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AOPI

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Originally appearing in Volume V27, Page 282 of the 1911 Encyclopedia Britannica.

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AOPI , P20p3, p3OB and AOP3, P80p2, p3 P3 p2OB satisfy the conditions of similarity sides; thus P2, P3 represent the roots See also:

cos js(0+2w)+t See also:

sin 1 (0+22r), cos 1 (0+41r)+t sin 3(0+42r) respectively. If B coincides with A, the problem is reduced to that of finding the three See also:

CUBE (Gr. K46os, a cube)

cube roots of unity. One will be represented by A and the others by the two angular points of an equilateral triangle, with A as one angular point, inscribed in the circle. The problem of determining the values of the nth roots of unity is See also:

EQUIVALENT

equivalent to the geometrical problem of inscribing a See also:

REGULAR

regular The nth See also:

POLYGON (Gr. rroXus, many, and ywvia, an angle)

polygon of n sides in a circle. See also:

GAUSS, KARL FRIEDRICH (1777-1855)

Gauss has'shown in his Roots of Disquisitions arithmeticae that this can always be done unity, by the See also:

COMPASS (Fr. corn pas, ultimately from Lat. cum, with, and passus, step)

compass and ruler only when n is a See also:

PRIME, PRIMER AND PRIMING

prime of the See also:

FORM (Lat. forma)

form 2n+I. The determination of the nth See also:

root of any complex number requires in addition, for its geometrical See also:

SOLUTION (from Lat. solvere, to loosen, dissolve)

solution, the See also:

DIVISION (from Lat. dividere, to break up into parts, separate)

division of an See also:

ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle into n equal parts. 19. We are now in a position to factorize an expression of Factorisa- the form x"-(a+tb). Using the values which we tions. have obtained above for (a +0)0', we have s= n-1 = 8+25>< 8+2s~r x"-(a+tb)=P [x-r" (cos n +t sin- n )]• (I) s=o If b=o, a=I, this becomes s=n—1 [ 2Sa 2Sir x"—1=P x-cos n -tsin n =(x-I)(x+I)P )n—1 (x—cos 2SIr t t sin 2n s=1 2Sa =(x— I)(x+i)P s=.n—1(x2—2x cos 2- +I) (n even). (2) s=0 s=1(n—1) x"—I=(x-I)P (x2—2xcos2 +I) (n See also:

ODD (in middle English odde, from old Norwegian oddi, an angle of a triangle; the old Norwegian oddamann is used of the third man who gives a casting vote in a dispute)

odd). (3) s=1 If in (I) we put a=-1, b=o, and therefore 0=3r, we have =n—1 x"+I =P s =n-1 2- n I1r —t sin 2S+n I1r1 s=o 1 s=}(n—2) 2S+I7r =P [x2—2x cos n +1] (n even). s=o s=i(n—3) 2S+I7r x"+I = (x+i)P [x2—2x COS n +I] (n odd). (5) s=0Also x2"-2x"y" cos ne+y2" = (xn - yn cos n6 let sin nb) (x"- y" cos nO -t sin n6) s=n-/ 0 =P ( B+2ssr t sin —=-1 s=o .x-y cos \ n n J s=n- =P [x2-2xy cos B+2n"+y2] .

(6) s=0 See also:

AIRY, SIR GEORGE BIDDELL (1801-1892)

Airy and See also:

Adams have given proofs of this theorem which do not involve the use of the See also:

SYMBOL (Gr. o-uµ(3oXov, a sign)

symbol t (see See also:

COMB (a word common in various forms to Teut. languages, cf. Ger. Kamm, the Indo-Europ. origin of which is seen in yoµros, a peg or pin, and Sanskrit, gambhas, a tooth)

Comb. Phil. Trans., vol. xi). A large number of interesting theorems may be derived from De Moivre's theorem and the factorizations which we have Example of deduced from it; we shall See also:

NOTICE

notice one of them. Example In See also:

EQUATION (from Lat. aequatio, aequare, to equalize)

equation (6) put y1/x, take logarithms, and then re Theorem. differentiate each See also:

side with respect to x, and we get 2n(x2n-1—x—2n—i) s _n.--1 2(x—x 3) x2i-2 cos n0+x-2"-ss=0 2 2Ssr x -2 cos 6+—+x n Put x22=See also:

ALB (Lat. alba, from albus, white)

alb, then we have the expression n (a2n - b2tt) (See also:

A2(z)

a2 - b2) (a'n -2 a,b, cos n0+b2") for the sum of the See also:

SERIES (a Latin word from serere, to join)

series s=n—1 s=o 2 2S!r a-2ab cos 0+ n +b2 20. Denoting the complex number x+iy by z, let us consider the series 1 +z+z2/2 ! + ... +z'/n! +. . . This series converges uniformly and absolutely for all values of z whose Then.. moduli do not exceed an arbitrarily chosen See also:

POSITIVE (or PORTABLE) ORGAN

positive potlal number R.

