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Therefore 2 sine® A-1-cos 2 A
Siñe 3 A=3 sine A-4 sine' A.(K. 123.)
Hence 4 sines A=3 sine A-sine 3 A
Cos 4A=8 sine+ A-8 sine A+1 (0. 123.)
Consequently 8 sinea=cos 4A+8 sine A-1=
Cos 4A-4 cos 2A +3, by substituting for 8 sine A its value
4-4 cos 2A, obtained from the equation 2 sine A:

1-cos 2A, and by following the same method, we shall obtain

Sine A=sine A
Sine A=1-4 cos 2A
Sine A= sine A-4 sine 3A
Sinet A=-cos 2A +1 cos 4A
Sines A=1: sine A-sine 3A +16 sine 5A
Sine A=;-*cos 2A + cos 4A-3' cos 6A
Sine? A=#sine A-4 sine 3A +67 sine 5A-7's sine

7A, &c. 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. 124. in the same manner as above.

Cos A=COS A
Cose A=} + { cos 2A
Cos® A=&COS A + 4 cos 3A
Cost A= + cos 2A +} cos 4A
Coss A=1: COS A tio cos 3A +cos 5A
Cos A=%+cos 2A + cos 4A + cos 6A

Cos? A=HCOS A+*+cos3A+cos 5A +o's cos 7A, &c. The law of continuation is the same as S. 125, only all the terms here are positive.

(U) The sine and cosine of any arc, or the sine and cosine of any multiple of that arc, may also be derived by substituting

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z, in the exponential expression e=1+ 1 +

this+

+

+

+, &c.

1•2•3•4

Where e is the number whose hyperbolic logarithm is 1.* The addition of the two new equations, obtained by substitution, if divided by 2, will give a series expressing the sine; and the subtraction, if divided by 2 V-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, er 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; it 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 cr are supposed nearly to coincide, so that ts may be considered as a straight line, the

tst a right angle, and the str= LCTA. Lastly, the sectors com and cst are similar. Hence we deduce the following proportions :

(W) cm : cP::mo : on, hence mo=(cm.on)-CP
(X) cm : ct::mo : Ts, or, Ts=(cr.mo)+cm
(Y) CA=cm ; ct::TS : Tt, hence mo=(cm.Tt) - CT?
(Z) CA : AT::TS : st, hence mo=(cm2.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 354. Vince's Trigonometry, pro, 27. and 28. page 79. and 80. 2d edite

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tities. If, therefore, we put z=the arc Am, om will be represented by ž, and if At=t, Pm=X, AP=v, and ct=s, then rt will be represented by t, no by ić, VP (=nm) by ö, and st by s, consequently

ri ri I.

(W. 126. and M. 104.). ; CP

Nr2 22 t

g2 t II. Ź =

(Y. 126. and P. 105.) po2 + t2

go2 § gol s III. Ż

(Z. 126. and N. 104.) AT. CT

SNS22

riv IV.2 =

(A. 126. and L. 111.) Рт

V2ru-02 (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.*

ri I. Thus 2 =

3x4 3.5.26

+ Nr go

2r3

2.4.75 2.4.6°g? 3.5.728

zl 3.xti 3.5.2 &c.)=i+

357•c 2.4.6.8

+

+ 2r2

2.40r4 2.4.6r6' 2.4:6:8.789

23 the fluent of which is x = x+

3.25

3:50x7
to-
2.3 •? 2-4.5.4

+

2.4.6.7.96 3:5•72009 &c. and by reversion of the series X = 2

23 2.4.6.8.9.7.89

23.7% 25 +

&c. 2.9.4:50p4 2-3-4-5-6.7°pobe rot 227

t? t4 II. Ż

tttt r2+t2

.22
gob

202 toi

731 t5
&c. the fluent of which is z=t-

+
372 574

225 1727 ,6229 by reverting the series t=%+

+
-t

&c. 3r! 1574 315762835788 In a similar manner the rest

may

be found. But 7p? - x?cosine z; rad -cos =versed sine z; 22

22 =sec %,

cosec z. tang 3

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&c.;

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(

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cot %;

COS

sine z

sine3 A

+

+

+

(D) Hence, if Asany arc, and r=radius, then

3 sines A 3.5•şine? A 3:5•7•sine A A = sine At

2.372

2.4.5974 2.4.6776 2.4.6.8.9p8? &c.

