# 1911 Encyclopædia Britannica/Trigonometry

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​TRIGONOMETRY (from Gr. τρίγωνον, a triangle, μέτρον, 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 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
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 π＝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 1/2α＝√{1719(3438−cosα)} 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 Sūrya-Siddhānta, 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°＝890′, sin 7° 30′＝449′, sin 3° 45′＝225′ were thus obtained. Now the angle 3° 45′ is itself 225′; thus the arc and the sine of 1/96 of the circumference were found to be the same, and consequently special importance was attached to this arc, which was called the right sine. From the tables of sines of angles at intervals of 3° 45′ the law expressed by the equation
sin (n + 1.225′) − sin (n.225′)＝sin (n.225′) − sin (n−1 . 225′)−sin(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° 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 1°＝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 um, 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 φ from the equation sin φ/cos φ＝k, and he also made a table of the length of shadows of a vertical object of height 12 for altitudes 1°, 2°, . . . of the sun; this is a sort of cotangent table. He was acquainted not only with the triangle formulae in the Almagest, but also with the formula cos a＝cos b cos c + sin b sin c cos A for a spherical triangle ABC. Abū’l-Wafā 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 Yūnos 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 Vest Arabs, Geber (q.v.), who lived ​at Seville in the 11th century, wrote an astronomy in nine books, which was translated into Latiri 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 Müller (1436–1476), known as Regiomontanus, was a pupil of Purbach and taught astronomy at Padua; he wrote an exposition of the Almagest, and a more important work, De triangulis planis et sphericis cum tabulis sinuum, which was published in 1533, a later edition appearing in 1561. He reinvented the tangent and calculated a table of tangents for each degree, but did not make any practical applications of this table, and did not use formulae involving the tangent. His work was the first complete European treatise on trigonometry, and contains a number of interesting problems; but his methods were in some respects behind those of the Arabs. Copernicus (1473–1543) gave the first simple demonstration of the fundamental formula of spherical trigonometry; the Trigonometria Copernici was published by Rheticus in 1542. George Joachim (1514–1576), known as Rheticus, wrote Opus palatinum de triangulis (see Tables, Mathematical), which contains tables of sines, tangents and secants of arcs at intervals of 10″ from 0° to 90°. His method of calculation depends upon the formulae which give sin na and cos na. in terms of the sines and cosines of (𝑛−1)α and (𝑛−2)α; 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. François Viète or Vieta (1540–1603) employed the equation (2 cos 1/2φ)$^3$−3(2 cos 1/2φ)＝2 cos φ to solve the cubic 𝑥$^3$−3𝑎$^2$𝑥＝𝑎$^2$𝑏(𝑎>1/2𝑏); 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.
Vieta gave 𝑦＝2 sin 1/45φ, where 𝑐＝2 sin φ, 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 (1590–1634), published at the Hague in 1626, contains the theorems which give areas of spherical triangles and polygons, and applications of the properties of the supplementary triangles to the reduction of the number of different cases in the solution of spherical triangles. He used the notation sin, tan, sec for the sine. tangent and secant of an arc. In the second half of the 17th century the theory of infinite series was developed by John Wallis, Gregory, Mercator, and afterwards by Newton and Leibnitz. In the Analysis per aequationes numero terminorum infinitas, which was written before 1669, Newton gave the series for the arc in powers of its sine; from this he obtained the series for the sine and cosine in powers of the arc; but these series were given in such a form that the law of the formation of the coefficients was hidden. James Gregory discovered in 1670 the series for the arc in powers of the tangent and for the tangent and secant in powers of the arc. The first of these series was also discovered independently by Leibnitz in 1673, and published without proof in the Acta eruditorum for 1682. The series for the sine in powers of the arc he published in 1693; this he obtained by differentiation of a series with undetermined coefficients.
In the 18th century the science began to take a more analytical form; evidence of this is given in the works of Kresa in 1720 and Mayer in 1727. Friedrich Wilhelm v. Oppel’s Analysis triangulorum (1746) was the first complete work on analytical trigonometry. None of these mathematicians used the notation sin, cos, tan, which is the more surprising in the case of Oppel, since Leonhard 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 √−1; he published in 1712 the general formula for tan 𝑛φ in terms of tan φ, which he obtained by means of transformation of the arc into imaginary logarithms. The greatest advance was, however, made by Euler, who brought the science in all essential respects into the state in which it is at present. He introduced the present notation into general use, whereas until his time the trigonometrical functions had been, except by Girard, indicated by special letters, and had been regarded as certain straight lines the absolute lengths of which depended on the radius of the circle in which they were drawn. Euler’s great improvement consisted in his regarding the sine, cosine, &c., as functions of the angle only, thereby giving to equations connecting these functions a purely analytical interpretation, instead of a geometrical one as heretofore. The exponential values of the sine and cosine, De Moivre’s theorem, and a great number of other analytical properties of the trigonometrical functions, are due to Euler, most of whose writings are to be found in the Memoirs of the St Petersburg Academy.
Plane Trigonometry.
* Fig. 1.
1. 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 positiveConception of Angles of any Magnitude. 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 degree is divided into sixty equal parts called minutes; and the minute into sixty equal parts called seconds;Numerical Measurement of Angular Magnitudes. 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 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—(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. If 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 π. We shall indicate later on some of the methods which have been employed to approximate to the value of this number. Its value to 20 places is 3·14159265358979323846; its reciprocal to the same number of places is 0·31830988618379067153. In circular measure every angle is measured by the ratio which it bears to the unit angle. Two right angles are measured by the number π, and, since the same angle is 180°, we see that the number of degrees in an angle of circular measure θ is obtained from the formula 180×θ/π. The value of the radian has been found to 41 places of decimals by Glaisher (Proc. London Math. Soc. vol. iv.); the value of 1/π, from which the unit can easily be calculated, is given to 140 places of decimals in Grunerts Archiv (1841), vol. i. 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.
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 Sm of is convenient to have a simple mode of indicating in portions of which direction on the straight line the length AB has an Infinite been described; this may be done by supposing that a Straight point moving in one specified direction 15 describing Lina a positive length, and when moving in the opposite direction a negative length. Thus, if a point moving from A to B is moving in the positive direction, we consider the length AB as positive; and, since a point moving from B to A is moving in the negative direction, we consider the length BA as negative. Hence any portion of an infinite straight line is considered to be positive or negative according to the direction in which we suppose this portion to be described by a moving point; which direction is the positive one is, of course, a matter of convention.
If perpendiculars AL, BM be drawn from two points, A, B on any straight line, not necessarily in the same plane with AB, the 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 each other projection of BA being ML has the oppositeProjections of Straight Lines on each other.sign to the projection of AB. If two points A, B be joined by a number of lines in any manner, the algebraical sum of the projections of all these lines is LM-that is, the same as the projection of AB. Hence the sum of the projections of all the sides, taken in order, of any closed polygon, not necessarily plane, on any straight line, is zero. This principle of projections we shall apply below to obtain some of the most important propositions in trigonometry.
4. Let us now return to the conception of the generation of an angle as in fig. 1. Draw BOB′ at right angles to and equal to AA′. We shall suppose that the direction from A′ to A is the 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 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 A 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 α are denoted by sin α, cos α, tan α, cot α, sec α, cosec α 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 α. 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 π+α 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 α 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 2nπ+α, 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 2nπ+α is positive if α is between 0 and π and negative if α is between π and 2π, and the cosine of the same angle is positive if α is between 0 and 1/2π or 3/2π and 2π and negative if α is between 1/2π and 3/2π
Fig. 2.
In fig. 2 the angle POA is α, the angle P‴OA is -o., P'OA is π−α., P″A is π+α, POB is évr-a. By observing the signs of the projections we see that
sin(−α) = −sin α, sin(π−α), sin α, sin (π+α)-= −sin α, cos(−α) =cos α, cos(1'r−α)= −cos α, cos(1r+α) = −cos α, sin(1/2π−α) =cos (1, cos(π'l""(1)= sin α. Also sin(1/2π+α) =sin(1r-Qvr−α) = sin(é1r−α)= cos α, cos(1/2π+α) = −cos(π−1/2π−α) = −cos(1/2π−α) = −sin α.
From these equations we have tan(-a)=-tan a, tan(1r-a)= ~tan (l, t21fl(7f+U.)=*ft1l] a, tan(%1r-o.)=C0t a, tan (%'ll"j“U.>= -cot a, with corresponding equations for the cotangent.
The only angles for which the rejection of OP on B′B is the same as for the given angle POA (=α) are the two sets of angles bounded by OP, OA and OP', OA; these angles are 2nπ+α. and 2nπ+(π−α), and are all included in the formula nr-l-(-1)'a, where 1 is any integer; this therefore is the formula for all angles having the same sine as α. The only angles which have the sante cosine as a are those bounded by OA, OP and OA, OP′”, and these are all included in the formula 2nπ±α. Similarly it can be shown that nπ±α includes all the angles which have the same tangent as α.
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$^2$α+cos$^2$α=1, and from this by the help of the definitions of the other functions we deduce the relations 1 + tan$^2$α = sec$^2$α, I + cot$^2$α = cosec$^2$a. 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, 1/2π, π, 3/2π, 2π, and the signs of the functions of angles between these values; I denotes numerical increase and D numerical decrease:—
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 I,
π/12 15° /6;/2 V6-L-f'2 750 i§§ 1r
π/10 18° 'ir-1 v"50+2vs ...Q 2.
π/6 30° 2 .7 60 371
π/5 36° — . 5 ° 2.