Consequently the See also:

FUNCTION

function E(z), defined serienes. 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(zi+z2). In accordance'with a known theorem, since the series for E(zi) E(z2) are absolutely convergent, we have E(zi) XE(z2) =E(z1+z2). From this fundamental relation, we deduce at once that {E(z) l' =E(nz), where n is any positive integer. The number E(i), the sum of the convergent series I+I+1/2!+1/3! is usually denoted by e; its value can be shown to be 2.718281828f59. . .. It is known to be a transcendental number, i.e. it cannot be the root of any algebraical equation with rational coefficients; this was first established by Hermite. See also:

Writing z=1, we have E(n)=e°, where n is a positive integer. If z has as a value a positive fractionp/q, we find that{E(See also:

pig) }4=E(p)=e5; hence E(pig) is the real positive value of &I Q. Again E(-p/q) XE(p/g) =E(o) =1, hence E(-p/q) is the real positive value of e p'Q. It has been thus shown that for any real and rational number x, the value of E(x) is the See also:

PRINCIPAL

principal value of es. This result can be extended to irrational values of x. if we assume that e5 is for such a value of x defined as the limit of the sequence ei1, e:2 , where x2,.

. . is a sequence of rational See also:

numbers of which x is the limit, since E(x1), E(x2) . . ., then converges to E(x). Next consider (1+z/m)"', where m is a positive integer. We have by the See also:

BINOMIAL (from the Lat. bi-, bis, twice, and nomen, a name or term)

binomial theorem,} r (1+m) m- i+z+ (I m/ z2. + (I m) (I m) .. . (I—Sm1)'+..+1.m) m (I m) (I J) ... (I—'m 1 \ lies between 1 and + (m+m+... +s ml) ; hence the product equals i -0,s . S- I/2m where 03 is such that o<0 <i. We have now (I+m) m=i+z+ (I-m) zj+...+[I-See also:

BES, or BESAS (Egyp. Bes or Besa)

Bes.2m 1]Si+... +[i-8 m-Il '' 2 = 1+z +22/2 !+ . . .

+Zs/s ! +Rs, where ,s+i z"` z2 z Rs=(s+I)!+...+nt!-zm I+B3I+... zs—2 zm—2 +0,(s-2) !+. +‘(,- 2) Since the series for E(z) converges, s can be fixed so that for all values of m >s the modulus of zs+i/(s+i) ! +. . . +z'"/m! is less than an arbitrarily chosen number le. Also the modulus of I+0sZ/I +...+0n,z"-2/(m-2)! is less than that of 1+11 zI/i! +1z 12/2! +..., or of emods hence mod Rs<le+(I/2m). mod (z2e') <e, if m be chosen sufficiently See also:

GREAT

great. It follows that limm_,o(i+z/m)"'=E(z), where z is any complex number. To evaluate E(z), write I +x/m = sr) cos 4>, y/m = p sin 0, then s=1 (4) Also E(z) = limn _~ {p" (cos m~ +i sin mo) }, by De Moivre's theorem.

Since p"`= (I+m) ") I+(/ m-} x/ Jm)2 , we have =es. limm... ) I+m(Jm+x/ Jrn)2 Let r be a fixed number less than Jm+x/.,See also:

IM(K2+a2+h2—2 ah cos 0)

Im, then limn,_ I+m(Jm+xJm)2'sra lies between i and limm... ) 1+n- „ y2 ) 2 ' or between i and See also:

eve 2r2; hence since r can be taken arbitrarily large, the limit is r. The limit of m4, or m tan-'{y/(x+m)} is the same as that of my/(x+m) which is y. Hence we have shown that E(z) =es(cos y+i sin y). 21. Since E(x+iy) =i'(cos y+sin y, we have cos y+i sin y =E(iy), and cos y-i sin y=E(-iy). Therefore cos y=z{E(iy) Exponential +E(-iy)}, sin y=zi{E(iy)-E(-iy)}; and using the series defined by E(iy) and E(-iy), we find that Values of cos y = I - y2!2 ! + y4 /4 ! - .. , sin y = y - y2/3 ! Trigono- + y'/5!

