AOPI, P20p3, p3OB and AOP3, P80p2,
p3 P3 p2OB satisfy the conditions of similarity
sides; thus P2, P3 represent the roots cos js(0+2w)+t 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 cube roots of unity. One will be represented by A and the others by the two angular points of an equilateral triangle, with A as one angular point, inscribed in the circle.
The problem of determining the values of the nth roots of unity is equivalent to the geometrical problem of inscribing a regular The nth polygon of n sides in a circle. Gauss has'shown in his Roots of Disquisitions arithmeticae that this can always be done
unity, by the compass and ruler only when n is a prime of the
form 2n+I. The determination of the nth root of any complex number requires in addition, for its geometrical solution, the division of an angle into n equal parts.
19. We are now in a position to factorize an expression of Factorisa the form x"(a+tb). Using the values which we tions. have obtained above for (a +0)0', we have
s= n1 = 8+25>< 8+2s~r
x"(a+tb)=P [xr" (cos n +t sin n )]• (I)
s=o
If b=o, a=I, this becomes
s=n—1 [ 2Sa 2Sir
x"—1=P xcos n tsin n
=(xI)(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=(xI)P (x2—2xcos2 +I) (n 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 =n1 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 .xy cos
\ n n J
s=n
=P [x22xy cos B+2n"+y2] . (6)
s=0
Airy and Adams have given proofs of this theorem which do not involve the use of the symbol t (see 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 notice one of them. Example
In equation (6) put y1/x, take logarithms, and then re Theorem.
differentiate each side with respect to x, and we get
2n(x2n1—x—2n—i) s _n.1 2(x—x 3)
x2i2 cos n0+x2"ss=0 2 2Ssr
x 2 cos 6+—+x n
Put x22=alb, then we have the expression
n (a2n  b2tt)
(a2  b2) (a'n 2 a,b, cos n0+b2") for the sum of the series
s=n—1
s=o 2 2S!r a2ab 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 positive potlal number R. Consequently the 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. 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(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 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 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 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 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/(m2)! is less than that of 1+11 zI/i! +1z 12/2! +..., or of emods hence mod Rs, 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/.,Im, then limn,_ I+m(Jm+xJm)2'sra lies
between i and limm... ) 1+n „
y2 )
2 ' or between i and 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 yi 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 v3ellknown expansions of cos
Functions. y, sin y in 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've`v). These are known as the exponential values of the cosine and sine. It can be shown that the symbol e' as defined here satisfies the usual laws of combination for exponents.
22. The two functions cos z, sin z may be defined for all complex or real values of z by means of the equations cos y='2(E(z)+ E(z) }, sin z = (,,){E(z) E(z) }, where E(z) represents the sumfunction of i+z+z2,2!+...+z"jn!+... For real values of z this is in accordance with the ordinary definitions, as appears from the series obtained above for cos y, sin y. The fundamental properties of cos z, sin z can be deduced from this definition. Thus
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 22sin 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 aifl would then also be a root, and E(2a) _ E(a+iO)E(ai$)=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 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 diameter of a circle by considering the case of real values of z.
23. Consider the binomial theorem
(a+b)"=+na"'b+n(2 i I) a"2b2+.. . +n(nI)...(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(nI)2COSn40+ 2l ..
+n(n I) ...(n r+t)2 cos(n2r)B+.. . r
When n is odd the last term is 2n(n')' ' (n+3) cos B, 1(nI)
and when n is even it is n(n1)i .. (Zn+i) an l
If we put a=eie, b= e 'e, we obtain the formula (1).n(2sino)"=2 cosn62ncos(n2)0+n(ni) 2cos (11—4)0
I
+(—I)"rn(nI)• r (n—r+I) 2cos(n—2r)0...
when n is even, and
( I)1(ni)(2 sin B)" = 2sin nBn . 2 sin(n — 2)B+n(. 21) 2 sin(n 4)8
Iwhen 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(I px)+log(I qx) = Expansion log(1p+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(nrI)(nr2)... (n2r+I) Arc. r!
