the formulae just given: 2tanJ? =cotri+cotr 2 +cotr 3 cotr, 2tan.Ri =cot r -j-cot r 2 +cot rs cot r t , tan r tan r\ tan r 2 tan r a = n 2 , sin 2 5 = cot r tan ri tan r 2 tan r s , sin 2 (sa) =tan r cot r\ tan r 2 tan r a .

15. If = ./4+B + C IT, it may be shown that 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.

We have Formulae for Spherical Excess.

hence cos \(A + B)

sin \C sin \(C - E)

sin \C sin %C - sin \(C - E)

cos j(a + 6) sin \(A + B) __ cos J(a - b)

COS jC COS JC COS Jc cos J(a + b) , cos J(C - E) cos J(a - b).

cos Jc a cos JC ~" ' cos \c ' sin JC +sin J(C )

cos J(o -f b) + cos ji(a + b) ' therefore tan^(C-E) =tlin & tan i( J ~ c Similarly tan IE tan 2 J(C-) =tan |(s-a) tan |0-Z>); therefore tan JE = {tan Js tan J(.s a) tan K$ &) tan $(s c)ji (25) This formula was given by J. Lhuilier.

Also sin JCcos JE-cos ^C sin. |= cos ^ (a 1 + ^ sin *C; COS gC i ^ i r^ i i ^-- i 1-* cos 4 (a &) cos JC cos JE+sm JC sin | = ^ lg cos JC; whence, solving for cos JE, we get , l+cos o+cos b -(-cos c cos JE = ! - 1 ! - nr"^ i (26)

4 cos ja cos J6 cos Jc This formula was given by Euler (Nova acra, vol. x.). If we find sin JE from this formula, we obtain after reduction sin JE = ; 2 cos \a cos Jft cos Ji a formula given by Lexell (Ada Petrop., 1782)

CVr\m f-h* *iniiatir\nc f o T ^ foo^ ftt\ ftA\ \\rf formula given by Lexell (Ada Petrop., 1782).

From the equations (21), (22), (23), (24) we obtain the following formulae for the spherical excess : sin 2 jE = tan R cot RI cot RI cot R$ 4(cot ri+cot >-2+cot ; hence cos j = cos M N sec j<z.

(cot r cot n+cot r 2 +cot r 3 ) (cot r+cot n cot r 2 +cotr 3 )X (cot r+cot n+cot r 2 +cot r,).

The formula (26) may be expressed geometrically. Let M, N be the middle points of the sides AB, AC. Then we find cos MN i +cos a+cos 6+cos c 4 cos 56 cos jC 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 i E.

16. Let ABCD be a spherical quadrilateral inscribed '* ,in a small circle; let a, b, c, d denote the sides AB, BC, oadri CD> DA respectively, and *, y the diagonals AC, BD.

It can easily be shown by joining the angular points Inscribed of the Quadrilateral to the pole of the circle that la Small A + C = B +P' , ll , we . use the last expression in (23)

Circle e radii of the circles circumscribing the triangles BAD, BCD, we have sin A cos Ja cos jo" cosec j;y = sin C cos j6 cos jC cosec Jy; whence sin C cos \b cos \c cos \a cos {d This is the proposition corresponding to the relation A-\-C = trlor a. plane quadrilateral. Also we obtain in a similar manner the theorem sin \x sin Jy sin B cos j6~sin A cos {d' analogous to the theorem for a plane quadrilateral, thac the diagonals arc proportional to the sines of the angles opposite to them. Also the chords AB, BC, CD, DA are equal to 2 sin Ja, 2 sin J6, 2 sin %c, 2 sin %d 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 sin \x sin Jy = sin \a sin Jc+sin j& sin \d.

It has been shown by Remy (in Crelle's Journ., vol. iii.) that for any quadrilateral, if z be the spherical distance between the middle points of the diagonals, cos o+cos 6+cos c+cos <i =4 cos J* cos \y cos \z. This theorem is analogous to the theorem for any plane quadrilateral, 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 Gudermann (in Crelle's Journ., vol. xlii.).

Analytical Trigonometry.

17. Analytical trigonometry is that branch of mathematical analysis in which the analytical properties of the trigonometrical Peiiodl- functions are investigated. These functions derive their city of importance in analysis from the fact that they are the simFuactioas. P^ est sin g 1 y periodic functions, and are therefore adapted to the representation of undulating magnitude. The sine, cosine, secant and cosecant have the single real period 2ir; i.e. each is unaltered in value by the addition of 2ir to the variable. The tangent and cotangent have the period jr. 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 functions, since they remain unaltered when the sign of the variable is reversed.