- ., where y is any real number. These metrical are the v3ell-known expansions of cos Functions. y, sin y in See also:

POWERS, HIRAM (1805-1873)

powers of the circular measure y. Where z 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 e'v, e-'v are E(iy), E(-iy). The above expressions for cos y, sin y may then be written cos y = z (e'v+e`v), sin y = Zi(e'v-e`v). These are known as the exponential values of the cosine and sine. It can be shown that the symbol e' as defined here satisfies the usual See also:

LAWS

laws of See also:

COMBINATION (Lat. combinare, to combine)

combination for exponents. 22. The two functions cos z, sin z may be defined for all complex or real values of z by means of the equations cos y='-2(E(z)+ E(-z) }, sin z = (,,){E(z) -E(-z) }, where E(z) represents the sum-function of i+z+z2,2!+...+z"jn!+... For real values of z this is in accordance with the See also:

ORDINARY (med. Lat. ordinarius, Fr. ordinaire)

ordinary See also:

DEFINITIONS

definitions, as appears from the series obtained above for cos y, sin y. The fundamental properties of cos z, sin z can be deduced from this See also:

DEFINITION (Lat. definitio, from de-finire, to set limits to, describe)

definition. Thus cos z + i sin z = E(z), cos z - i sin z = E(- iz) ; therefore cos2z+sin2z = E (iz) . E( — iz) =1.

Again cos (zi +z2) is given by z (E (iz, +iz2) + E (- iz, -iz2) } = z { E (izI) E (iz2) + E (- izI) E (-iz2) } or ;{EUzi) +E(-iz,)}{E(iz2)+E(-iz2)}+'-44{E(izi)-E(-iz,)} E(iz2) -E(-iz2) }, whence we have cos (11±z2) = cos z, cos 22-sin 2, sin z2. Similarly, we find that sin (zl+z2) =sin z, cos z2+ cos z, sin z2. Again the equation E(z) =1 has no real roots except z =o, for ev> 1, if z is real and >o. Also E(z) =1 has no complex root a+ifl, for a-ifl would then also be a root, and E(2a) _ E(a+iO)E(a-i$)=I, which is impossible unless a=o. The roots of E(z) = i are therefore purely imaginary (except z =o) ; the smallest numerically we denote by 2 ir, so that E(2i7r)=1. We have then E(2iar)={E(2i,r)}r=1, if r is any integer; therefore 21irr is a root. It can be shown that no root lies between 2irr and 2(r+I)iir; and thus that all the roots are given by z= =2irr. Since E(y+2i,r) =E(z)E(2i1r) =E(z), we see that E(z), is periodic, of See also:

period 2ivr. It follows that cos z, sin z are periodic, of periods 27r. The number here introduced may be identified with the ratio of the circumference to the See also:

DIAMETER (from the Cr. &a, through, µErpov, measure)

diameter of a circle by considering the See also:

case of real values of z. 23. Consider the binomial theorem (a+b)"=+na"-'b+n(2 i I) a"-2b2+..

. +n(n-I)...(n—r+I)a"-rbr+...+b”. r Putting a=eie, b=e ie, we obtain (2 cos o)"=2 cos 150+112 COS n—20 +n(n-I)2COSn-40+ 2l .. +n(n- I) ...(n- r+t)2 cos(n-2r)B+.. . r When n is odd the last See also:

TERM

term is 2n(n-')' ' (n+3) cos B, 1(n-I) and when n is even it is n(n-1)i .. (Zn+i) an l If we put a=eie, b= -e 'e, we obtain the See also:

formula (-1).n(2sino)"=2 cosn6-2ncos(n-2)0+n(n-i) 2cos (11—4)0 I +(—I)"-rn(nI)• r (n—r+I) 2cos(n—2r)0... when n is even, and (- I)1(n-i)(2 sin B)" = 2sin nB-n . 2 sin(n — 2)B+n(. 21) 2 sin(n -4)8- Iwhen n is odd. These formulae enable us to See also:

EXPRESS (through the French from the past participle of the Lat. exprimere, to press out, by transference used of representing objects in painting or sculpture, or of thoughts, &c. in words)

express any positive integral See also:

POWER [WILLIAM GRATTAN] TYRONE (1797-1841)

power of the sine or cosine in terms of sines or cosines of multiples of the See also:

ARGUMENT

argument. There are corresponding formulae when n is not a positive integer. Consider the identity See also:

LOG (a word of uncertain etymological origin,possibly onomatopoeic; the New English Dictionary rejects the derivation from Norwegian lag, a fallen tree)

log(I -px)+log(I -qx) = Expansion log(1-p+qx+pgx2). Expand both sides of this ofSinesand equation in powers of x, and equate the coefficients of Cosines of x", we then get Multiple p"+4" = (p+q)"-n(p+q)"-2pq Arcs in Powers of +n(2, 3) (1,+4)"-4p2g2+... - p Sines +qx+pqx+(-I)'n(n-r-I)(n-r-2)...