π/4 45° V? Y 2 45° 3 1
4 cosine sine 4 p
These are obtained as follows. (1) 1/4π. The sine and cosine of this angle are equal to one another, since sin in-=cos (évr-iw); and since the sum of the squares of the sine and cosine is unity each is 1/√2. (2) 1/4π and 1/3π. 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 1/2 or cos 1/2π=1/2, and from this the sine is found. (3) π/10, π/5, 2π/5, 3π/10. In the triangle constructed in Euc. iv. 10 each angle at the base is 2/5π, and the vertical angle is 1/5π. If a be a side and b the base, we have by the construction a(a−b)=b$^2$; hence 2b=a (1/5-I); the sine of -/r/Io is b/za or § (/5-1), and cos gr is a/2b=§ (/5-{-1) (4) fyr, ,521r. Consider a right-angled triangle, having an angle § -ir. Bisect this angle, then the opposite side is cut by the bisector in the ratio of /3 to 2; hence the length of the smaller segment is to that of the whole in the ratio of x/3 to x/3-1-2, therefore tan,131r={w/3/(/3-l-2)} tan évr or tan 1/12π=2−√3, and from this we can obtain sin, lsr and cos #svn
Fig. 3.
5. Draw a straight line OD making any angle A with a fixed straight line OA, and draw OF making D an angle B with OD, this 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 ofFormulae for Sine and Cosine of Sum and Difference of Two Angles. 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 + -}-ir with OD, and is therefore equal to OF sin B; hence
OF cos (A -l-B) =-'OE cos A-I-EF cos (%1r+A) =OF (cos A cos B-sin A. sin B),
or cos (A +B) ==cos A cos B-sin A sin B. The angles A, B are absolutely unrestricted in magnitude, and thus this formula is perfectly general. We may change the sign of B, thus cos (A -B)=cos A cos (~B)-sin A sin (-B), or cos (A -B) =cos A cos B»|-sin A sin B. If we projected the sides of the triangle OEF on a straight line making an angle-I-if with OA we should obtain the formulae sin (A iB) =sin A cos Bicos A sin B, which are really contained in the cosine formula, since we may put %'ll""'B for B. The formulae
tanA=f=tanB cotAeotB=»=I
im (A *B> immma '=°' <A *Bl =@a§ 7r are immediately deducible from the above formulae. The equations HC-it-D) cos %(C-D),
§ (C-D) cos'§ (C-l-D),
sin C-{-sin D=2 sin
sin C-sin D=2 sin
cos D-l-cos C=2 cos § (C-l-D) cos é(C-D), cos D-cos C=2 sin %(C-l-D) sin § (C-D), may be obtained directly by the method of projections. Take two equal straight lines OC, OD, making angles C, D, with OA, and draw OE perpendicular to CD. The angle which OE makes c with OA is HC-I-D) and that which E DC makes is'1r C D'the anle
if -l' 'l' ly S
COE is %(C~D). The sum of the pro-
jections 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 C4-cos D=2 cos § (C+D)
cos %(C-D). The difference of the
0 A projections of OD and OC an OA
is equal to twice that of ED, hence we have the formula cos D-cos C
D The other two formulae will
FIG. 4.
=2 sin %(C-l-D) sin %(C-).
be obtained by projecting on a straight line inclined at an angle j“%7l' to OA.
As another example of the use of projections, we will find the sum of the series cos a.-I-cos (a-|-H)-l-cos (a+2;3)+, . ~}-cos (a-l-n-IB). S of § uppose an unclosed polygon each angle of which 52:6 of is -r-B to be inscribed in a circle, and let A, A, , A2, cosines In A3, . . ., A., be n~}-I consecutive angular points; Amhmeucal let be the diameter of the circle; and suppose a progression straight line drawn making an angle a. w1th AA1, then a-l-B, a-l"2B, . are the angles it makes with A1 A3, /12, A3, ...; wtiave by projections AA" cos (a-I-inf 1B)=AA1{cos a+ cos a+B+...+cosa+(n-1)B}, also ' AA1=D sin id, AA, ,=D sin %7lB; hence the sum of the series of cosines is cos (a.~l-én-I 13) sin $1113 cosec é/3. By a double application of the addition formulae we may obtain the formulae
sin (A, -i-A24-A3) =sin A1 cos A2 cos A3 F°""""" f°' ~l-cos A1 sin A3 cos A34-cos A, cos A3 sin A3 Slneand -A -A -A
Cosmeof -sm, sin gsm 3,
Sum of cos (Al-l-A3-l-A3) =cos A3 cos A3 cos A3 Angles -cos A1 sin A3 sin A3-sin A1 cos A2 sin A3 -sin A1 sin A3 cos A3.
We can by induction extend these formulae to the case of n angles. Assume sin (A1+A2+ . . +A, ,)=S1-S3+Sf, -, . . cos (A1+A3+ . . +A, ,)=S3-S2+S4- . . . where S, denotes the sum of the products of the sines of r of the angles and the cosines of the remaining 11-1 angles; then we have Sill . . . *l'An'l'A, ,+1)=Cf)S An+1(S1"S3"j'S5" . . -l-S111 An+1(S0-'S2“l"S4'- .
The rght-hand side of this equation may be written (S, cos A, .,1+.S'., sin A, ,+1) - (S3 cos A, ,+1-{-S3 sin A, .+1)+ . ., Of SQ-S'3+ .
where SC denotes the quantity which corresponds for n-H angles to Sf 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
F°"'""'<"° cos 2A =cos” A -sin2A =2 cos' A - I = I -2 sin” A, for Multiple
t A
Jndsub' sin 2A =2 sin A cos A, tan 2A =»-1-%-, Mumple ' *tan
Angles. sm 3 4 =3 sin A -4 sin* A, cos 3A =4 cos3 A -3 cos A, sin nA = n cos"'1 A sin A - cos”'° A sin' A + . . +<- nf"—-l" ' bl, jg, ff' "”) cw-'=f-1 A Sin2f+1 A, n- 1 .
cos nA =cos"A -MET) cos"" A sin' A + . . +f) Cosa-ar A Sinn A+ 5
may all be deduced from the addition formulae by making the angles all equal. From the last two formulae we obtain by division A ntanA - ¢an2A+...+<-1>f ¢an2f+1A+.., tan n f ' ' -
1 -7%-f-Qtan2A-l-...+(~1>f >¢an2fA+... 3 tan A -tan" A
1 -3 tan2 A
The values of sin é/1, cos § A, tan %A are given in terms of cos A by the formulae
-, A t 1 A 3
sm;/1=<-1>@ 1-¥-, ¢0s», eA=<-na iE'f-, In the particular case of n=3 we have tan 3A = - A
where P is the integral part of A/2-/r, q the integral part of A/21r--ir, and r the integral part of A /1r.
Sin %A, cos 5A are given“in terms of sin A by the formulae 2 sin 35A = (-1)l>'(I -l-sin A)%+(-I)q'(I —sin A)l, 2 cos %A = (-1)i>'(1--sin AH-(-1)'1'(I -sin A)l, Xfliere ji' is the integral part of A /27l"'j'% and q' the integral part of zur-~.
6. Iniany plane triangle ABC we will denote the lengths of the sides BC, CA, AB by a, b, c respectively, and the angles BAC, ABC, A CB by A, B, C respectively. The fact that the projections of b and 1: on a straight line perpendicular to the P"°p"“" side a are equal to one another is expressed by the equa- of Triangles 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-l-c cos B expresses the fact that the side a is eqlual to the sum of the projections of the sides b and 6 on itself; t us we obtain the equations
a=bcos C+c cos B*
b=ccosA-1-acos Cc=acosB-|-bcosA
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 122-|-of-a2=2bc cos A or cos A =(b2-{-cl-112)/2bc, which gives the cosine of an angle in terms of the sides. From this expression for cos A the formulae
Sin %A= (S-bgés-c) E, COS %%1= 5(sZ;a) g é, . tan;/1 = Sggfgi i, Sin A =, %;§ <s-a><s-bm-¢>}*, where s denotes Ha-l-b-l-6), 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+1'C, rA +pB+qC, QA -l-VB-1-pC for A, B, C respectively, where p, q, 1' are any quantities such that p-l-q-I-r-l-I is a positive or negative multiple of 6, provided that we change the signs of all the sines. Suppose p-I-q-l-r-l-I =6n, then the sum of the three angles 2mr - (pA +qB+rC),2n1r - (rA +pB +qC),21z1r - (gA -I-1B-I-pC) is 1r; and, since the given relation follows from the condition A+B-l-C =-fr, we may substitute for A, B, C respectively any angles of which the sum is 1r; thus the transformation is admissible. B. 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 sidjes of the triangle ABC are respectively a cos A, b cos B, c cos C,1r-2A,1r-2B, '/r-2C; we may therefore substitute these expressions for a, b, 6, 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, -L cos C cos 2C and the angles are 4A -1r, 4B~jvr, 4C—/r; We may therefore substitute these expressions for the sides and angles of the original triangle; for example, we obtain thus the formula COS A a2 cos” A cos” 2A -122 cos' B cos? 2B-02 cosz C cos' 2C 4 2bc cos B cos C cos 2B cos 2C
This transformation obviously admits of further exten-SIOU- Solution of
(I) The three sides of a triangle ABC being given, Triangles. the angles can be determined by the formula L tan éA = I0-j-§ log (s-b)+§ log (s-c)-é log s-§ log (s»a) and two corresponding formulae for the other angles. ​cos c=cos a cos b-l-sin a sin b cos C
(2) The two sides a, b and the included angle C being given, the angles A, B can be determined from the formulae A -1-B = 7F - C, 4
L tall § (A -B) =log (a-b)-log (a+b) + L cot %C, and the side 1: is then obtained from the formula log c=log a-l-L sin C-L 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 Lsin B=L sin A-Hog b-log a;
this gives two supplementary values of the angle B, if b sinA < a. If b sinA > athere 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 b > a, 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=1r-A -B and the side b from the formula log b= log a-Q-L sin B-L sin A. The area of a trian le is half the roduct of g P
f; ";'Z 'Es a side into the perpendicular from the opposite agdoiadrpangle on that side; thus we obtain the expressions Metals ébc sin A, {s(s-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 af Mathematics for X885 by M. Baker.
