&c. Sine 3A 3 sine A 4 sine3 A. Sine 4A (4 sine A-8 sine3 A)√√1 —sine2 ▲ Sine 6A (6 sine A-32 sine3 A+ 32 sine' A)/1-sine▲, (L) Sine A=√1-cos2 A OR, Sine 2A 2 Cos A/1-cos2 A Sine 3A (4 cos2 A-1)/1-cos2 A Sine 4A=(8 cos3 A-4 cos a)√/1-cos2 A Sine 5A (16 cos4A-12 cos2 A+1)/1-cos2 A Sine 6A(32 coss A-32cos3 A+6cos A)/1-cos2 A,&c. (COS B.COS A)-(sine A sine B) (M) Cosine (A+B)= rad -(D.109). Now if в 2A, and radius=1 as before, we shall obtain (1-cos 2A).cos A (P. 102.)=cos A-(cos A.cos 2A) therefore Cos 3 A(cos 2A.cos A) - COS A+(cos A.COS 2A)=(2 cos A.cos 2A)-COS A. And by making B=3A, and pursuing the same method, we shall find cos 4A=2 cos A.cos 3 A-cos 2A. (N) *Cos A=cos A Cos 2A=2 cos A.cos A I Cos 3 A=2 cos A.cos 2A - cos A Cos 4▲ 2 cos A.cos 3A-cos 2 A Cos 5A-2 cos A.cos 4A-cos 3A Cos 6A=2 cos A.cos 5A-cos 4A, &c. OR, (0) By substituting the value of cos A=√1—sine2 A, cos 2A, found above, &c. +Cos A=1- sine2 A Cos 2A=1-2 sine2 A Cos 3A (1-4 sine2 A)√ 1 — sine2 A Cos 4A 8 sine+ A-8 sine2 ▲+1 Cos 5A Cos 6A (16 sine* A-12 sine2 A+1)1-sine2 A 1-18 sine2 A+ 48 sine+ A-32 sine A, &c. (P) Cos A=cos A OR, * Cagnoli, page 27. Legendre, page 358. Cos Cos 4A8 cos4 A-8 cos2 A+1 Cos 5A16 cos5 A-20 cos3 A+ 5 cos A Cos 6A 32 cos A - 48 cos^ A+ 18 cos2 A-1, = tang A.tang B rad2-(tang A.tang B) (Q) Tang (A+B): Now if B=24, and rad stitution, and tang 3A= *Tang A=tang A &c. (C.109,or K.112.) 2 tang A by sub -Hence, =1; tang 2A= 3 tang A-tang3 A 1-3 tang2 A Also if 1 sec A 4 cot3 A-4 cot A 5 cot A-10 cot2 A+1 &c. be substituted for cos A (N. 117.) the secant of the multiple of any arc may be obtained. OF THE SINES AND COSINES OF THE POWERS OF ARCS.t (S) Cos 2A=1-2 sine2 A (0.117.) *Cagnoli, page 29. Crakelt's translation of Mauduit, page 34. Therefore Therefore 2 sine2 A=1-COS 2 A Sine 3A 3 sine A-4 sine3 A (K. 117.) Cos 4A 8 sine1 A-8 sine2 A+1 (O. 117.) Consequently 8 sine' Acos 4A+8 sine2 A-1= Cos 4A-4 cos 2A +3, by substituting for 8 sine2 A its value 4-4 cos 2A, obtained from the equation 2 sine2 A= 1-cos 2A, and by following the same method, we shall obtain cos 4A 5 76 6 Sine A 35 7A, &c. I 7 64 sine The law of continuation is obvious, for the odd powers are expressed in terms of the sines, and the even powers in terms of the cosines of the multiples of A; and the signs are alternately + and The numerators of the co-efficients (reckoning from the right hand towards the left), are the co-efficients of a binomial whose power is the same as that of the sine of A; except in the even powers, where the term in which a is not found, has the numerator of its co-efficient only one half of the corresponding co-efficient of the binomial, and the denominators are 2 involved to the power of the sine −1. (T) To deduce formulæ for the successive powers of the cosine of any arc, we must apply P. 118, in the same manner as above. Cos5 A= IO Cos? A I I 5 6 32 COSA +24 COS 3A+ cos 5A + 64 cos 6A cos 7A, &c. The law of continuation is the same as S. 119, only all the terms here are positive. of (U) The sine and cosine of any arc, or the sine and cosine any multiple of that arc, may also be derived by substituting the imaginary quantities -1 and-x√-1, successively for Z3 z, in the exponential expression =1+ + + it 1.2 1.2.3 Z4 1.2.34 + &c. Where è is the number whose hyperbolic logarithm is 1* The addition of the two new equations, obtained by substitu tion, if divided by 2, will give a series expressing the sine; and the subtraction, if divided by 2-1, will give a series for the cosine of any arc. THE DETERMINATION OF THE VALUE OF THE SINE AND OF THE COSINE, &c. OF ANY ARC, IN TERMS OF THAT ARC. PROPOSITION XXIII. (Plate IV. Fig. 1.) To determine the increment of an arc, in terms of the increment of the sine, tangent, secant, &c.; and thence to deduce several useful formula. Let Am be any arc; Pm its sine, CP its cosine, AT its tangent, CT its secant, &c. Take the arc mo indefinitely small, draw onv parallel to pm, and mn parallel to AC; also, from the centre c, with the radius CT, describe the arc ST. Then, mo is the increment of the arc am; mn is the decrement of the cosine, or the increment of the versed sine; on is the increment of the sine; Tt the increment of the tangent, and st the increment of the secant. The triangles mno, and cpm, are equiangular and similar; for the arc mo, being extremely small, may be considered as a straight line. Likewise the triangles CAT and Tst are equiangular and similar, for the lines ct and CT are supposed nearly to coincide, so that Ts may be considered as a straight line, the Tst a right angle, and the 4 stT=4CTA. Lastly the sectors com and CST are similar. Hence we deduce the following proportions. (W) cm: CP:: mo: on, hence mo= (cm.on)+CP =(cm2.Tt)÷CT3 (Z) CA: AT:: Ts: st, hence mo=(cm3.st)÷AT.CT (A) Pm: cm::mn: mo, hence mo⇒(cm.mn)÷Pm (B) Now it is shewn by writers on fluxions, that the limiting ratio of the cotemporary increments or decrements of any two quantities, will be the ratio of the fluxions of those quan ❤ Éléments de Géométrie, par A. M. Legendre, 6th edit. page 654. Vince's Trigonometry, krop. 27 and 28, page 79 and 80, 2d edit. tities. tities. If, therefore, we put z=the arc am, om will be represented by ż, and if AT=t, Pm=x, AP=v, and CT=s, then Ti will be represented by t, no by , vr(nm) by v, and st by s, consequently rx CP rx II. III. z= (Z. 120, and N. 98.) √2rv - v2 (A. 120, and L. 98.) (C) If these formulæ be expanded, and the fluents of each term be taken, we shall obtain the common series for the arc, in terms of its sine, tangent, &c.* 3x4 3.5.6 :+ + 2r3 2·4°r5 2.4.6.7 x2x 3x4x 3.5.xx 3.5.7.xx =rx (=== + &c.; 203 3x5 + 2.3.2 2•4•5r 2·4·6·7·6 3.5.x7 + &c. and by reversion of the series x=x In a similar manner the rest may be found. But &c. and &c. * Vide Baron Maseres' Trigonometry, page 424, &c. R (D) Hence, |