(n-2r+I) Arc. r! (P+q)"-l'•pr4'+.. . If we write this series in the See also:

REVERSE

reverse See also:

order, we have I); 7211( +q"=2(—I);[(pq)pq) n2 1 (1,2 g) 2 +n2(n4l ?2) (pq)2 2 (±) a n2(n2— 6)(n2—42)(1,4)2 3 (P42 -q) z(1,+q)nJ when n is even, and p"+4"=2(-)n2i[n(pq)2 (1,2 q) - n(3'-I2(pq)n23 (1,2 4) 2 +n (n2 — 15 I(n2 - 32) (1,4)"4 (1,24) 6+ ... + (- )'''.21(p +q)"] when n is odd. If in these three formulae we put p=ei%,q=e-ie,we obtain the following series for cos ne: 2 cos no= (2 cos 0)"—n(2 cos o)"-2+n( 2T 3)(2 COS 0)"-4—... +(-1)rn(n-r- I)(n-ryl2)... -2r+I)(2 cos o)"-2r+... (7) when n is any positive integer; ( — I )Los uB = - 2 titcos2B+n2(n24l-22) cos4B -n2(n2—226)(ln2—42) cos8 I B +... +(-I)22"-' cos no when n is an even positive integer; n-1 (n2- I2) (n2 — I2)(n2- 32) (-I) 2 cos nO=n cos nO- n 3l cosaB+n 5l cos B.- n-I ... +(—I) 2 2 n-1 COS "0 (9) when n is odd. If in the same three formulae we put p=ee, q= -e-0B, we obtain the following four formulae:—(-1)22 cos n0= (2 sin O)"-n(2 sin 0)"-'-+n(2! 3) (2 sin o)"-4-.

. . +(-1)rn(n-r I)..,.(n- 2r-i) 2 sin 0)n-2r r ( +.(a even); (Io) n-1 (- I)2 2 sinnB=the same series (n odd); (II) 12. 2 n2(n2 — 22) 4 n2(n2 -- 22) (n2 — 42) g cos n0 = l - lsinB+ 4 • sing- ,~6 sinB +... +2"-' sin "0 (n even); (I2) sin nO = n sin 0 - n (n23 l~ 12) sin3B + n (n'- - 15 (n2 - 32) sinSB - .. . n-1 +(—I) 2"-' sin"o (n odd). (13) Next consider the identity 1, - q = p-q 2 I - px i - qx I - (p + 4)x + pqx Expand both sides of this equation in powers of x, and equate the coefficients of x"-', then we obtain the equation f p"-qn=(p+q)"-'-(n-2) (P+q)"-a 1,4 p-q See also:

ANALYTICAL

Analytical Definitions of Trigonometrical Functions. Expansion of Powers of Sines and Cosines in Series of Sines and Cosines of Multiple See also:

ARM (a common Teutonic word; the Indo-European root is ar, to join or fit; cf. the Lat. armus, shoulder, and the plural word arma, weapons, Gr. apphs, joint, and the reduplicated apapilKSV, to join)

Arm +(-') 2rL (ti_ i)2n (sn-}-i) (8) +(n-3) (n-4) (p+q)n-bp2g2-... (—I)r (n-r-I) (n-r-2) ... (n- 2r) If, as before, we write this in the reverse order, we have the series (-I)2 [n (PI i) (pq)2 1 -n(n32 -22) 1'p-1-g\ 2, )2-n-3 +(-)Tn(n-I)-(n+3) 1(n—l)! sin B +n(n2-25)(n2-42)(1, g11(in)2~~+...+(-,)21(1, +q)"-i when it is even, and n-1 n1 n-3 (—I) 2 [(pq)2 - n 2; I'(~24)2(pq)2 + (n2—12)(n2—32) (p2 q) a(pq)n2• +...+(—I)n21(p+q)"—1J when n is odd. If we put p=eie, q=e-ce, we obtain the formulae (77 sin no= lino (2cosB)^-1. -(n-2)(2cos B)^-'+(n - 3)2 - 4)(2 +(—I)r(n—r—I)(it-r—2 (n—2Y) (2 cos 0)"-2r-1 + .