The theory of the trigonometrical functions is intimately connected with that of complex numbers that is, of numbers of the form *+ty(i = V -i). Suppose we multiply together, by the connexion rules of ordinary algebra, two such numbers we have wlia Taeory (xi + tyi) (xt + ty 2 ) = (xiXiyiyt) + i(#iy 2 + * 2 yi). of Complex We observe that the real part and the real factor of the Qaaautles. 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 Xi = n cos : , y^ = n sin 0i, * 2 = f z cos 2 , yi r 2 sin 2 , the above equations becomes ri(cos 0i+t sin 0j) Xr 2 (cos 2 + 1 sin 2 ) = n r 2 (cos0i+0 2 + 1 sin 0!+0 2 ).

We may now, in accordance with the usual mode of representing complex numbers, give a geometrical interpretation of the meaning of this equation. Let Pi be the point whose co-ordinates referred to rectangular axes Ox, Oy are xj, yi ; then the point PI is employed to represent the number Xi+iyi. In this mode or representation real numbers 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, AI represent the numbers =t i, the points a, Oi the numbers t. Let P 2 represent the expression *2+ty 2 and P the expression (*i+'yi)(x 2 +iy 2 ). The quantities ri, 0], r 2 , 2 are the polar coordinates of PI and P 2 respectively, referred to O as origin and Ox as initial line; the above equation shows that n r 2 and 0i+02 are the polar co-ordinates of P; hence OA : OPi : : OP 2 - OP and the angle POP 2 is equal to ' IG ' 8 ' the angle PiOA. Thus we have the following geometrical construction for the determination of the point P. On OP 2 draw a triangle similar to the triangle O^Pi so that the sides OP 2 , OP are homologous to the sides OA, OPi, and so that the angle POP 2 is positive; then the vertex P represents the product of the numbers represented by PI, P 2 . If x 2 +ty2 were to be divided by Xi+iyi the triangle OP'P 2 would be drawn on the negative side of P 2 , similar to the triangle OA PI and having the sides OP', OP 2 homologous to OA, OPi, and P' would represent the quotient.

1 8. If we extend the above to n complex numbers by continual repetition of a similar operation, we have (cos 0i + t sin 0i) (cos 2 + t sin 2 ) . . . (cos n + i sin n ) * Molvre '* = cos (0! = 2 + . . . + n ) + t s in (0i + 2 + . . . +0 B ). " If 0i = 2 = . . . =0 n =0 1 , this equation becomes (cos 0+t sin 0)" = cos M0+i sin M0; this shows that cos +t sin is a value of (cos n0+t sin n9). If now we change into 8/n, we see that cos 0/n+t sin 0/n is a value of (cos + t sin 0)n; raising each of these quantities to any positive integral power m, cos me/n+i sin m0/n is one value of (cos 0+t sin 0)?. Also cos ( mS/n) + 1 sin (m6/ri) = cos m8/n + 1 sin m8/n' hence the expression of the left-hand side is one value of (cos + i sin 0)-"'". We have thus*De Moivre's theorem that cos ke+i sin k8 is always one value of (cos 0+i sin 0)*, where k is any rational number. This theorem can be extended to the case in which k is irrational, if we postulate that a value of (cos 0+t sin 0)* denotes the limit of a sequence of corresponding values of (cos 0+t sin 0)*,, where hi, fe. . .k,. . . is a sequence of rational numbers of which k is the limit, and further observe that as cos &0+i sin k8 is the limit of cos &s0+t sin k,0.

The principal object of De Moivre's theorem is to enable us to find all the values of an expression of the form (0+16)""", where m and n are positive integers prime to each other. _ If a = r cos 0, b = r sin 0, we require the values of '" enKoots r"" (cos 0+t sin )/. One value is immediately fur- J"V*2*** nished by the theorem ; but we observe that since the O nafl "v- expression cos 0+t sin 6 is unaltered by adding any multiple of 2jrto 0, the n/mth power of r mln (cos mj+asr/n+i sin j.9+2S7r/) is o+tfc, if i is any integer; hence this expression is one of the values required. Suppose that for two values Si and S 2 of s the values of this expression are the same; then we must have m.0+25iir/re TB.0+2j 2 7r/n; a multiple of 2jr, or Sist must be a multiple of n. Therefore, if we give s the values o, I, 2,. . . I successively, we shall get n different values of (0+16)""", and these will be repeated if we give s other values; hence all the values of are obtained by giving s the values o, i, 2, ... n i in the expression r ml " (cos m . + 2sir/n + i sin m.O + 2sir/n), where r = (a 2 +6 2 )i and 0= arc tan 6/a.