Let a, b, 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 S of the quadrilateral in terms of the sides and the angle a.
We have 2S=ad sin A -l-bc sin(2a.-A) and $(a2-l-dz-b'-c ) =ad cos A -bc cos (aa.-A); hence 4S2+§ (a2-l-d2-b”-c2)'l=a2d'¢+b2c”-zabcd cos za. If 25 = a + b-l-c +d, the value of S may be written in the form S= {s(s-a)(s-b)(s-c)(s-d) -abcd cos2a}é. Let R denote the radius of the circumscribed circle, 1' of the inscribed, and ri, 72, ra of the escribed circles of a triangle Rad” °f CW" ABC; the values of these radii are given by the follow°""'5°"”'°d» ing formulae:-
”S;';"°db d R=abc/4S=a/2 sin A,
""' SC” ° r=S/s=(s-a)tan 5A =4R sin § A sin %B sin %C € m;;l';f'" r1=S/(5-a)=s tan § A=4R sin § A cos $B cos § C. Spherical T 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, Definition AB denote these arcs, the circular measure of the “spherical angles subtended by these arcs respectively at the Tris” le centre of the sphere are the sides a, b, c of the spherical g ° 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, Crespectively are the angles A, B, C of the spherical triangle. It is not necessary to consider triangles in which a side is greater than -/r, since we may replace such a side by the remaining arc of the great circle to Associated which it belongs. Since two great circles intersect »n Lmg, es each other 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 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 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 vx hich 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 1r-/l, 1r-B, 1r-C, and the angles vr-a,
-/r-b, -/r-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 0 Bangle and each angle into the supplement of the opposite side.
L 8. Let O be the centre of the sphere Q on which is the spherical triangle FIC” 5 ABC. Draw AL perpendicular to OC and AM perpendicular to the plane
A
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 plane OBC, this gives angles. V 5:21121 OA cos c =OL cos a-I-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 b+sin a sin b cos C. Sides am; We may obtain similar formulae by interchanging the Angles, letters a, b, c, thuscos
a=cos b cos c-I-sin b sin c cos A
cos b=cos c cos a-l-sin c sin a cos B (I) 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 -l-sin C sin A cos b (2) cos C= -cos A cos B-i-sin A sin B cos 6 In the figures we have AM=AL sin Ci=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 A]lI= r sin c sin B,
therefore sin B sin a=sin C sin b. v By interchanging the sides we have the equation sin A sin B sin C
afiarsrz; =a;7 -k (S)
we shall find below a symmetrical form for k. If we eliminate cos b between the first two formulae of (I) we have cos a sin2c=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 -f-cos c cos B =sin B cot A +cos c cos B.
We thus have the six equations
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 -I-cos c cos A cot c sin b=cot C sin A -I-cos b cos'A (4) cot c sin a=cot C sin B-1-cos a cos B cot a sin c=cot A sin B-{-cos c cos B . When C=§ 1r formula (1) gives
cos c=cos a cos b (a)
a11d (3) gives sin b =sin B sin c ( sin a=sin A sin c B)
from (4) we get tan a =tan A sin, b=tan c cos B tan b =tan B sin a =tan c cos A (7) The formulae cos c =cot A cot B (e) and cos A -cos A sin B
cos B=C0S b sin A ff)
follow at once from (a), (B),
used for the solution of
(v). These are the formulae which are 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 A 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 FIG. 6.
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 Formula bisectors of the angles A, B, C, and if DE be drawn- farSlue perpendicular to BC, it may be shown that BE and Coslue = § (a +C - b) and CE = § (a + b - C), and that 01'H-Hlf the angles BDE, ADC are supplementary. We have Anil”sinc sinADB sinb sinADC '-2 lA
also sinBD ' sin § A ' sin CD 'sin § A ' therefore Sm 2
    Sm BD Sm Sm. CDE Sm BDE. But sin BD sin BDE=sin BE
sin b sin c
=sin § (a-l-c-b), and sin CD sin CDE=sin CE-=sin § (a+b-c); A ' l -b ' 1 b- 1 s
therefore SIUE: 2 2 g (5)
Apply this formula to the associated triangle of which -r-A, 1r-B, C are the angles and ar-a, 1r-b, c are the sides; we obtain A jsin l(b-l-cf-a) sin 1(a+b-l-c)) the formula cos; - I- Sin b Siuzc) 2 (6) ​ By division we have
A sin $(a+c-b) sin %(a+b-c) fi
tan; S sin § (b-|-c-a) sin § (a-I-b-l-c) (7) and by multiplication
sinA =2{sin (a-~b+c) sin § (b+c-a)sin § (c-l-a-b) sin § >(a+b-c)}§ sin b sin C=lI —cosfa-coszb-coszc +2 cos a cos b cos c}5 sin b sin 0. Hence the quantity k in (3) is
{1 -cos'a-cos' b-COS2C+2 cos a cos b cos c}l/sin a sin b sin c. (8) 0fHalf- Apply the polar triangle transformation to the formulae sides. (5), (6), (7) (82'a.nd we obtaiin B C a A
cos '(A-|- B) cos (+
COS2 i 2 sin B sin cf i (9)
a -cos l(B-l-C-A) cos l(A+B~|-C lf
S'“5= 2 sin B Sin cz 5 (10)
a -cos %(B-l-C-A) cos § (A +B+C i H) WE" COS;<A+c-B) ws;<A +B~c
lfk' ={1 -cos?/I -cos2B-cos2C-2cosA cosBcosC}%/sinA sinBsinC, we have. k/€'=I (12)
10. Let E be the middle point of AB; draw ED at right angles to D ' AB to meet AC in D; then DE
Del b, bisects the angle ADB.
C Fm;ZI;:s Let CF bisect the angle
° DCB and draw FG perpendicular
to BC, then
CG=%(a-b, L FBE=§ (A--B),
LFCG=9O°-%C.
From the triangle CFG we have
cos CFG=cos CG sin FCG, and
from the triangle FEB cos EFB=
A Bcos EB sin FBE. Now the angles
3 CFG, EFB are each supplementary
FIG. 7. to the angle DFB, therefore cos§ (a - b)cos%C = siné (A -I-B)cos%c. (13) Also sin CG=sin CF sin CFG and sin EB=sin BF sin EFB; therefore sin§ (a-b)cos§ C=sin%(A -B)sin§ c. (14) Apply the formulae (13), (14) to the associated triangle of which a, 1r-b, 1r-c, A, 1r-B, vr-C are the sides and angles, we then have
sin§ (a+b)sin%C==cos§ (A -B)sin-$0 (15) cos%(a+b)sin§ ;C=cos§ (A -{-B)cos§ c. (16) The four formulae (13), (14), (15) (16) werehrst given by Delambre in the Connaissance des Temps for 1808. Formulae equivalent to these were given by Mollweide in Zach's Monatliche Correspondenz 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 S lug f to logarithmic calculation. The solutions will be unique T; 01:0 except in the two cases (1) where two sides and the angle aug 5° 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 ca obtuse. Then C and c are determined from the ses equations
Ambiguous
1 . 1
"ffl-bl ~(A-ltan
§ C= cot %(A -B), tan § tan § (a-b). I
Now tan § C, tan ic, 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 at; 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 § sin a and the two parts A, a are both acute or both obtuse, these bgng 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 bsin A>sin a. Thirdly, if sin b=sin a then B=A or -/r=A. If a is acute, 11-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=%1r, there is no solution unless A =%-/r, 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 bgsin a was given by Professor Lloyd Tanner (Messenger of Math., vol. xiv.). 14. If 1 is the angular radius of the small circle inscribed in the triangle ABC, we have at once tan 1=tan 5A sin (J-2s=a.+b+c; from this we can derive the formulae a), where
tan 1=1L cosec s=§ N sec § A sec %B sec %C= Refill “f sin a sin § B sin § C sec 'EA (21) cimles where 11, N denote the expressions ';7"t°'} t° rang es.
{sinssin (s-a) sin (s-b) sin (s-c)}%, {-cos S cos (S-A) cos (S-B) cos (S-C)}§ . Formulae. 2
for November 1808. They were also given by Gauss (Theoria moms, 1809), and are usually called after him. 11. From the same figure we have
Napiefs tan FG=tan FCG sin CG=tan FBG sin BG; Analogies. therefore cot4;Csin§ (a-b)tan§ (A -B)sin%, (a-Hz), 1 b
or tan§ (A-B) = cotéC. (17)
Apply this formulae to the associated triangle (vr-a, b, 1r-c, -/r-A, B, 1I'“C), and we have
¥ b
cot %(A -l-B) = § C,
1 b
or fan;(A -1-B) = ac. (is)
If we a l these formulae (17), (18) to the polar triangle, we have PP Y
1 = 1
tan, (a b) Sin é(/1+B)tan, c (19)
L .