. } (14) r where n is any positive integer; (-1)2 'sinno=sine {ncoso_n(n31 22)cos3o+n(n2-2"-5 5! n2-42)coss-... +(—I)2 (2 cos o)"-' (n even); (15) (-I)n2lsin nO =sin B j I—ii22!'' cos2B--(n2-12) (2-32) cos°9-. +(—1)2 1(2 cos B)"—' (n odd). (16) If we put in the same three formulae p=coo, q= -co, we obtain the series n-2 (-I) 2 sin no=cos0[sin"-'o-(n-2)sin"-3o+(n-3)2jn-4)sin"-SB-... +()r(n—r—I)(n —rI—2)...(n—2r)s.nn tr-1B-I-...,(n even); (17) n-I (S- I) 2 cos no =the same series (n odd) ; (18) sin n9=cos 0) n sin 0(n2-22) sin'o+ n(n2—22) ~(n2—42) sinso+ l 3• 5• ...+(—1)2 I(2 sin o)"—' (n even); (19) cos nB=cos 0) I -n22 T-sin2o+(ns-I24(nz-32)sin'o-.. +(2 sin 0)"-' (n odd). (20) We have thus obtained formulae for cos nO and sin nO both in ascending and in descending powers of cos 0 and sin 0. See also:

VIETA (or VIETE), FRANCOIS, SEIGNEUR DE LA BIGOTIERE (1540-1603)

Vieta obtained formulae for chords of multiple arcs in powers of chords of the See also:

SIMPLE

simple or complementary arcs equivalent to the formulae (13) and (19) above. These are contained in his See also:

WORK

work Theoremata ad angulares sectiones. Jacques See also:

BERNOULLI

Bernoulli found formulae.equivalent to (12) and (13) (Mem. de l' See also:

ACADEMIC

Academic See also:

des Sciences, 1702), and trans-formed these series into a form equivalent to (10) and (II). See also:

JEAN

Jean Bernoulli published in the Acta eruditorum for 1701, among other formulae already found by Vieta, one equivalent to (17).

These formulae have been extended to cases in which n is fractional, negative or irrational ; see a See also:

paper by D. F. See also:

Gregory in Camb. Math. Journ. vol. iv., in which the series for cos nO, sin nO in ascending powers of cos 8 and sin 8 are extended to the case of a fractional value of n. These series have been considered by See also:

EULER, LEONHARD (1707-1783)

Euler in a memoir in the Nova acta, vol. ix., by See also:

LAGRANGE, JOSEPH LOUIS (1736-1813)

Lagrange in his Calcul des fonctions (r8o6), and by See also:

POINSOT, LOUIS (1777–1859)

Poinsot in Recherches sur l'analyse des sections angulaires (1825). 24. The See also:

general definition of Napierian logarithms is that, if ex-,y=a+,b, then x+cy=log (a+cb). Now we know that Theory of exr,Y=excos y Lex sin y; hence ex cos y=a, ex sin y =b, or ex= (a +b2) 3, y=arc tan b/a ' mir, where m Logarithms. is an integer. If b =o, then m must be even or odd according as a is positive or negative; hence loge (a+cb) =loge (a2+b2) j+ c (arc tan b/a' 2nir) or log, (a+eb) =log, (a2+b2)+ e (arc tan b/aa2n+7r), according as a is positive or negative. Thus the See also:

LOGARITHM (from Gr. Abyos, word, ratio, and apx0µos, number)

logarithm of any complex or real quantity is a multiple-valued function, the differ- ence ence between successive values being 27rc; in particular, T perbo the most general form of the logarithm of a real positive merry. quantity is obtained by adding positive or nega- tive multiples of 22n to the arithmetical logarithm. On this subject, see De See also:

Morgan's See also:

TRIGONOMETRY (from Gr. rpiywvov, a triangle, /ATpov, measure)

Trigonometry and See also:

Double See also:

ALGEBRA (from the Arab. al-jebr wa'l-mugabala, transposition and removal [of terms of an equation], the name of a treatise by Mahommed ben Musa al-Khwarizmi)

Algebra, ch. iv., and a paper by See also:

PROFESSOR (the Latin noun formed from the verb profiteri, to declare publicly, to acknowledge, profess)

Professor See also:

CAYLEY, ARTHUR (1821-1895)

Cayley in vol. ii. of Proc.