We now return to the geometrical representation of the complex numbers. If the points Bi, 5 2 , B 3 ,...B n represent the expres-; sion x+iy, (x+iy) 2 , (x+iy) 3 , ' . . . (x-\-iy) n respectively, the triangles OABi, O5iJ5 2 , . . . O5n_iB are all similar. Let (x+ty)" = a+tb, then the converse problem of finding the nth root of a-\-ib is equivalent to the geometrical problem of describing such a series of triangles that OA is the first side of the first triangle and OB n the second side of the wth. Now it is obvious that this geometrical problem has more solutions than one, since any number of complete revolutions round O may be made in travelling from Bi to B n . The first solution is that in which the vertical angle of each triangle is B n OA jn; the second is that in which each is (B n OA +2ir)/, in this case one complete revolution being made round O; the third has (B n OA +4?r)/n 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 + 1 sin 0)i. Suppose B to represent the expression cos 0+ t sin 0. If the angle AOPi is |0, PI represent the root cos |0+t sin J0; the angle AOB is filled up by the angles of the three similar triangles AOPi, PiOpi, piOB. Also, if P 2 , P 3 be such that the angles PiOP 2 PiOP 3 are fa-, f * respectively, the two sets of triangles AOPi, PiOps, p 3 OB and AOP,, PsOpt, piOB satisfy the conditions of similarity and of having OA, OB for the bounding sides; thus Pi, PS represent the roots J(0+2ir), cos f(0+4a-)+i sin FIG. 9.

cos J(0+27r)+i sin _ 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 problem of determining the values of the nth roots of unity is equivalent to the geometrical problem of inscribing a regular The th polygon of n sides in a circle. Gauss has shown in his Knots of Disquisitiones arithmeticoe that this can always be done Unity ky the compass and ruler only when n is a prime of the form 2 f +i. The determination of the nth root of any complex number requires in addition, for its geometrical solution, the division of an angle into n equal parts.

19. We are now in a position to factorize an expression of Factoriza- the form_ *" (a+tb). Using the values which we tlons. have obtained above for (o+t6)' /n , we have If 6=0, = * I "^ I X^~T I COS 5=0 L \ I, this becomes s = {n 1 / ,, \ (x-i)(x+i)P (x* - 2x cos =^ r +i)( even). (2)

S=i \ / +i (nodd). (3) ' If in (i) we put a= i, 6 = 0, and therefore = ir, we have (neven). (4)

fa odd). (5)

X+I= (X+1)P 5=0 Also x*" 2x"y cos nB+y* 1 = (x" y n cos tie+i sin nff) (x n y cos nO i sin nff)

0+2SV . e+2SK\ x 31 cos - =tisin - - I n n J Airy and Adams have given proofs of this theorem which do not involve the use of the symbol t (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.

In equation (6) put y = l/x, take logarithms, and then * differentiate each side with respect to x, and we get Theorem.

it 2 " 2 cosn6+x~^ n ~ s=0 Put x 2 = a/b, then we have the expression (a 2 - ft 2 ) (a 2 " - 2a"6" cos n8 + 6 2 ") for the sum of the series a 2 -2a& cos 0+ 20. Denoting the complex number x+iy by z, let us consider the series l+z+z 2 /2 ! + . . . +z"/! + . . . This series converges uniformly and absolutely for all values of z whose _ moduli do not exceed an arbitrarily chosen positive '" ejr " number R. Consequently the function (z), defined P a f ntlal as the limiting sum of the above series, is continuous Se " es in every finite domain. The two series representing (zi) and (zj), when multiplied together give the series represented by (01+22). In accordance with a known theorem, since the series for (zi) (35) are absolutely convergent, we have (zi)X(z2) = (21+22). From this fundamental relation, we deduce at once that j(z)j" = E(nz), where n is any positive integer. The number (i), the sum of the convergent series 1 + 1 + 1/21+1/3!..., is usually denoted by e; its value can be shown to be 2-718281828459. ... It is known to be a transcendental number, i.e. it cannot 6e the root of any algebraical equation with rational coefficients; this was first established by Hermite. Writing z = i, we have (n)=e n , where n is a positive integer. If z has as_ a value a positive fraction p/q, he real positive hence E(p/q)

we find that \E(p/g)}" = E(p)=e*>; hence E(p/q) is the real positive value of e"'". Again E(-p/g)XE(p/q)=E(o) = l, hence E(-plq) is the real positive value of e"*!*. It has been thus shown that for any real and rational number x, the value of E(x) is the principal value of e*. This result can be extended to irrational values of *, if we assume that e x is for such a value of x defined as the limit of the sequence e 11 , e",. . ., where xi, x*,. . . is a sequence of rational numbers of which * is the limit, since E(XI), E(x 2 ) . . ., then converges to (*).