nm l(A+B)=9 @tan lc. (20)
2 cos § (A -1-B) 2
The formulae (17), (18), (19), (20) are called Napier's “ Analogies ”; they were given in the Mirzf. logar. canonis de.rc1iptio. 12. If we use the values of sin $11, sin éb, sin éc, cos éa, cos éb, cos § c, given by (2), (10) and the analogous formulae obtained by interc anging the letters we obtain by multiplication 5°h"'°'“"°'3 sin éa cos éb sin C=sin lc cos § (B-l-C-A) The escribed circles are the small circles inscribed in three of the associated triangles; thus, applying the above formulae to the triangle (a,1r-b, -ir-c, A, 1r-B,1r-C), we have for 11, the radius of the escribed circle opposite to the angle A, the following formulae tan 1, =tan § A sin s=n cosec (s-a) =§ Nsec 5/l cosec § B cosec § C =sin a cos § B cos %C sec § A. (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 Rl, RQ, R the radii of the circles circumscribing the associated triangles, we have by writing évr-R for 1, in-R1 for 11, 1r-a for A, &c., in the above formulae cot R=cot be cos (S-A) =§ n cosec éa cosec ib cosec § c=-N sec S =sin A cos ib cos sc cosec ia (23)
cot R1= - cot ia cos S=%1'l cosec sa sec $11 sec éc=N sec (S-A) =sin A sin éb sin éc cosec éa. (24) The following relations follow from the formulae just given:- 2tanR =cot 11-l-cot 12-I-cot 13-cot 1, 2tanR1 =cot 1 +cot 124-cot 13-cot 11, tan r tan 11 tan 12 tan 13 =n2, sinz .S =cot 1 tan 11 tan 12 tan 13, sin2 (s-U.) =tan r cot 1, tan 12 tan 13. 15. If E =A +B'l'C*1F, it may be shown that E cos éa cosébsin C=coséccos§ (A -l-B-C) . sin ia sin éb sin C=cos iccos § (A +B~}-C) These formulae were given by Schmiesser in C1elle's Joum., vol. x. The relation sin b sin c-l-cos b cos c cos A=sin B sin Coos B cos C cos a was given by Cagnoli in his Trigonomelry (1786), ca Mrs and was rediscovered by Cayley (Phil. Mag., 1859) Ffluhm It follows from (1), (2) and (3) thus: the right-han ° side of the equation equals sin B sin C -1-cos a (cos Asin B sin C cos a) =sin B sin C sin* a-l-cos a cos A, and this is equal to sin b sin c -l-cos A (cos a-sin b sin c cos A) or sin b sin c + cos b cos c cos A.
ti
multiplied by the square of the radius is the area of £°'m"l'” the triangle. Ne give some of the more important Srherkal the quantity E, which is called the' E§ cess We have
cos %'(A -i- B): cos § (a + b) ard sin § (A + B) = cos *Ha - b), sin 5C cos ie ' cos éC cos éc
sin § (C - E) Cos é(a il- b) cos § (C - E) cos § (a - b) or sin %C = T cos és and cos § C " cos éc he sin § C - sin § (C - = cos $6 - cos § (a -1- b) nee sm °}C + sm %(C - E) cos éc + cos § (a + b); ​ therefore $\qquad {\frac {\tan {\tfrac {1}{4}}E}{\tan ^{2}{\tfrac {1}{4}}(C-E)}}=\tan {\tfrac {1}{2}}s\tan {\tfrac {1}{2}}(s-c).$
Similarly $\quad \tan {\tfrac {1}{4}}E\tan ^{2}{\tfrac {1}{4}}(C-E)=\tan {\tfrac {1}{2}}(s-a)\tan(s-b);$
therefore
tan § E={tan és tan § (s-a) tan § (s-b) tan § (s-c)}§ (25)
This formula was given by J. Lhuilier.
cos 1 a+b
Also sin 2C cos 2E-cos 2C sin § E= sin %C; 2
cos Ha - 11)
cos %C cos § E+sin 2C sin %E= -#qi cos éC; CO5 QC
whence, solving for cos § E, we get
I cos a cos b-l-cos c
COS %E= ~%—f (26)
4 cos ia cos 2b cos it
This formula was given by Euler (Nova acra, vol. x.). If we find sin 2E from this formula, we obtain after reduction iE= l.. .l,
Sm 2 2 cos ia cos éb cos és
a formula given by Lexell (Acta Petrop., 1782). From the equations (2I), (22), (23), (24) we obtain the following formulae for the spherical excess zsin22E =tan R cot R1 cot R2 cot R1
4(cot r1+cot r2+cot ra)
(cot r-cot r1-I-cot 12-I-cot ra) (cot r+cot r1-cot 12-}-cot r3)X (cot r+cot 11-1-cot r2-{~cot 13).
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 -l-cos a+cos b-l-cos c
hence cos lE=cos MN sec la.
4 cos éb cos in ' 2 2
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 pzéssing thropgh the middle points of the sides of the trian le are e es is sin 2 .
16. Let ABCD be a spherical quadrilateral inscribed in a small circle; let a, b, c, d denote the sides AB, BC, CD, DA respectively, and x, y the diagonals AC, BD. 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) Properties of Spherical Quadrilateral inscribed in Small Circle. for the radii of the circles circumscribing the triangles BAD, BCD, we have
$\sin A\cos {\tfrac {1}{2}}a\cos {\tfrac {1}{2}}d{\text{ cosec}}{\tfrac {1}{2}}y=\sin C\cos {\tfrac {1}{2}}b\cos {\tfrac {1}{2}}C{\text{ cosec}}{\tfrac {1}{2}}y$;
Whence ${\frac {\sin A}{\cos {\tfrac {1}{2}}b\cos {\tfrac {1}{2}}c}}={\frac {\sin C}{\cos {\tfrac {1}{2}}a\cos {\tfrac {1}{2}}d}}$
This is the proposition corresponding to the relation A+C= for a plane quadrilateral. Also we obtain in a similar manner the theorem
sin éx sin %y
sin B cos lb sin A cos 'dy
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 éa, 2 sin éb, 2 sin éc, 2 sin id respectively, and the plane quadrilateral formed b 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 ix sin %y=sin éa sin és-l-sin éb sin id.
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 11-1-cos b+cos c+cos d=4 cos éx cos éy cos iz.
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 CreIle's ]ourn, vol. xlii.)
Analytical Trigonometry.
17. 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 analysis from the fact that they are the simplest singly periodic functions, and are therefore adapted to the representation of undulating Periodicity of Functions.magnitude. The sine, cosine, secant and cosecant have the single real period 2π; 'i e. each is unaltered in value by the addition of 2π to the variable. The tangent and cotangent have the period π. 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 x+ιy(ι=√−1). Suppose we multiply together, by the rules of ordinary algebra, two such numbers we haveConnexion with Theory of Complex Quantaties.
$(x_{1}+\iota y_{1})(x_{2}+\iota y_{2})=(x_{1}x_{2}-y_{1}y_{2})+\iota (x_{1}y_{2}+x_{2}y_{1}).$
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 $x_{1}=r_{1}\cos \theta _{1},y_{1}=r_{1}\sin \theta _{1},x_{2}=r_{2}\cos \theta _{2},y_{2}=r_{2}sin\theta _{2},$ the above equations becomes
$r_{1}(\cos \theta _{1}+\iota \sin \theta _{1})\times r_{2}(\cos \theta _{2}+\iota \sin \theta _{2})=r_{1}r_{2}(\cos {\overline {\theta _{1}+\theta _{2}}}+\iota \sin {\overline {\theta _{1}+\theta _{2}}}).$
We may now, in accordance with the usual mode of representing complex numbers, give a geometrical interpretation of the meaning of this equation. Let P1 be the point whose co-ordinates referred P to rectangular axes Ox, Oy are x1, y1; then the point P1 is employed to represent the number x1+ιy1. In this mode or representation real numbers are measured along the axis of x and imaginary ones along the axis of y, additions J, being performed according to the parallelogram law. The. points lg A, A1 represent the numbers ±1, the points a, a1 the numbers ±ι Let P2 represent the expression x2+Ly2 and P the expression (x1+ Ly1)(x2+Ly2). The quantities r1, 01, r2, 02 are the polar coordinates of P1 and P2 respectively, referred to O as origin and Ox as initial line; the above equation shows that r1 fg and are the polar co-ordinates of P; hence OA: OP1:: OP2r OP and the angle POP2 is equal to the angle P$_1$OA. Thus we have the following geometrical construction for the determination of the point P. On OP2, draw a triangle similar to the triangle OAP$_1$ so that the sides OP2, OP are homologous to the sides OA, OP1, and so that the angle POP2 is positive; then the vertex P re resents the product of the numbers represented by P1, P2. lip x2-E-Ly2 were to be divided by x1+Ly1 the triangle OP'P2 would be drawn on the negative side of P2, similar to the triangle OAP1 and having the sides OP', OP2 homologous to OA, OP1, and'P' would represent the quotient.
18. If we extend the above to n complex numbers by continual repetition of a similar operation, we have—
$(cos\theta _{1}+\iota sin\theta _{1})$ (cos 02 + L sin 02) (eos 0,1 + L sin 02) gi =cos(01= 02 +. + e, .> + L Sin <01 +02 + +o, ,) -De Moivre’s Theorem.
If 01=02=. .=021=01, this equation becomes (cos 0-I-L sin 0)" =cos 110-I-L sin 110; this shows that cos0 +L sin 0 is a value FIG. 8.