See also:

Long afterwards Euler employed the series obtained from 4a =arc tan 3+ arc tan 3, which, however, gives less rapidly converging series (Introd., Anal. infin. vol. i.). T. F. de Lagny employed the formula arc tan I/-/3=7r/6 to calculate 7r to 127 places; the result was communicated to the See also:

Paris See also:

Academy in 1719. G. See also:

Vega calculated 7r to 140 decimal places by means of the series obtained from the equation '-17r=5 arc tan 3 +2 arc tan A. The formula **vr=arc tan 2+arc tan 3+arc tan 3 was used by J. M. Z. Dase to calculate 7r to 200 decimal places. W. See also:

Rutherford used the equation 2r=4 arc tan - arc tan .'o + arc tan 61,. If in (23) we put y = 3 and 3, we have 2r=8 arc tan 3+4 arc tan +=2.4I+3 10 3.5 Toe a 2 2.4 2 +'S6 I+3 100 3.5 (too) +.

, a rapidly convergent series for in which was first given by See also:

Hutton in Phil. Trans. for 1776, and afterwards by Euler in Nova acta for 1793. Euler gives an ,equation deduced in the same manner from the identity x=20 arc tan ++8 arc tan A. The calculation of 7r has been carried out to 707 places of decimals: see Proc. See also:

ROY, WILLIAM (1726-1790)

Roy. Soc. vols. xxi. and xxii. ; also CIRCLE. o =sin o sin' B 3 sin' 0 1.3.5 - sin7 B (21) Aagle +2 3 2.4 5 +2.4.6 7 +' ) In Powers 0 must See also:

lie between the values t Zir. This equation of its Sine. may also be written in the form I x' I.3 xs I.3.5 x7 arc sinx=x+2 3+2.4 5+2 4.6 7+'•• (22) arc sin x 27. We shall now obtain expressions for sin x and cos x as See also:

INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

infinite products of rational factors. We have Factorize- sin x=2 sin 2sinx 2 --=23 sin-sinx See also:

don of Sine 4 and Cosine. x+2r . x+3r sin sm 4 4 proceeding continually in this way with each See also:

FACTOR (from Lat. facere, to make or do)

factor, we obtain x x-{-r x+2r x+n-Ir sin x =2"-1 sin nsin n sin n ... sin n where n is any positive integral power of 2. Now sinx+rrsinx+n-rr x+rr rr-x r II n sin n sin n n r =sin2 -sin2n, x+ins x _ sin n =cos-.

Hence the above may be written sinx=2n-1 sine (sing -n-sin2n) (sing 91 -sin2 rx-L) ' .' ( 2kr 2x) x sin n -sin n cosh, where k = = I. Let x be indefinitely small, then we have 2"-1 r 2r kr =— I sin' rIsin2 n... sing n; hence sin2 x/n si x/n sin x =n Sinn cos n (I sin2 r/n) ( sin' 2r/n/n) sin2^k /n) ' We may write this in x =n sin x cos x (I -sin2 x/n) (I —sin, sin2 x/n sin ) R, n n sin2 r n mr/n where R denotes the product /_ sin2 x/n r/n) ( sin" x/n \J in2 x/n _ I sing m+2r/n/ (I sin2 kr/n)' and to is any fixed integer See also:

INDEPENDENT

independent of n. It is necessary, when we make n infinite, to determine the limiting value of the quantity R; then, since the limit of sin x is sin x, and that of n sin x/n cos x/n x sinx=(1-') ( x2) ( x2 ) lim R. x I 22x2 m r m= The modulus of R — I is less than (I+sin2 m+Ir/n) (I+sin2 m+2r/n) ... (I+singpkr/n) -I' where p =mod. sin x/n. Now eAP2> I +A p2, if A is positive ; hence mod. (R-I) is less than exp. p2(cosec2 m+Ir/n+ ... + cosec2 kr/n)-I, or than exp. ;ping{I/(m+I)2+... +I/k2}-I, or than exp. { p2n2/4m2 } - I. Now p2 =sin2 a/n.cosh2 /i/n+cos2 a/n. sinh2 fl/n, if x=a+xfl; or p2=sin2 a/n+sinh2 /3/n.