Next consider (i +z/m) m , where m is a positive integer. We have by the binomial theorem, I \ /"g \ ' -- nr)7\+--- + (m)

lies between, and i+ (++. . . +*-/)

hence the product equals iB^.s i/2tn where 0, is such that o<0,< i. We have now ' 2m m il 2 + [i-* m J^ where z"> z 2 ( z n + . . .+^-i -] I+03- + P. ^m\ 2m ( " 3 r ^ Since the series for (z) converges, s can be fixed so that for all values of m>s the modulus of z'+'/fc + i)! + . . . +z m /ml is less than an arbitrarily chosen number |. Also the modulus of i+03Z/i + ...+0 m z- 2 /(z-2)! is less than that of i+i|z|/l! + |z| 2 /2! +..., or of e mod ', hence mod R,<%t+(i/2m). mod (zV)<e, if m be chosen sufficiently great. It follows that lim m _ 00 (i+z/n)'"=(z), where z is any complex number. To evaluate (z), write i+x/m = p cos <t>, y/m=p sin <j>, then (z) = lim m _co {p m (cos m<t>-\-i sin m<t>)\, by De Moivre's theorem.

-m ^ i j -{ } i i I -^ t we hcivG lim o m \ ml ( m(V+*/V0 2 )

. Let r be a fixed number less than V*+*/Vl> then lim m _co lies between i and linim-m j i-| ^-5 f , or between i and e 2 ' 2rt ; hence since r can be taken arbitrarily large, the limit is i. The limit of m<j> or m tan~ l {y/(x+m)\ is the same as that of my/(x+m) which is y. Hence we have shown that (z) =e*(cos y+i sin y).

21, Since E(x+iy)i I (cos y+sin y, we have cos y+i sin y = E(iy), and cos y i sin y=E(iy). Therefore cos y = i{(iy) +E(-iy)\, sin y = %i\E(iy)-E(-iy)\; and using Exponential the serjes d e fi nec l by (i'y) and (-iy), we find that Values of Trigonometrical Functions.

cos y = i - y 2 /2 ! + y 4 /4! - .

y = y /3 ! + y 5 /5 !.., where y is any real number. These are the well-known expansions of cos y, sin y in powers of the circular measure y. Where z is a complex number, the symbol e z may be defined to be such that its principal value is E(z) ; thus the principal values of e' v , tr*" are E(iy), E(iy). The above expressions for cos y, sin y may , then be written cos y = %( e ivJf-e- iv ), sin y = Jt(e' e~' v ). These are known as the exponential values of the cosine and sine. It can be shown that the symbol e? as defined here satisfies the usual laws of combination for exponents.

22, The two functions cos z, sin z may be defined for all complex or real values of z by means of the equations cos y = j((z) + , ,i__t (-z)),sinz = (k)((z)-(-z)),whereE(z)represents the sum-function of I + z+ z 2 /2! + . . . + z"/re! + . . . For real values of z this is in accordance with the ordinary definitions, as appears from the series obtained above for cos y, sin y. The fundamental properties of cos z, sin z can be deduced from this definition. Thus sin z=E(z). cos zi sin z=( iz); therefore cos 2 z+sin 2 z = (iz). ( iz) = i. Again cos (Zi+z 2 ) is given by Analytical Definitions of Trigonometrical Functions.

COS |(zz 2 )E(Z2)j, whence we have cos (zi+z 2 ) = cos Zi cos z 2 sin Zi sin Z 2 . Similarly, we find that sin (z 1 +z 2 ) =sin Zi cos Zj + cos Zi sin z 2 . Again the equation (z) = l has no real roots except z=o, for e">i, if z is real and >o. Also E(z) = i has no complex root a+i0, fot o if) would then also be a root, and (20) = (a+i/3)(o iff) = 1, which is impossible unless a = o. The roots of E(z) = i are therefore purely imaginary (except z = o); the smallest numerically we denote by 2 iv, so that (2iV) = i. We have then (2tVr) = |(2tV)) r = i, if r is any integer; therefore 2iirr is a root. It can be shown that no root lies between 2iVr and 2(r+i)zV; and thus that all the roots are given by z== t 2Vr. Since (y+2iV) = (z)(2zV) =(z), we see that (z) is periodic, of period 2iV. It follows that cos z, sin z are periodic, of periods 2ir. The number here introduced may be identified with the ratio of the circumference to the diameter of a circle by considering the case of real values of z.

23. Consider the binomial theorem Expansion of Powers of Sines and Cosines I" Series of p uttinga = Sines and Cosines of (2 COS 0)" = Multiple Arc.

+ n(n-i).^(n-r+i) 2cos(w _ 2r)9+ When n is odd the last term is 2 and when re is even it is n( - n ~ ' i ! cose, If we put a=e l9 , 6= e~' fl , we obtain the formula +(-i)^*- I )-- r -f- r + I >acoB(ii-.2r)a. . .