of (cos 110-l-L sin n0)§ . If now we change 0 into 0/n, we see that cos 0/n+L sin 0/n is a value of (cos 0-l-L sin 0)Ti; raising each of these quantities to any positive integral power m, cos m0/n-l-L sin m0/n is one value of (cos 0-|~L sin 0)Lfl. Also cos (- m0/n) + L sin (-m0/n) =;
hence the expression of the left-hand side is one value of (cos 0+ L sin 0)'”'/“ We have thus De l/1oivre's theorem that cos k0-I-L sin k0 is always one value of (cos 0-I-L 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+L sin 0)" denotes the limit of a sequence of corresponding values of (cos 0-l-L sin 0)", , where k1, k2. k .. is a sequence of rational numbers of which k is the limit, and further observe that as cos k0-l-L sin /20 is the limit of cos k,0+L 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 $(a+\iota b)^{m/n}$, where m and n are positive integers prime to each other. lf a=r cos 0, b=r sin 0, we require the values of r"'“" (cos 0+L sin 0)"'/" One value is immediately furnished by the theorem; but we observe that since theThe n Roots of a Complex Quantity. expression cos 0+¢ sin 0 is unaltered by adding any multiple of 21r to 0, the n/mth power of r""" (cos m.0+2s1r/n+L sin m.0+2s1r/n) is a-l-Lb, if s is any integer; hence this expression is one of the values required. Suppose that for two values 51 and 52 of s the values of this Expression are the same; then we must have m 0+2s11r/n-m.0+2'§ 1T/n; a multiple of 21r, or S1-52 must be a multiple of 11. Therefore, if we give s the values o, I, 2, .n- I successively, we shall get n different values of (a+Lb)"'"', and these will be repeated if we give s other values; hence all the values of ​ (a+¢b)'“/'* are obtained by giving s the values 0, 1, 2, ...nin the expression r"'/" (cos m.0 + 2s1r/n + L sin m .0 + 2S1I'/71), where r=(a2-I-b”)} and 0=arc tan b/a. We now return to the geometrical 'representation of the complex numbers. If the points B, , B2, B3, ...B, , represent the expression x+¢y. (x+»y)”. (x+¢y)“. B# . . (x-I-¢y)" respectively, the triangles OAB1, OBIB2, . . OB, , 1B, , are all similar. Let (x-+-Ly)"=a-l-vb, then the converse problem of finding the nth root of a~|-Lb 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., the second side of the nth. 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 B1 to B". The first solution is that in which the vertical angle of each triangle is B, ,OA/n; the second is that FIG in which each is (B, .0A +2-/r)/n, 9' in this case one complete revolution bein made round O the third has B OA r h g; <.. -l-41)/"fo r e 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 04-L sin 0)?s. Suppose B to represent the expression cos 0+ r. sin 0. If the angle AOP, is § 0, P1 represent the root cos § 0-Q-L sin § 0; the angle AOB is filled up by the angles of the three similar triangles AOP1, P1Op1, PIOB. Also, if P¢, P3 be such that the angles P10P¢ P10193 are § r, $77 respectively, the two sets of triangles AOP2, P20p., p30B and AOP3, PsOp2, pzOB satisfy the conditions of similarity and of having OA, OB for the bounding sides; thus P2, P, represent the roots cos § (6+21r)-I-L sin § (0+21r), cos § (6-{-41;-) +L sin § (0+41r) 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 circ e. The problem of determining the values of the nth roots of unity is equivalent to the geometrical problem of inscribing a regular The nth polygon of n sides in a circle. Gauss has'shown in his Roots of Dzsquzsztzones amthmetzcae that this can always be done Unity by the compass and ruler only when n is a prime of the form 2f'+I The determination of the nth root of Q B 1; 1% R 0 A P. fa FIG. ro. V 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 sition to factorize an expression of F, ¢, -¢0|-|, , g- the form x"-(Ii10+ib). Usin the values which we tions. have obtained above for (a +tb§ I /", we have s= -1 3 x”-(a+Lb)=P n [x-1“(cos¥&r+isin@)]- (1) s=0 If b =o, a=1, this becomes ="'1 2s1r 2s1r x”-I=1;0 x-cos?-rsins=i" 1 2S1r . 2S7r =(x-I)(x-|-I)P 1 x-cos-7T=»=tsmT = s=%n-1 251. =(x-I)(x+I)P x2-2x cos-I-+I (n even). (2) 5=I =l(n-1) x"- 1 =(x-1)€' 1 x2-2x cos?%'+I (n odd). (3) If in (I) we put a= vi, b=o, and therefore 0=1r, we have =n~1 T-' °"" x*-{»-1:12 x c0S LSin =0 n n =i(-2) =i;' 0 n x'-2x cos 23-1%-r+I (n even). (4) s=i(n~3) x~+i =(x+1>P0 x2-2x cos il;-¥'+r (n odd)- (5) = Also x2"-2x”y" cos n0-l-y2" = (x"-yn cos n6+¢ sin n0)(x"-y" cos n0-L sin nt?) ="'1' "i-T"""f"T;'; =P x-ycos-i-lr=¢=1.sin-:Qls=0 77' n "="“'1 2 2S1r (6) Airy and Adams have given proofs of this theorem which do not involve the use of the symbol 1, (see Cafnb. Phil. Trans., vol. xi). A large number of interesting theorems may be derived from De Moivre's theorem and the factorizations which we have ==P x -2xycos0-—-y2 . s=0 M deduced from it; we shall notice one of them. Example if In equation (6) put y=I/x, take logarithms, and then?;:;?;:;1re 3 differentiate each side with respect to x, and we get 2n x2"'1-x'2"'1) § =""1 2(x-x'3) x2"-2 cos n0-1-x“2" s:0 x2-2 cos 6+-774-x" Put x2 =a/b, then we have the expression n(a2n b2n> (112 - b2l(a2"— 2a"b" cos n0+b2 ) for the sum of the series s=n- 1 I Z, 5:0 a2-2ab cos 0-|-%;1-r-I-b2 20. Denoting the complex number x-I-iy by z, let us consider the series 1+z+z2/2!+...+z"/nl+. . This series uniformly and absolutely for all values of z whose converges moduli do not exceed an arbitrarily chosen positive Theejfgl number R. Consequently the function E(z), defined QZZEZ as the limiting sum of the above series, is continuous in every finite domain. The two series representing E(z1) and E(z2), when multiplied together give the series represented by E(z1+z1). In accordance'with a known theorem, since the series for E(z1) E(z2) are absolutely convergent, we have E(z1)><E(z»1) =E(z, +z2). From this fundamental relation, we deduce at once that {E(z)1" =E(nz), where n is any positive integer. The number E(1), , the sum of the convergent series 1 +1-1-1/2!+1/gl., ., is usually denoted by e; its value can be shown to be 2-718281828459 .... It is known to be a transcendental number, Le. it cannotbe the root of any algebraical equation with rational coefficients; this was 'first established by Hermite. Writing z=I, we have E(n)=e", where n is a positive integer. If z has as a value a ositive fraction p/q, we find that{E(/5/g)}q=E(p) =e"; hence E(p/gg is the real positive value of ef'/'1. gain E(-11/g)><E(p/9) =E(0)=I, hence E(-p/g) is the real positive value of e'1'/41. It has been thus shown that for any real and rational number x, the value of E(x) is the principal value of ez. This result can be extended to irrational values of x, if we assume that e' 'is for such a value of x defined as the limit of the sequence e", e”, , where xl, x2, . is a sequence of rational numbers of wluph x is the limit, since E(x1), E(x2) . ., then converges to E x Next consider (I -1-z/m)'", where m is a positive integer. We have by the binomial theorem, m 2 on ~+Z+<~.f.>a+~+<1~f:.> <1~.a>~. s-I za z '" “M H+ "tial 1 2 S-I Also . . (I-W » lies between 1 and 1+ 5-ki-}- -}-5%;, hence the product equals I-085.5-I/2711 where 0, is 0 < 0, < I We have now such that m 2 (14%) =r-|-2+ (1-i) E-, +. . + 1-o, § 77l I m “ +V i-0, ,, i-1%= I-}-z-l-z2/2!-i- +28/s!+R, ,, where Zs+l Zm Z2 2 R, =E;- *Z-Y!-l- I-lf-03?-l-. Zs-2 Zm-2 -l-Hsgjm-|'~. . . +0m . Since the series for E(z) converges, 5 can be fixed so that for all values of m >s the modulus of z*+1/(s+I)!+. +z'"/ml is less than an arbitrarily chosen number és. Also the modulus of I-|-Gaz/I+. -l'9mZ""'2/(1')1"2)l is less than that of I-i-'I |z| /I! + [z[”/25 -l- , or of e'11°d=, hence mod R.<%e-l-(I/zm). mod (z”e=)<f, if m be chosen sufficiently great. It follows that 11mm-, ,°(I +z/m)"'=E(z), 'where z is any complex number. To evaluate E(z), write H-x/m=p cos ¢, y/m=p sin 41, then ​E(z)=lim$_{m=∞}$ρ$^m$(cos m¢ +i sin m4>)l, by De Moivre's theorem. SiI'lC€ p"'= (I m§ I+ im, W8 have llITlm...mp'” =e'. 1im, ,. .¢, § i-4-m '/mix/vm Jim. Let r be a fixed number X . y? im
ess than x/m-l-x//rn, then lim, ,, ,, , 2 I+m('/m+x'/m 2 hes 2
between I and lim, ,., , I 4% g my or between I and 51/2 /212; hence since r can be taken arbitrarily large, the limit is I. The limit of m¢ or m tan'1{y/'(x+m)} is the same as that of my/(x+m) which is y. Hence we have shown that E(z) =e'(cos y-I-i sin y).