Hence limn=„ p2n2=a2+132, limn pn=mod. x. It follows that mod.,, (R - I) is between o and exp. { (mod. 4)2/ xm2 } - 1, and the latter may be made arbitrarily small by taking m large enough. It has now been shown that sin x=x(I -x2/See also:

R2(rt-s2)

r2)(I -x2/22r2) (I-x2/m2r2) (I +e,,,), where mod. em decreases indefinitely as m is increased indefinitely. When m is indefinitely increased this becomes xrz xrz n= ^° x2 sin x=x (I T2 — ) (I -22,2) ... =xP ' (I -ri2,n.2) (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 3r-2x by means of the equation cos x=2 sin -2x 4 cos 4 , or may In the formula for sin x as an infinite product put x=lr, we then get 1=5 a 2 : 3.3 556: : :. if we stop after 2n factors in the 4.4 numerator and denominator, we obtain the approximate equation r 12 .32.52 ... 2_ I -2 22.42.62 ... (2n)2 (2nd-1) or I .3.5.:.22 n I = 11nr, where n is a large integer. This ex- pression was obtained in a quite different manner by See also:

WALLIS, JOHN (1616-1703)

Wallis (Arithmetica infinitorum, vol. i. of Opp.). 28.

We have sin (x+y)=(x+y)P(I+xn -) sinx xP(I +-I) nr or cos y+sin y cot x (I+x) \I+x+0 (I+x+2r) (1+xy2r1 ... Equating the coefficients of the first power of y on both sides we obtain the series cot x=x+x+r+x rr+x+2r +x I2r (27) From this we may deduce a corresponding series for cosec x, for, since cosec x=cot ;x-cot x, we obtain cosec xxxnrx I rrx+2r+x I2r—x+3r—x-13r+ (28) By resolving cos (x+y) into factors we should obtain in a similar cos x manner the series 2 2 2 2 2 _ 2 tanx=2x r+2x+3r-2x 3r+2x+5r-2x 5r 2x+"" (29) and thence secx=tan(-+2)—tan x=?2x+r+2 2 2x 3r2 ._2x 3r+2x+""(3°) These four formulae may also be derived from the product formulae for sin x and cos x by taking logarithms and then differentiating. See also:

GLAISHER, JAMES (1809-1903)

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 = i {cot 2x+ cot 1(x+r) } (see a paper by Dr SchrSter in Schlomilch's Zeitschrift, vol. xiii.). Various series for In may be derived from the series (27), (28), (29), (3o), and from the series obtained by differentiating them one or more times. For example, in the formulae (27) and (28), by putting x=r/n we get sr =It tan n) I nI II+n+I—2n_I+2n+I' r=n sin n I+nn I I—n+I 211-I 2n+l' .. If we put n=3, these become r=3113 (I-2+--5+--8+...) , 3113 I I _1+1+1. By differentiating (27) we get cosec2 x=z2+(x+r)2 S+ (a. .r)2+(x+2,02+(x I2r)2+... putx=6,andwegetr2=9i I+52+7+Iiz+ ••• S• These series, among others, were given by Glaisher (Quart. Journ.

Math. vol. xii.). x2 29. We have sinh rx=rxP I+n2) , cosh rx=P (I+(2nx2+I)2) if we differentiate these formulae after taking logarithms we obtain the series Sums of and sin mr/n mr/n is unity, we have be deduced thus s sin 2X P (I n x2) = (1-4,r-) 2) () cos x=2 sin x P (1_ x I 3 rr I-5 r .. nEr2) ^=m 4x2 (26) =P O I-(2n+i)2r2) If we See also:

CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

change x into ix, we have the formulae for sinh x, cosh x as infinite products- sinh x=x'P o (I+) , cosh x=P (I+(2n)2r2)' Series for Cot, Cosec, Tan and Sec. Series for n derived from Series for Cot and Cosec. sin' Certain 2x coth rx- 2x2=I2+x2+22+x2+32+x2+.., Series. 2x tanh rx=l2+x2+32x2+52.x2+• .. These series were given by Kummer (in Crelle's Journ. vol. xvii.) The sum of the more general series I2n+x2n+2rz".+x2"+32n+xsn + ... , has been found by Glaisher (Prot. Lond. Math. Soc., vol. vii.) If U^ denotes the sum of the series I^'+2I—m+3 +..., V' r that of the series Im+3 +5 .., and W"' that of the series Sums of t +- - +., we obtain by taking toga- 3,,, 5 log (x cosec x) = U2 (x_) 2+2 U4 ( ) 4+ Us (x_) 6+ ... , I (3:) log (sec x) = V2 (3;1 r~ 2+2V44+3V6(.2) 6+.; and differentiating these series we get 2 Cot x =2x 2x— 4x3 — e — " • r V V4 6 2 tan x = 2222x+424x3+626x5 + ....