(-i);- when re is even, and (-l)^ n ~ I )(2sin9)"= 2sinn0-re . 2sin(re 2)0+ -2 sin (re 4 sin0 when n 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 formulae when n is not a positive integer.

Consider the identity log(l -/>x)+log(l -qx) = Expansion log(lp+qx+pqx*'). Expand both sides of this of Sines and equation in powers of x, and equate the coefficients of Cosines of X", we then get Multiple t., j % n _ 4J j 2 i (/>+<?)" W+.

Sines and Cosines of Arc.

If we write this series in the reverse order, we have when n is odd. If in these three formulae we put p = e^, g = e-< 9 , we obtain the following series for cos nO :

2 cos n8 = (2 cos e)"-n(2 cos e)"- i + n ^"7 (2 cos e)-<- . .

when n is any positive integer; + . ..+(-i)2 2 - 1 cos"9 when n is an even positive integer ; cos ^- . ..+(-l) 2 2 -l cos "9 when n is odd. If in the same three formulae we put 2= e-' e , we obtain the following four formulae:

(9)

(- 1)22 cos n9 = (2 sin 9)"- (2 sin ( i) 2 2 sin n0 = the same series (n odd); - ' cos 8, = I . sm 6 9 + .. . +2"~ l sin "0 (n even) ; sin n9 = re .3! ~ 5! +(-i)Ta lrt m(odd).

Next consider the identity (12) (13)

I - qx I - (p + q)x + pqx 1 ' Expand both sides of this equation in powers of x, and equate the coefficients of x"~ l , then we obtain the equation "- l -(n-2)(p+q)- 3 pq If, as before, we write this in the reverse order, we have the series (-if' [n (t2) to? -2^ (to) ' te) 3 B - 3 pq) ^ + ... + (- ^(p + fl ) > when n is even, and when n is odd. If we put p = e i9 , q e ie , we obtain the formulae sinnfl = sin0 j (2 cose)"- 1 - (n -2) (2 cos 9)"" 3 + ^" ~ 3 ffi ~^ (2 cos9)"~ 5 where n is any positive integer ; / (-1)

( 2 -2 2 ) . , n(*-2 2 )(n 2 -4 2 ). -- s - - 3 s - - 1 - z6 -i) 2 (2 cos 0)-> (n even); + (-i) r ~(2 cos 0)"- 1 (n odd). (16)

If we put in the same three formulae p = e lfl , q=e~ t9 , we obtain the series -2 2 L sm ^ n n 2! ; (17)

( i) 2 cos n0 = the same series (n odd); , cos 9 n sin (19)

(20)

. . . + ( i) 2 (2 sin 0)"~' f ( even); cos n0 = cos j i j sin 2 0+- ^1 ^sin 4 . . .

+ (2 sin0)-' |(nodd).

We have thus obtained formulae for cos nO and sin nd both in ascending and in descending powers of cos and sin 0. Vieta obtained formulae 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 sectiones. Jacques Bernoulli found formulae equivalent to (12) and (13) (Mem. de I' Academie des Sciences, 1702), and transformed these series into a form equivalent to (10) and (n). Jean Bernoulli published in the Acta eruditorum for 1701, among other formulae already found by Vieta, one equivalent to (17). These formulae have been extended to cases in which n is fractional, negative or irrational; see a paper by D. F. Gregory in Camb. Math. Journ. vol. iv., in which the series for cos nO, sin 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 acta, vol. ix., by Lagrange in his Calcul des fonctions (1806), and by Poinsot in Recherches sur I' analyse des sections angulaires (1825).

The general definition of Napierian logarithms is that, if then x+iy = \og (a+ib). Now we know that os y+ie* sin y; hence ex cos y-a, e" sin y =b ' or * = ("+&*)*- y = arc tan 6/o'*r, where m 'is an integer. If 6 = p, then m must be even or odd according as a is positive or negative ; hence log. 0+iJ) =log. (o j + &)%+ i (arc tan b/a2mr) or log. (a+ii) =log. (o 2 +6 2 )i+ i (arc tan b/a^2n+r), according as a is positive or negative. Thus the logarithm of any complex or real quantity is a multiple-valued function, the differ- H b lie ence Detween successive values being 2iri; in particular, yP e ' <f the most general form of the logarithm of a real positive quantity is obtained by adding positive or nega- tive multiples of 2iri to the arithmetical logarithm. On this subject, see De Morgan's Trigonometry and Double Algebra, ch. iv., and a paper by Professor Cayley in vol. ii. of Proc. London Math. Soc.