21. Since E(x-I-iy)=1§ '(cos y-i-sin y, we have cos y-I-i sin y =E(iy), and cos y-i sin y=E(-iy). Therefore cos y=%{E(iy) -l-E(-iy)}, sin y=§ i{E(iy)-E(-iy)l¥ and using ExP°"°"“"I the series defined by E('iy) and E(-fy), we find that V"'f""s °f cos y = I - y”/2! + y*/4! - ..., sin y = y -y“/3! T"'g";;'°; -i-y5/5! »- ., where y is any real number. These fetal” are the well-known expansions of cos y, sin y in powers "U 'ms' of the circular measure y. Where 2: is a complex number, the symbol e' may be defined to be such that its principal value is E(z); thus the principal values of ef#/, e"f/ are E(iy), E(-iy) The above expressions for cos y, sin y may then be written cos y=§ (ef"+e“f1'), sin y=§ 'i(ef1/-e"f”). These are known as the exponential values of the cosine and sine. It can be shown that the symbol ezas 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 eos y=§ {E(z) + E(-rz) }, sin 2 = (§){E(z) -E(-2) }, where E(z) represents the sum-function of 1 -Jr 2-l- z2, '2! -|-, + z"/nl + For real values of z this is in accordance with the ordinary definitions, as appears from the series obtained above for cos y, sm y. The fundamental properties of cos z, sin z can be deduced from this definition. Thus sin z = E(z), cos z -i sin z = E(-iz); therefore cos'zj+-sin2z=E(iz) E(-iz) =1. Again cos (z, -l-z2) is given by élE(12. + iz2) + E(- iz, -122) } =% {E(i21)E (wi) + E(7 121) E (-qz2)} Of ilE(1Z1)'l' E('“1Z1)}lE(¢Z2)'i'E<"1Z2)l'l'i E(¢Z1)"E(-'LZ1)l {E(zz2)-E(-122)}, whence we have cos (21-+122 = cos zl cos zzsin zl sin 22. Similarly, we find that sin (zl-4-z2)=sin z, cos 22+ cos z, sin zz. Again the equation E(z) = I has no real roots except
    =o, for e">1, if z is real and >o. Also E(z)=1 has no complex
root o.-HB, for o.-id would then also be a root, and E(2a) = E(a.+i8)E(a-iB) =I, which is impossible unless m=0. The roots of E(z) = I are therefore purely imaginary (except z=0); the smallest numerically we denote by 2 1l1r, so that E<2'iT)=I~ We have then E(2'i1rr)={E(2i'n')}'=I, if r is any integer; therefore 2i1rr is a root. It can be shown that no root lies between 2i1rr and 2(r+I)i1r; and thus that all the roots are given by z= =f=2i1rr. Since E'-l-217-rr)=E(z)E(2i1r)=E(z), we see that E(z), is periodic, of period 2'i7I'. It follows that cos z, sin z are periodic, of periods 21r 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 cos z+i
(LlTb)" = U"+nd" b+7E£%?l2 a"'”*b"+ . . . (n I) (+I)
W * n'-7 n r n
    nd gcijlnes + rl a by+' ' ' +I)
i" Serie' °f Putting a =eL9, b =e'|0, we obtain Sines and
cosines of (2 cos 6)"=2 cos m9-i-nz cos n-20 fgtlple +"-(Z I) 2 cos rl-{-. . +71—1—(n I)';'!(n r+I)2cos(n-2r)6+.. ”( I) ff"-l-3)
When n is odd the last term is eos 0, 2 ' ¢
and when n is even it is
5 .
If we put a=e, b= -e'»9, we obtain the formula (- i)%»(: sin0)"=2 cosn0-2ncos (n-2)0-l-'fl-IL? 2 cos (11-4)0- . +(- 1)"* 2 cos(n-2r)0.
+< I>gn<n~»>é#e»+1>
when n is even, and " A
(— 1)&( >(2 sin 0)" = 2sin m9-n 2 sin(n-2)0-|-$22 sin(n -4)0... +'3?n(n-1). %(n-Q-3)
~' 9
5(1)-I)! ”'“
when n is odd. These formulae enable us to express any positive integral power of the sine or cosine in terms of sines or cosines of multiples of the argument. There are corresponding formulae when n is not a positive integer.
Consider the identity log(1-px)+log(1-qx)= log(I -p+qx-l-pgxz). Expand both sides of this equation in powers of x, and equate the coefficients of x$^n$, we then get
12"-l-Q"' = (1>+q)"- n(1>+9)"'2l>q of (-3), , s
4% (P 'l'9) '1'292+- ° ° C2;'f:i;n:f +< I)'n(n-r-1)(n-VX2).. (n-21-1-1) Am (P“|'9)"'2'P'9'+- — If
we write this series in the reverse order, we have fl Il
i>'~+q"=2<- of <1>q>3-§ '§ <z>q>2"1(Q“i)2 2 2 2
+f'¥;! 3 (1>q>2
2 2 2 'L L1
f "f @(pq)2 3 G+. +(- 1)2 § (p+g)» when n is even, and
YL-I 1; i;
- 'E 1
+ (Pq) 2 "+. ..+<-1>"2;<1>+q>when n is odd. If in these three formulae we put P =e°9, q =e-'19, we obtain the following series for cos 110:- 2 eos n0= (2 eos 0)"-n(2 cos 9)” 2+ (2 cos 0)"'4-. +(I), n(n-r- 1)(n-152). (n-2r+I)(2 Cos 0>n 2' +' Q 1(7)
when n is any positive integer; 2 2 2 2 2(-I)2cos
u0= I-300520-i-n (n4, 22) c0s'0-fl (712 22 (nz 42) c0559 fl
+. +(-1)§ 2"“1 cos "0 (8)
when n is an even positive integer; n-1
(- I)T cos n0=n cos n6-n( ;, 12) cos"0+"(n2 12,02 32)cos56-1 +(-1)%" 2 "rl cos "0
(9)
when n is odd. If in the same three formulae we put p=e19, g= - e-19, we obtain the following four formulae:- fl
(- I)§ 2 eos n0= (2 sin 0)"-n(2 sin 0)"”'2-I-@ (2 sin 0)"'4n(n-r-1) (n 2r 1)
+(- I)' (2 Sin 0)"'”'+. . (1L even); (10) -1
(- I)%2 sin n6=the same series (n odd); (11) 2 'Z
cos nB;= I - sin?0 + sin40 - sin“0 +. . +2"“1 sin "6 (n even); (12) sin 116 = n sin 0 - n( 23T 12) sin30 'l' nw- - I;)in2 32) sin59 - . 1
+< o'%2»-1 Simo (n Odd). (13)
Next consider the identity f
<1 =, P q
I Px 1~Qx I”'(P+!l)x+PQx2
Expand both sides of this equation in powers of x, .and equate the coefficients of x"", then we obtain the equation
p$^n$-gi = (1>+a>"-'- (11 - 2) <1> +q>'~-3 pa (-)(-)
(1>+q)""'1>'¢1'- - .<-1>f( )
<”"Q2> "'( ' 2” <p+q>"“"-1p'gf+ If, as before, we write this in the reverse order, we have the series < [»(fL;1) M* (ey) 3<pq>f”
+n<n2 22)y(nz 42
~ .95 If-
r—g *'l(f;g)5(1>Q)' +-..+(-I)z (1>+q)"" ​
when n is even, and
Q '11, ,, , if
-1
<~1> 2 (0402 (902
n~5 n~1
+ '<1>q>T+. . . +<-1>T£1>+q>"-1 when n is odd. If we put p=e's, g=e-'6, we obtain the formulae sin 710 =sin0 (2 cos0)"'1 - (n-2) (2 cos 0)"T“+ (2 cos0)"““ +(-1)' " '-I)( r;T' ' '( '2')<2 cos o>~~=~-1+ . (14) where n is any positive integer; I 2- z 2 c 2 2 2
(-1)Z 1sinn0=sin0 ncos0-n(n3,2>cos30+n(n 23101 4)cos5-. "
+(-If 1(2 cos 0)"" g (n even); (15) n-I
(-1) 2 sin n0=sin 6 1 -Q; cm20+£ cos“9- . . 1-1
+(-1)2 (2 cos 0)"" (rr odd). (16)
If we put in the same three formulae p=e'9, q= -e“'9, we obtain the series
n~2
(-1) 2 sinn0=cos6 sin"'10- (n-2)sin""'0-l- (n”3;$n)sin““50~... +(l)r S§ n»4~10+ (n even); (17)
n-l
(-1) 2 cos n0=the same series (rt odd); (18) 2 z
sin n0=cos 0 n sin 0- sin30+ s1n'6+ 2-1
+(-1)' (2 sin 0)"“1 (n even); (19) cos nl? =cos 0 I -£§ ¥sin'0+ sin'0-. . -l-(2 sin 9)"'1 (11 odd). (20)
We have thus obtained formulae for cos rc0 and sin 119 both in ascending and in descending powers of cos 0 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 T heoremala ad angular es section es. lacques Bernoulli found formulae equivalent to (12) and (13) (Mém. de l' Académie des Sciences, 1702), and transformed these series into a form equivalent to (10) and (11). lean Bernoulli published in the Acta erudilorum 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. Joum. vol. iv., in which the series for cos rt0, sin 118 in ascending powers of cos 0 and sin 0 are extended to the case of a fractional value of n. These series have been considered b Euler in a memoir in the Nova acta, vol. ix., by Lagrange in his Calcul des functions (1806), and by Poinsot in Recherches sur Vanalyse des sections angulaires (1825).
24. The general definition of Napierian logarithms is that, if crfo'=a+¢b, then x+iy=log (a-l-tbl). Now we know that cx+'>'=exc0s y-l-tex sin y; ence ex cos y=a, ex sin y Zzeggfés =b, or e¢=(a2+b2)%, y=arc tan b/a=+=rn1r, where rn 3 'is an integer. If b=o, then rn must be even or odd according as a is positive or negative; hence log, (a-l-tb) =log, (a'+b2)%-l- L (arc tan b/aizrzvr) or log, (a-Hb) =log, (a'+b”)%-lf L (arc tan b/a=h2n+1r), according as a is positive or negative. Thus the logarithm of any complex or real quantity is a multiple-valued function, the difference between successive values being 21|'L; in particular, gzperbflk the most general form of the logarithm of a real positive me{°"° quantity is obtained by adding positive or negary five multiples of 2-m. to the arithmetical logarithm. On th1s subject, see De Morgan's Trigonometry artzl Double Algebra, ch. iv., and a paper by Professor Cayley in vol. 11. of Proc. London Malh. Soc.
25. We have from the definitions given in § 21, cos Ly-'= Hey-l-e-y) and sin ly=§ 1(ey-e-Lv). Theexpressions, § (ey-l-e-Lv), Hey-e-y) are said to define the hyperbolic cosine and sine of yand are written cosh y, sinh y; thus cosh y=cos ty, sinh y= -L sin ty. lhe 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
cos2(x-I-ty) -l-sin2(x-Hy) = 1,
cosh' y-sinh? y = I;
cos(x-|~iy) =cos x cosh y-i sin x sinh y, sin(x-l»-Ly) =sin x cosh y-l-1. cos x sinh y, cosh(a. -l-3) =C0sh a. cosh B-l-sinh a. sinh (3, sinh(a-l-3) =sinh a cosh, B-l-cosh a. sinh H. 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 rt on both sides of equation (13), this process requiring, however, a justification of its validity, we get I sinff 0 I 3 sins 0 I 3 S sin" 0 V Expansion = ' i. 4. . ofnnAngle
6 s1n0-l-2 3 -l-2 4 5 -I-2 4 6 7 -l-..., (2I)]npowers 0 must lie between the values is-ir. This equation Off” 5111* may also be written in the form
a s .1
arc sin x=x-P; 3%-l-Léi x
2 4 6
when x lies between =*= I.