(32) r sr In (31) X must lie between tz- and in (32) between =y'-r. Write equation (30) in the form forTrigono- This may be also easily shown as follows. Let metrical y =cos Al x, and let y', y"... denote the See also:

DIFFERENTIAL

differential Functions. coefficients of y with regard to x, then by forming these we can show that 4xy"+2y'+y=0, and thence by See also:

LEIBNITZ (LEIBNIZ), GOTTFRIED WILHELM

Leibnitz's theorem we have 4xy("+2)+(4n+2)y(*+I) ~-Yt*)=o. Therefore y.= — 2 —y%y,,, ; = — 2(2n + 1)- y(„+,4xyC„+2) hence -2-NIT cot x/x=—2— 4x 4x 4x -6— —IO— -14 Replacing I x by x we have tan x= x x x z I _ 3_ ~. Euler gave the continued fraction tan nx =n tan x (n2 — I) tan2x (n2 — 4) tan2x (n2 — 9) tan2x _ 3— 5— 7 this was published in Mem. de l'acad. de St Petersb. vol. vi. Glaisher has remarked (See also:

MESS (an adaptation of O. Fr. mes, mod. mets; Ital. messo; derived from the Late Lat. missum, past participle of mittere " to send or place in position ")

Mess. of Math. vols. iv.) that this may be derived by forming the differential equation (i — x2)y(" ),— (2m + 1)xy('"+I) + (n2 — m2) y(') = o, where y=cos (n arc cos x), then replacing x by cos x, and proceeding as in the former case. If we put n=o, this becomes tan x tan2x 4 tan2x 9 tan2x 1 ~- 3+ 5+ 7+ whence we have z 4 X2 z2 arc tan x = + 3+ 5+ 7.+ .'+2n+I +"' 31. It is possible to make the investigation of the properties of the simple circular functions See also:

REST (O. Eng. rest, reste, bed, cognate with other Teutonic forms, e.g. Ger. Rast, Riiste, rest, and probably Gothic Rasta, league, i.e. resting or stopping place)

rest on a purely analytical basis other than purely the one indicated in § 22. The sine of x would be Analytical defined as a function such that, if x= f dY Treatment J 011 (1 — Yz)' of Circular then y=sin x; the quantity r would be defined to Functions. 2 be the complete integral f of (Idy y2)• We should then have z —x = f t ( dy y2) Now change the variable in the integral to z, where y2+z2 =1, we then have z —x = f od (Idy z2)' and z must be defined as the cosine of x, and is thus equal to sin (1sr—x), satisfying the equation sin2 x+cost x=I. Next consider the differential equation J(dyy2)(I Z2)=0. This is equivalent to d{Pl(1—z2)+z,l (1—y2)} =0; hence the integral is y' (I —z2) -l-z-V (I —y2) = a See also:

CONSTANT, BENJAMIN (1845-1902)

constant.

The constant will be equal to the value u of y when z=o; whence pi (1 —z2)+z./ (I —y2) =is. The integral may also be obtained in the form yz— '(I—y2)J(I—z2)=iI(1—u2). y z Leta=r , J o, (1—y2)' R of (dz z2) =I. m' 0 —.2); we have a+/3=y, and sin y=sin a cos /3+cos a sin 13, COs y=Cos a COS /3—sin a sin /3, the addition theorems. By means of the addition theorems and the values sin 4zr=1, cos 44Tr=o we can prove that sin (za+x)= cos x, cos (vr+x) = —sin x; and thence, by another use of the addition theorems, that sin (lr+x) = —sin x cos (sr+x) = —cos x, from which the periodicity of the functions sin x, cos x follows: We have also f 11 (dyy2)=—s loge{ s/ (I—y2)+6y}; whence loge Ix/ (I — y2) + Lyl + loge k' (1 — z2) + sz} = a constant. Therefore { (I — y2)l + Ly{ d(1 — z2)+Lzl = (I — u2) + su, since u =y when z =o ; -whence we have the equation (cos a + sin a) (cos /3 + 6 sin 13) cos (a + /3) + c sin (a + 13), from which De Moivre's theorem follows.

End of Article: AOPI

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