25. We have from the definitions given in 21, cos iy = $(ey+e-y) and sin iy = \i(ey e-y). The expressions, \(ey+e-y), \(ey e-y) are said to define the hyperbolic cosine and sine of y and are written cosh y, sinh y; thus cosh y = cos iy, sinh y= i sin iy. 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 connected with the circle. We may easily show from the definitions that cos*(x+iy)+siri'(x-\-iy) = I, cosh 2 y sinh 2 y = I ; cos(x+iy) =cos * cosh yi sin x sinh y, sin(x+ty) =sin x cosh y+i cos x sinh y, cosh(o + /3) =cosh o cosh /3+sinh o sinh 0, sinh (a + 0) =sirih o cosh j3+cosh o sinh jl These formulae are the basis of a complete hyperbolic trigonometry. The connexion of these functions with the hyperbola was first pointed out by Lambert.

26. If we equate the coefficients of n on both sides of equation (13), this process requiring, however, a justification of its validity, we get must lie between the values may also be written in the form to flowers |T. This equation of Id Sine.

when x lies between By equating the coefficients of n 2 on both sides of equation (12) we get o 2 3-5 3 3-5-7 which may also be written in the form (22)

(arc sin x)* = 3. ,- 2> . ...,.,, 3 2 3-5 3 3-5-7 4 when x is between =*= I . Differentiating this equation with regard to x, we get arc sin x 3" '3-5 '35-7 if we put arc sin * = arc tan y, this equation becomes arc tan y = T j j i+- jTT^H "4 ( jT 2) +( (23)

This equation was given with two proofs by Euler in the Nova acta for 1793.

It can be shown that if mod x< I, then for any such real or complex value of x, a value of log. (i+*) is given by the sum of the series x 1 * 2 /2 +* 3 /3 ...

We then have 1 \ a ii^ = v-(- J- 4- 4- Gregory's 2 6 I x 357 Series.

put iy for x, the left side then becomes zjlog (i+oO log (i iy)| or i arc tan y =*= mis ; 5 ,,7 2-4- 3 ' 5 7 + The series is convergent if y lies between i ; if we suppose arc tan y restricted to values between Jir, we have hence arc tan ynir=y arc tan y=y (24)

which is Gregory's series.

Various series derived from (24) have been employed to calculate the value of ir. 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 = ir/6, Sertes * Br y = l/V3 in (24). The calculation is to be found in Calculation Sherwin's Mathematical Tables (1742). About the same ' time J. Machin employed the series obtained from the equation 4 arc tan J arc tan ,,fa = Jir to calculate ir to 100 decimal places. Long afterwards Euler employed the series obtained from Jir = arc tan 3+ arc tan J, which, however, gives less rapidlyconverging series (Introd., Anal, infin. vol. i.). T. F. de Lagny employed the formula arc tan i/V3=ir/6 to calculate ir to 127 places; the result was communicated to the Paris Academy in 1719. G. Vega calculated ir to 140 decimal places by means of the series obtained from the equation Jir = 5 arc tan $+2 arc tan y 3 . The formula Jir = arc tan |+arc tan t+arc tan i was used by J. M. Z. Dase to calculate irto 200 decimal places. W. Rutherford used the equation ir = 4 arc tan J arc tan , J + arc tan 5"^.

If in (23) we put y = J and $, we have ir = 8 arc tan 3+4 arc tan ^ =2-4 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 the identity T = 2o arc tan $ +8 arc tan / 9 . The calculation of ir has been carried out to 707 places of decimals ; see Proc. Roy. Soc. vols. xxi. and xxii.; also CIRCLE.

27. We shall now obtain expressions for sin x and cos x as infinite products of rational factors. We have x . x+ir . x . x+ir Factorlza- sin x = 2 sin rsm = 2 3 sin-sin - lion of Sloe aodCoslae. sJn X+2 *sin X+3r - 4 ' 4 proceeding continually in this way with each factor, we obtain , . X . 3C-T-T . X + 2TT . X + nIlT sin x = 2"~ l sin -sin sin - . . .sin , where n is any positive integral power of 2. Now . x+rir . x+n rir . x+rir . rirx . .fir . ,x sin - sin - = sm - sin - = sm 2 -- sin 2 -, n n n n n n and sin Hence the above may be written , . x / . IT . x\ I . 2ir . . x\ sin x = 2 n sm-lsirr - sin 2 - I Ism 2 -- sm 2 rl ... n \ n n/ \ n n/ (. , for . , x\ x sm 2 --sm 2 -Jcos-, where k = Jn = i. Let x be indefinitely small, then we have 2**~^ TT 27T klT i =---81^ -sin 2 -...sin'-; hence . x x f sin 2 x/n\ / sin 2 */ \ / sin 2 x/n \ sin * = n s,n- cos - (i -ftrffc) (i -.. 2lr/n j . - . (i -3 We may write this . x x I sin 2 x/n\ / sin 2 x/n \ _ sin z = n sm - cos - ^i-gy^ . . (i -^ mr/n ) R, where R denotes the product (sin 2 x/n \ I _ sin 8 x/n \ f ^ sin 2 x/n \ "sin 2 m + iT/n/ V~sin 2 m+2K/nJ ' ' \ I ~sin 2 k*/n} ' 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 sin m*]n .