By equating the coefficients of rt' on both sides of equation (12) we get
622. 2 § sm40 QSIDSB 2.4.6 s1nB0 U Sm H+; 2 +3-5 3 +5-5-7 4) 'l' " (22) which may also be written in the form A ~ 2:2 EE iff? 2-4-65
(arcsinx) x-l-3 2-l-3 5 3-l-3 5 7 4+... when x is between =*=1. Ditferentiating this equation with regard to x, we get
ar .'n x
if we put arc sin x=arc tan y, this equation becomes L Emi L4 2
arctany I, ' y, 1-l-3 I+y2+3 5 l | y, +... gfhis equation was given with two proofs by Euler in the Nova acta or 1793.
It can be shown that if mod x< I, then for any such real or complex value of x, a value of log, (1 +x) is given by the sum of the series xl-x2/2 -l-x3/3-.
We then have
~ (23)
1 1+x x3 x5 x7 g' »
5'°€r:¢-”+3+g+';+~—» sliillf”
put ty for x, the left side then becomes éllog (1-l-ty)-log (1 -f.y)] 01' I. 8.I'C tan y=|=Ln7|";
s s 1
hence arc tan yin-1r=y-;:;¢+%-3-§ +. .. The series is convergent if y lies between i 1; if we suppose arc tan y restricted to values between =b}1r, we have yfl 3,5
arc tan y=y-3—l-E-..., (24)
which is Gregory's series.
Various series derived from (24) have been employed to calculate the value of rr. At the end of the I7th century -fr was calculated to 72 places of decimals by Abraham Sharp, by means of the series obtained by putting arc tan y=1r/6, Seff" f°" y=I//3 in (24). The calculation is to be found in C"I"'""“°" Sherwin's Mathematical Tables (1742). About the same °f" time ]. Machin employed the series obtained from the equation 4 arc tan § -arc tan 2§ ;, =i1r to calculate 'll' to 100 decimal places. Long afterwards Euler employed the series obtained from § 1r=arc tan %-l- arc tan é, which, however, gives less rapidly converging series (Introd., Anal. infirz. vol. i.). T. F. de Lagny employed the formula arc tan 1//3=1r/6 to calculate rr to 127 places; the result was communicated to the Paris Academy in 1719. G. Vega calculated vr to 140 decimal places by means of the series obtained from the equation %r=5 arc tan 2.-I-2 arc tan 30. The formula § 1r=arc tan é-I-arc tan § -l-arc tan § was used 'by ]. M. Z. Dase to calculate vrto 200 decimal places. W. Rutherford used the equation -r=4 arc tan é - arc tan 710 -le arc tan 519. If in (23) we put y=% and %, we have 1r=8 arc tan %+4arc tan1}=2-4 I+;-~é~|-gil; 2 2 2 4 2 2
+56 1+ . + (wo) + §
3 100 3. 5 ' ' ' »
a rapidly convergent series for rr which was first given by Hutton in Plril. Trans. for 1776, and afterwards by Euler in Nova acta for 1793. Euler gives an, equation deduced in the same manner from the identity 1r=20 arc tan ~}-l-8 arc tan sag. 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 Factorlza- sin x =2 sin;~csin3%'=2'* siniisinii lion of sm 4 4 andCoslne. ~ x'f“21f - x'l'3"f S111 T-SIHTQ proceeding continually in this way with each factor, we obtain x. x+1r. x+21r x+n-r1r in x==2"'1Sll'l 'Sl[1°*°'l - . .smi- n rr S n n n where n is any positive integral power of 2. Now x-l-r1r x+n-r1r x-I-r-/r . nr-x rvr . x in sm =sxn sm = in”—sinh S n n n n S n rt x-l-émr x nd sin-i=cosa n n
Hence the above may be written x . 1r . x 2-rr . x sm x=2"" sm; sm' 5-smz 5) <s1n' -5—sm' z . . in” kr in” x cosx s n s n .n, where k = én = I. Let x be indefinitely small, then we have I-2” lsin'7r 'n“ 2” in? kr - n nsi n .s n, hence n x n x COS x I sm' x/n X sin' x/rt I sm” x/n = ' - - . -w-* - . . sl Smn n smi 7Ff1l sm' 21r/n sm' Er/n
We may write this x x sin' x/n sin' x/n sm x=n smzcosz 1-<-T— I- R, sm rr/n ' ' ' sin* m1r'/n
where R denotes the product
sin” x/n > (L siwn) (I sin* x/n sin* m+11r/n sin* m+21r/n " ' SUI2 k"/11 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 sin x . sin xy R, then, since the limit of n Sin x/n COS x/n is? and that of Sin mt/n is unit we have m1r/n y sin x < (lim R. -T - 1-T2 I-2,72 I-WT, mfw The modulus of R-1 is less than 1+-1.=e;=—> -1 sin' m+I1r/n sinz m+2n/n ' ' ' S1112 IQ#/rl where p=mod. sin x/n. Now eAP2>I+Ap“, if A is positive; hence mod. (R-I) is less than exp. p2(cosec' rn-f-Ivr/11+ -lcosec kr/n)-I, or than exp. § p2n'{r/ m+1)2+. .. +I/k“}-I, or than exp. {p'n'/4m'}-I. Now p2=sin2 a./n.cosh2 B/n+cos2 a./n. sinh' B/n, if x=a+1;6; or p'=sin2 u./rt-l-sinh” /3/n. Hence lim, ,=, p'n'=a'-H82, hmnm, pn=mod. x. It follows that mod, ,=° (R-I) is between 0 and exp. {(mod 4)2/ xm”}-I, and the latter may be made arbitrarily small by taking m large enough. It has now been shown that sin x=x(r-x2/1r')(I-x2/2”1r2) (1 -:cz/m11r') (I -l-em), where mod. em decreases indefinitely as rn is increased indefinitely. When m is indefinitely increased this becomes
sin x=x<1-%) <1-é ...=x;l (1-7% ., (25)
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 by means of the equation cos x=2 sin #cos W, or may be deduced thus
P (ae) C0Sx=srr 2x= nit = I 4x“> (I 4x”> (I i1, x2> 2 sm x P (x :i) rr” 3272 5212 nh- ggas xg
=5=0 I-Z-J-2n+I), "2 (26)
If we change x into tx, we have the formulae for sinh x, cosh x as infinite products—
sinhx x=0 I+-174; coshx 5:0 1+(2n+I),1r,
In the formula for sin x as an infinite product put x=1/2π, we after 2n factors in the numerator and denominator, we obtain the approximate equation
then get I if we stop 2:e z| 2 Hz =§ ~<2"+'> or =Vnw, 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
I @.' Series for sin (x-ty) = (x+y)P (I + mr cor, cone, sinx xp <I+%F> ' gxrand or cos y-l-sin y cot x 3' 3/ 3' 3' 5 = (HT) (HW-l-rr) <I+x-rr) (I+x+21r> (HW-21f
Equating the coefficients of the first power of y on both sides we obtain the series
cot x=1/x +¢+¢+, $, +, i-, +. . (27)
From this we may deduce a corresponding series for cosec x, for, since cosec x=cot éx-cot x, we obtain
cosec x=1/x − 12-36-l-1r x-1r+x+21r+x-21r x+31r x—31r+'" (28)
By resolving S(-éégi into factors we should obtain in a similar manner the series 2 2 2 2 2 2
tan x:1r-2x 1r+2x+31r-2x 3-/r+2x+51r -2x'51r-l-2x +"" (29)
and thence
secx-tan<;-l-5) -tan x-T 2x+T+2x-3T 2x-Q-556+ .... (30)
These four formulae may also be derived from the product formulae for sin x and cos x by taking logarithms and then differentiating. Glaisher has proved them by resolving the expressions for cos x/sin x and I/sin x as products into partial fractions (see Quart. Journ. Math., vol. xvii.). The series for cot x may also be obtained by a continued use of the equation cot x=${cot %x+ cot %(x -li-1r)} (see a paper by Dr Schroter in Schl6milch's Zeitschrift, vo . xm.
Various series for π 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=1r/rt we get I I I I Serlesior = lr i .. i. fr derived T 'mann I n-1+n+1 211-I+2n+1"' ' fmmgenyeg 1f 1 1 1 1 iorCotand ~"'" Smn I+n-1 n+I 2n-1+2n+1"' ' C°"'° If we put n=3, these become - I I 1 I I 1r-3x/3 i-E+;-E+;-§ +... sa/2;;; 1 1 1r- 2 I-l-2 4 54-7-l-S...
By differentiating (27) we get
cosec$^2$ x=E+<x+1r>2+(x-1f>2+<x+21f>2+<x-21f>2+- - -1 putx=;:, and we get π=9{ 1-i-$2-I-%+$+}
These series, among others, were given by Glaisher (Quart. fourrr. Math. vol. xii.).
29. We have sinh 1rx=1rxP <1 +;$>, cosh 7l'x=P' I+; if we differentiate these formulae after taking logarithms we obtain the series
π/2x I I I 1 gigs, "
zx wth, rx 2x2 1'+x2+2'-I-x2+32-|-x2+' ' " ll' I I I 22 tanh ...
These series were given by Kummer (in Crelle’s Journ. vol. xvii.) The sum of the more general series + . . ., has been found by Glaisher (Proc. Land. Math. Soc., vol. vii.)