(P+q)"l'•pr4'+.. . If we write this series in the reverse 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=eie,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(nr I)(nryl2)... 2r+I)(2 cos o)"2r+... (7) when n is any positive integer;
( — I )Los uB =  2 titcos2B+n2(n24l22) cos4B n2(n2—226)(ln2—42) cos8
I B
+... +(I)22"' cos no when n is an even positive integer;
n1 (n2 I2) (n2 — I2)(n2 32)
(I) 2 cos nO=n cos nO n 3l cosaB+n 5l cos B.
nI
... +(—I) 2 2 n1 COS "0 (9)
when n is odd. If in the same three formulae we put p=ee, q= e0B, 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(nr I)..,.(n 2ri) 2 sin 0)n2r
r ( +.(a even); (Io)
n1
( 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  .. .
n1
+(—I) 2"' sin"o (n odd). (13)
Next consider the identity 1,  q = pq 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)"'(n2) (P+q)"a 1,4
pq
Analytical Definitions of Trigonometrical
Functions.
Expansion of Powers of Sines
and Cosines in Series of Sines and Cosines of Multiple
Arm
+(') 2rL (ti_ i)2n (sn}i)
(8)
+(n3) (n4) (p+q)nbp2g2...
(—I)r (nrI) (nr2) ... (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'p1g\ 2, )2n3
+()Tn(nI)(n+3) 1(n—l)!
sin B
+n(n225)(n242)(1, g11(in)2~~+...+(,)21(1, +q)"i
when it is even, and
n1 n1 n3
(—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=ece, we obtain the formulae
(77
sin no= lino (2cosB)^1. (n2)(2cos B)^'+(n  3)2  4)(2
+(—I)r(n—r—I)(itr—2 (n—2Y) (2 cos 0)"2r1 + . . } (14)
r
where n is any positive integer;
(1)2 'sinno=sine {ncoso_n(n31 22)cos3o+n(n22"5 5!
n242)coss...
+(—I)2 (2 cos o)"' (n even); (15)
(I)n2lsin nO =sin B j I—ii22!'' cos2B(n212) (232) 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
n2
(I) 2 sin no=cos0[sin"'o(n2)sin"3o+(n3)2jn4)sin"SB... +()r(n—r—I)(n —rI—2)...(n—2r)s.nn tr1BI...,(n even); (17)
nI
(S I) 2 cos no =the same series (n odd) ; (18)
sin n9=cos 0) n sin 0(n222) 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 Tsin2o+(nsI24(nz32)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. Vieta obtained formulae for chords of multiple arcs in powers of chords of the simple or complementary arcs equivalent to the formulae (13) and (19) above. These are contained in his work Theoremata ad angulares sectiones. Jacques Bernoulli found formulae.equivalent to (12) and (13) (Mem. de l' Academic des Sciences, 1702), and transformed these series into a form equivalent to (10) and (II). Jean Bernoulli published in the Acta eruditorum for 1701, among other formulae already found by Vieta, one equivalent to (17). These formulae have been extended to cases in which n is fractional, negative or irrational ; see a paper by D. F. Gregory in Camb. Math. Journ. vol. iv., in which the series for cos nO, sin 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 Euler in a memoir in the Nova acta, vol. ix., by Lagrange in his Calcul des fonctions (r8o6), and by Poinsot in Recherches sur l'analyse des sections angulaires (1825).
24. The 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 logarithm of any complex or real quantity is a multiplevalued 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 Morgan's Trigonometry and Double Algebra, ch. iv., and a paper by Professor Cayley in vol. ii. of Proc. London
Math. Soc.
25. We have from the definitions given in § 21, cos 0)= 3(eY+eY) and sin cy=3i(eYeY). The expressions, 3(eY+eY), i(eYeY) are said to define the hyperbolic cosine and sine of y and are written cosh y, sinh y; thus cosh y=cos cy, sinh y=c sin Ly. The functions cosh y, sinh y are connected with the rectangular hyperbola in a manner analogous to that in which the cosine andsine are connected with the circle. We may easily show from the definitions that
cos2(x+0)) +sin2(x+cy) = I,
cosh2 ysinh2 y=1;
cos(x+Ly) =cos x cosh yi sin x sinh y,
sin(x+cy) =sin x cosh y+c cos x sinh y, cosh(a+ll) =cosh a cosh 13+sinh a sinh
sinh (a+ =sinh a cosh 13+cosh a sinh j.