n sin x/n cos xjn is unity, we have . sin x , , . is and that of The modulus of R i is less than V~*~sin 2 m-r-iT/n/ \ I+ sin 2 m+2*/n) '" V+sin 2 kit In) "'' where /> = mod. sin x/n. Now e P i >i+Ap 1 , if A is positive; hence mod. (R i) is less than exp. jp^cosec 2 m + iv/n+ ... + cosec 2 kv/n) i, or than exp. ip 2 2 |i/(7n-|-i) 2 + . - + !/#) i, or than exp. (p 2 n 2 /4m 2 ) i. Now p 2 = sin 2 a/n.cosh 2 jfJ/n+cos 2 a/n. sinh 2 0/n, if x = a+ifl; or p 2 = sin 2 o/n+sinh 2 0/n. Hence lim B=01 P 2 n 2 = o 2 +/S 2 , lim B = <>> pn = mod. x. It follows that moc K=< (R i) is between o and exp. {(mod. 4) 2 / xnf ) i , and the latter may be made arbitrarily small by taking m large enough. It has now been shown that sin x = x(i A^/?r 2 )(i * 2 /2V) ... (i xP/mtir 2 ) (i+m), where mod. e m decreases indefinitely as m is increased indefinitely. When m is indefinitely increased this becomes This has been shown to hold for any real or complex value of x. The expression for cos x in factors may be found in a similar manner , ., . T 2x $ir2x by means of the equation cos x = 2 sin cos " , or may be deduced thus cos x = (26)

If we change x into ix, we have the formulae for sinh x, cosh * as infinite products / r 2 \ " = " / h+iJT-*)- cosh x = p I 1 n=0 V In the formula for sin * as an infinite product put * = lir, we , 7T I T 7 ^ 5 then get ' - J 2 ^> 4 4 6 ' ' " we stop a '' ter 2n " actors m tne numerator and denominator, we obtain the approximate equation I= 2 2 2 .4 2 .6 2 ... (2n) 2 ( 2n + I )

2.4.6. . . 2tl :

or 1 , , 2ni = * nir ' where n is a large integer. This expression was obtained in a quite different manner by Wallis ( Arithmetica infinitorum, vol. i. of Opp.). 28. We have Series for Cot, Cosec, Tan and Sec.

or cos y+sin y cot x Equating the coefficients of the first power of y on both sides we obtain the series From this we may deduce a corresponding series for cosec x, for, since cosec x = cot %x cot *, we obtain 1 -5=B+- i i i '~ - By resolving into factors we should obtain in a similar manner the series 2 2 2 2 2 +...,(29)

IT 2X JT + 2X ' 3ir 2X 3x4-2* ' SIT 2X 5X+2X These four formulae may also be derived from the product formulae for sin x and cos x by taking logarithms and then differentiating. Glaisher has proved them by resolving the expressions for cos x/sin x and I /sin x ... as products into partial fractions (see Quart. Journ. Math., vol. xvii.). The series for cot * may also be obtained by a continued use of the equation cot * = J|cot %x+ cot i(*+x) ) (see a paper by Dr Schroter in Schlomilch's Zeitschrift, vol. xiii.).

Various series for x may be derived from the series (27), (28), (29), (30), and from the series obtained by differentiating them one or more times. For example, in the formulae (27)

and (28), by putting x = ir/n we get f Series for Tr = ntan-ii- I - * i * f ^derived n( n I^n + l 2n l^2n + l' ' ' V fromSerles r _.x( I I I i ) for Cot and n sin i i -{- - .. i . ~ r I i . ( ; Cosec.

ni If we put n=3, these become 2 \ ' 2 4 By differentiating (27) we get put* These series, among others, were given by Glaisher (Quart. Journ. Math. vol. xii.).

/ x*\ / x ! \ 29. We have sinh rx = irxP 1 1 +r) , cosh irx = P(i+, -J-iVV if we differentiate these formulae after taking logarithms we obtain the series Certain Series.

These series were given by Kummer (in Crelle's Journ. vol. xvii.)

The sum of the more general series 1 2 n+x 2 n + 22n + x *>+.f+ x 2 n + . . . , has been found by Glaisher (Proc. Lond. Math. Soc., vol. vii.)

If U m denotes the sum of the series ;+;+T+ . . ., V m that of the series rs+rs+-r^+..., and W m that of the series * o o Sums of -s-T^+rs-rs-l-..., we obtain by taking loga- Powers of * * 3 ' Reciprocals rithms in the formulae (25) and (26)

ot Natural / x \> , I ... /x\ I ., /x\ .