If U$^m$ denotes the sum of the series 1/1$^m$, +1/2$^m$+1/3$^m$+ ..., V$^m$ that ​of the series ${\frac {1}{1^{m}}}+{\frac {1}{3^{m}}}+{\frac {1}{5^{m}}}+\cdots$, and $W^{m}$ that of the series Sums of Powers of Reciprocals of Natural Numbers. ${\frac {1}{1^{m}}}-{\frac {1}{3^{m}}}+{\frac {1}{5^{m}}}-{\frac {1}{7^{m}}}+\cdots$, we obtain by taking logarithms in the formulae (25) and (26)
$\log(x{\text{ cosec }}x)=U_{2}\left({\frac {x}{\pi }}\right)^{2}+{\frac {1}{2}}U_{4}\left({\frac {x}{\pi }}\right)^{4}+{\frac {1}{3}}U_{6}\left({\frac {x}{\pi }}\right)^{6}+\cdots$
$\log(\sec x)=V_{2}\left({\frac {2x}{\pi }}\right)^{2}+{\frac {1}{2}}V_{4}\left({\frac {2x}{\pi }}\right)^{4}+{\frac {1}{3}}V_{6}\left({\frac {2x}{\pi }}\right)^{6}+\cdots$
and differentiating these series we get
${\frac {1}{2}}\cot x={\frac {1}{2x}}-{\frac {U_{2}}{\pi ^{2}}}x-{\frac {U_{4}}{\pi ^{4}}}x^{3}-{\frac {U_{6}}{\pi ^{6}}}x^{5}-\cdots$
${\frac {1}{2}}\tan x={\frac {V_{2}}{\pi ^{2}}}2^{2}x+{\frac {V_{4}}{\pi ^{4}}}2^{4}x^{3}+{\frac {V_{6}}{\pi ^{6}}}2^{6}x^{5}+\cdots$
In (31) $x$ must lie between $\pm \pi$ and in (32) between $\pm {\tfrac {1}{2}}\pi$. Write equation (30) in the form
$\sec x=\sum (-1)^{n}{\frac {(2n+1)\pi }{({\overline {2n+1}}\cdot {\frac {1}{2}}\pi )^{2}-x^{2}}}$
and expand each term of this series in powers of $x^{2}$, then we get
$\sec x={\frac {2^{2}W_{1}}{\pi }}+{\frac {2^{4}W_{3}x^{2}}{\pi ^{3}}}+{\frac {2^{6}W_{5}x^{4}}{\pi ^{5}}}$
where $x$ must lie between $\pm {\tfrac {1}{2}}\pi$. 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 $U_{2},U_{4}\dots$ $V_{2},V_{4}\dots$ $W_{1},W_{3}\dots$. When $U_{n}$ has been found, $V_{n}$ may be obtained from the formula $2^{n}V^{n}=(2^{n}-1)U_{n}$.
For Lord Brounker's series of π, see Circle. It can be got at once Continued Factors for π. by putting $a=1$, $b=3$, $c=5$,$\dots$ in Euler's theorem $={\frac {1}{a}}-{\frac {1}{b}}+{\frac {1}{c}}-\cdots ={\frac {1}{a\;+}}\;{\frac {a^{2}}{b-a\;+}}\;{\frac {b^{2}}{c-b\;+}}\cdots$
Sylvester gave (Phil. Mag., 1869) the continued fraction
${\frac {\pi }{2}}=1+{\frac {1}{1\;+}}\;{\frac {1\,.\,2}{1\;+}}\;{\frac {2\,.\,3}{1\;+}}\;{\frac {3\,.\,4}{1\;+}}\cdots$
which is equivalent to Wallis's formula for π. This fraction was originally given by Euler (Comm. Acad. 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 for $\cos x$ and ${\frac {\sin x}{x}}$ that $\tan x={\frac {x}{1\;-}}\;{\frac {x^{2}}{3\;-}}\;{\frac {x^{2}}{5\;-}}\;{\frac {x^{3}}{7\;-}}\cdots$ Continued Fractions for Trigonometrical Functions. This may be also easily shown as follows. Let $y=\cos {\surd x}$, and let $y',y''\dots$ denote the differential coefficients of $y$ with regard to $x$, then by forming these we can show that $4xy''+2y'+y=0$, and thence by Leibnitz's theorem we have
$4xy^{(n+2)}+(4n+2)y^{(n+1)}+y^{(n)}=0$
Therefore ${\frac {y}{y'}}=-2-{\frac {4x}{y'/y''}}$, ${\frac {y^{n}}{y^{(n+1)}}}=-2(2n+1)-{\frac {4x}{y^{(n+1)}/y^{(n+2)}}}$;
hence $-2{\sqrt {x}}\cot {\sqrt {x}}=-2-{\frac {4x}{-\;6\;-}}\;{\frac {4x}{-\;10\;-}}\;{\frac {4x}{-\;14\;-}}\cdots$
Replacing $\surd x$ by $x$ we have $\tan x={\frac {x}{1\;-}}\;{\frac {x^{2}}{3\;-}}\;{\frac {x^{3}}{5\;-}}\cdots$
Euler gave the continued fraction
$\tan nx={\frac {n\tan x}{1\;-}}\;{\frac {(n^{2}-1)\tan ^{2}x}{3\;-}}\;{\frac {(n^{2}-4)\tan ^{2}x}{5\;-}}\;{\frac {(n^{2}-9)\tan ^{2}x}{7\;-}}\cdots$
this was published in Mém. de l'acad. de St Pétersb. vol. vi. Glaisher has remarked (Mess. of Math. vols. iv.) that this may be derived by forming the differential equation
$(1-x^{2})y^{(m+2)}-(2m+1)xy^{(m+1)}+(n^{2}-m^{2})y^{(m)}=0$
where $y=\cos {(n\arccos x)}$, then replacing $x$ by $\cos x$, and proceeding as in the former case. If we put $n=0$, this becomes
$x={\frac {\tan x}{1\;+}}\;{\frac {\tan ^{2}x}{3\;+}}\;{\frac {4\tan ^{2}x}{5\;+}}\;{\frac {9\tan ^{2}x}{7\;+}}\cdots$
whence we have
$\arctan x={\frac {x}{1\;+}}\;{\frac {x^{2}}{3\;+}}\;{\frac {4x^{2}}{5\;+}}\;{\frac {9x^{2}}{7\;+}}\cdots +{\frac {n^{2}x^{2}}{2n+1\;+}}\cdots$
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 Purely Analytical Treatment of Circular Functions. defined as a function such that, if $x=\int _{0}^{y}{\frac {dy}{\surd (1-y^{2})}}$, then $y=\sin x$; the quantity ${\frac {\pi }{2}}$ would be defined to be the complete integral $\int _{0}^{1}{\frac {dy}{\surd (1-y^{2})}}$. We should then have ${\frac {\pi }{2}}-x=\int _{y}^{1}{\frac {dy}{\surd (1-y^{2})}}$. Now change the variable in the integral to $z$, where $y^{2}+z^{2}=1$, we then have ${\frac {\pi }{2}}-x=\int _{0}^{z}{\frac {dy}{\surd (1-z^{2})}}$, and $z$ must be defined as the cosine of $x$, and is thus equal to $\sin({\frac {1}{2}}\pi -x)$, satisfying the equation $\sin ^{2}x+cos^{2}x=1$.
Next consider the differential equation
${\frac {dy}{\surd (1-y^{2})}}+{\frac {dz}{\surd (1-z^{2})}}=0$
This is equivalent to
$d\{y\surd (1-z^{2})+z\surd (1-y^{2})\}=0$
hence the integral is
$y\surd (1-z^{2})+z\surd (1-y^{2})=$
The constant will be equal to the value $u$ of $y$ when $z=0$; whence
$y\surd (1-z^{2})+z\surd (1-y^{2})=u$
The integral may also be obtained in the form
$yz-\surd (1-y^{2})\surd (1-z^{2})=\surd (1-u^{2})$
Let $\alpha =\int _{0}^{y}{\frac {dy}{\surd (1-y^{2})}}$, $\beta =\int _{0}^{z}{\frac {dz}{\surd (1-z^{2})}}$, $\gamma =\int _{0}^{u}{\frac {du}{\surd (1-u^{2})}}$; we have $\alpha +\beta =\gamma$, and $\sin \gamma =\sin \alpha \cos \beta +\cos \alpha \sin \beta$, $\cos \gamma =\cos \alpha \cos \beta -\sin \alpha \sin \beta$, the addition theorems. By means of the addition theorems and the values $\sin {{\frac {1}{2}}\pi }=1$, $\cos {{\frac {1}{2}}\pi }=0$ we can prove that $\sin {({\frac {1}{2}}\pi +x)}=\cos x$, $\cos {({\frac {1}{2}}\pi +x)}=-\sin x$; and thence, by another use of the addition theorems, that $\sin {(\pi +x)}=-\sin x\cos(\pi +x)=-\cos x$, from which the periodicity of the functions $\sin x$, $cosx$ follows:—
We have also $\int {\frac {dy}{\surd (1-y^{2})}}=-\iota \log _{e}\{\surd (1-y^{2})+\iota y\}$; whence $\log _{e}\{\surd (1-y^{2})+\iota y\}+\log _{e}\{\surd (1-z^{2})+\iota z\}=$ a constant. Therefore $\{\surd (1-y^{2})\}+\iota y\{\surd (1-z^{2})+\iota z\}=\surd (1-u^{2})+\iota u$, since $u=y$ when $z=0$; whence we have the equation
$(\cos \alpha +\iota \sin \alpha )(\cos \beta +\iota \sin \beta )=\cos(\alpha +\beta )+\iota \sin(\alpha +\beta )$
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 Braunmühl's Vorlesungen über Geschichte der Trigonometrie (Leipzig, 1900).  (E. W. H.)