These formulae are the basis of a complete hyperbolic trigonometry. The connexion of these functions with the hyperbola was first pointed out by Lambert.
26. if we equate the coefficients of n on both sides of equation (13), this process requiring, however, a justification of its validity,
we get Expansion
when x lies between = I.
By equating the coefficients of n2 on both sides of equation (12) we get
g2 = sine 0+ sin' 8+22.4 sin' 8+2.4.6 sin' B+
3 2 3.5 3 3.5 7 4 which may also be written in the form
_ 2x4 24x•8 2.4.6 xe
(arc sin x)2  x2+ 3 2 +3.5 3 +3.5.7 4 +' ' '
when x is between =I. Differentiating this equation with regard to x, we get
 xz=x{3x'+3—5xs+3.5.7x7+... ,
if we put arc sin x=arc tan y, this equation becomes 2 2 2. 2 2
arc tany=ly I+ I+y3.5 1+y2) +... . (23)
This equation was given with two proofs by Euler in the Nova acta for 1793•
It can be shown that if mod x< 1, then for any such real or complex: value of x, a value of log, (1+x) is given by the sum of the series x' x2/2 +x3/3  ..
We then have
I log 1+x =x+~3+x5, x+... ; Gregory's
2 I x 3 5 7 Series. put cy for x, the left side then becomes log (I+0))log (Iry)) or c arc tan y=cnir;
y3 ys y7 hence arc tan y=n7r=y3 +5 7+..
The series is convergent if y lies between= ; if we suppose arc tan y restricted to values between = 47r, we have
3 6
arc tan y=y3+5..., (24)
which is Gregory's series.
Various series derived from (24) have been employed to calculate the value of 7r. At the end of the 17th century a was calculated to 72 places of decimals by Abraham Sharp, by means of the series obtained by putting arc tan y=7r/6, Series for y=1hl3 in (24). The calculation is to be found in Calculation Sherwin's Mathematical Tables (1742). About the same of n.
time J. Machin employed the series obtained from the equation 4 arc tan J arc tan 234 ='i7r to calculate 2r to loo decimal places. 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 Paris Academy in 1719. G. 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. 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 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. 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 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 infinite products of rational factors. We have
Factorize sin x=2 sin 2sinx 2 =23 sinsinx
don of Sine 4
and Cosine. x+2r . x+3r
sin sm
4 4
proceeding continually in this way with each factor, we obtain
x x{r x+2r x+nIr
sin x =2"1 sin nsin n sin n ... sin n
where n is any positive integral power of 2. Now sinx+rrsinx+nrr x+rr rrx r
II n sin n sin n n r
=sin2 sin2n, x+ins x
_
sin n =cos.
Hence the above may be written
sinx=2n1 sine (sing nsin2n) (sing 91 sin2 rxL) ' .' ( 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 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. (RI) 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/r2)(I x2/22r2) (Ix2/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 3r2x
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 (2nd1)
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 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 ;xcot x, we obtain
cosec xxxnrx I rrx+2r+x I2r—x+3r—x13r+ (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+3r2x 3r+2x+5r2x 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. 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 211I 2n+l' ..
If we put n=3, these become
r=3113 (I2+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) = (14,r) 2) ()
cos x=2 sin x P (1_ x I 3 rr I5 r .. nEr2)
^=m
4x2 (26) =P O I(2n+i)2r2)
If we 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 differential Functions. 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("+2)+(4n+2)y(*+I) ~Yt*)=o.
Therefore y.= — 2 —y%y,,, ; = — 2(2n + 1) y(„+,4xyC„+2)
hence 2NIT 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 (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 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) lzV (I —y2) = a 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