Numbers. log (x cosec x) = t/ 2 ^-j +- [7 4 (^ - j + - Z7, ^-j + . . . , , , T . /2X\ 2 I /2X\ * . I T , /2X\ log (sec x) = V, (-) +-V< (-) +-F 6 (-)+...; and differentiating these series we get i i Ui U t In (31) x must lie between =*=* and in (32) between equation (30) in the form (31)

(32) = iir. Write sec ~-r i "* I and expand each term of this series in powers of x 2 , then we get 'IT 7T 7T^ where x must lie between 1^. 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 Ui, U t ... F 2 , V t ... and Wi, W 8 When U a has been found, V may be obtained from the formula For Lord Brounker's series of *-, see CIRCLE. It can be got at once Continued b V putting = 1,6=3, =5.... in Euler's Factors /or IT.

+b-a+c-b+" Sylvester gave (Phil. Mag., 1869) the continued fraction which is equivalent to Wallis's formula for jr. This fraction was originally given by Euler (Comm. Acod. Petropol. vol. xi.) ; it is also given by Stern (in Crelle's Journ. vol. x.).

30. It may be shown by means of a transformation of the series . sin x , xx 2 x 2 x 3 Continued for cos x and 5 that tan x = :;- = -^ -^-^ -7^... Fractions , , r n T .

for Tiigono- This mav "* a ' so eas y shown as follows. Let metrical y = cos V*, and let y', y*... denote the differential Functions, coefficients of y with regard to x, then by forming these we can show that $xy"-\-2y'+y = o, and thence by Leibnitz's theorem we have Therefore ,= - 2 - hence zVx cot Vx= 2 Replacing Vx by x we have tan x=-j-^ - -r-^-. . .

Euler gave the continued fraction n tan x (n 1 I ) tan'.r (n 2 4) tan 2 * (n 2 9) tan'x tciri 71 .v ' . . . j *5 ~~ / ^ this was published in Mem. de I'ocod. de St Petersb. vol. vi. Glaisher has remarked (Mess, of Math. vols. iv.) that this may be derived by forming the differential equation (i x 2)j<m-M) ( 2m _|_ i) x y*i+l) -J- ( n * _ fH 2 )jlC") = O, where y = cos (n arc cos x), then replacing x by cos x, and proceeding as in the former case. If we put n =o, this becomes tan'x 7 + whence we have X X? 4.X^ Q K^ fJ^X^ arctan *= f+ J+ 5+ F-"+5HF7T-" 31. It is possible to make the investigation of the properties of the simple circular functions rest on a purely analytical basis other than _ ! the one indicated in 22. The sine of x would be Analytical defined as a function such that, if x= I -77 - ^-, Treatment J 0> U - JTJ _ tan x tan'x 4 tfln'x = i + 3+ 5+ Treatment of Circular then Functions.

s ; n x . tne quantity * would be defined to } 2 /I fa i (. _ 5Y- We should then have n d * \~ x= \ V (i -f\' Now change the variable in the integral to z, where y 2 +z 2 = i, we then have | x = j z . / y _ z2 y and z must be defined as the cosine of x, and is thus equal to sin (iir x), satisfying the equation sin 2 x+cos 2 x = l. Next consider the differential equation dy dz This is equivalent to hence the integral is yV(i z 2 )+zV(i >*)= a constant. The constant will be equal to the value u of y when z = o; whence yV(i z 2 )+zV(i y 2 )=.

The integral may also be obtained in the form j v(i-y 2 ) = ~' loge ' "^ ~ and sin -y = sin a cos /3+cos a sin /3, cos 7 = cos a cos ft sin a sin ft, the addition theorems. By means of the addition theorems and the values sin iir = i, cos JT=O we can prove that sin (^ir+x) = cos x, cos (|TT-|-X) = sin x; and thence, by another use of the addition theorems, that sin (TT+X) = sin x cos (ir+x) = cos x, from which the periodicity of the functions sin x, cos x follows:

We have also J 'V V.' whence log e | V(i y 2 ) + iy) + log, j \'(i z 2 ) + iz| = a constant. Therefore j V(i - y 2 )| + ij{V(l - z 2 )-hz) = V(i - 2 ) + '. since =y when z = o; whence we have the equation (cos a + i sin a) (cos /3 + i sin /3) = cos (a + ft) + t sin (o + ft), from which De Moivre's theorem follows.

REFERENCES. Further information will be found in Hobson's Plane Trigonometry, and in Chrystal's Algebra, vol. ii. For further information on the history of the subject, see Braunmuhl's Vorlesungen iiber Geschichte der Trigonometrie (Leipzig, 1960). (E. W. H.)

Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)

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