AY Sine A = A

2.3.p? 2.3.4.5or 2.3.4.5.6•7•.6+

A3

AS

+

A9

A

A8

&c. 2.3.4.5.6.7.8.9789

The same series will apply to the chord of any arc by substituting the diameter d, or 2r instead of r.*

(E) If 90°- A be substituted for A, then we shall obtain 90°-A=sine (90°- A)+ sine3 (90°- A), 3 sine (90° — A)

-,&.c.; 2•3•gel

2.4.5.904

COS3 A 3 cosS A 3.5 cos? A hence =90° — COS A

m&c.; or, 2.3.22 2.4.5°94 2.4.6736 g -COS A73

COS3 A 3 25 cos5 A 3.5 A+ +

+
1
2.3.72 2.4.5

r4

2.4.6.7. 907 - cos? A

&c. zoo A?

A4 Cosaar +

+ 2r 2.3.4•p3 2.3.4.5.6°rs

tang3 A tangs A tangy a tang' A (F) A=tang A

&c. 3.2

9r8 (C. 127.)

A3 2A5

17A7 62A9 Hence tang A=A+

Зr2

1574 315r6 2835r155925r10 218844 A 13 +.

&c.t 6081075r19 ge2 ? 24

ző A= -t

+

&c. cot A 3 cot3 A

5 cots A 7 cot? A 9 cot' A A A3 2A5

A7

2A9 Cot A =

А*

3 4572 945r4 472576 9355578 1382A11

&c. 6385128757109

2-3-4-5-6-7-8077;&c.

+

+

574

7,6

1382A11

+

+

+

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A3 * Viz, chord A=A

hence the difference between an arc and its chord,

2:3-4p2' the radius being 1, is the 24th part of the cube of the length of the arc nearly.

24 See note, Chap. IV. (G. 83.)

+ Emerson's Trigonometry, 2d edit. page 32, &c.

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&c. or,

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22
git
37.6

3.5708
A=

+

+ coseca

2.9.cosec A' 24.5 cosecS A 2:4•6•7•cosec? A &c. A? 5A 61 A 6

27749 Sec Art

50521A 10 +

+
.t

+ 2r

720r5 806477 9628800rg 540553A12

&c. 95800320r17 22 A 7A3 31A5

127A? +

+ A 6 360r2 1512084 604800r8 73A9

&c. 3421440r 8

vers A 3 verso A 9.5 vers: A (H) A=»2r. vers A

1+ +

3.22 op ' 23.4.5.ro 24.4.6•7p3 &c.

Cosec A =

+

+

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Vers A=

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A8

&c. 2r 2.3.4.r3 2.3.4.5.6.75 2.3.4.5.6.7.8077

A 3
A5

A?
Covers A=r - A+

+
2•3•p 2 2.3.4.5•p4 2.3.4.5.6.776
Ao

&c. 2.3.4.5.6•7.8.9 •p 8

THE CONSTRUCTION OF A TABLE OF SÍNES, &c. (I) There are various methods of constructing a table of sines, but the following, though not the least laborious, is the most simple. Sine 30°= | radius=cos 60° (K. 31.)

rad
Sine 15°=} chord 300= N2 - N 3

2

rad Sine 71°=} chord 15o =

2

.

2 These sines are found thus. From the square of the radius subtract the square of the sine, the square root of the difference is the cosine(M.104). Theradius diminished by the cosine leaves the versed sine (R.105); and verso + sine’=chord (T. 105), the half of which is the sine of the next arc, &c.

From the sine of 70.30', find the sine of 3o.45', and so on continually, till the sines are as the arcs, which will be found at the end of the twelfth division from 30°: that is, at the end the twelfth division, the arc will be 52.44" 3"".45"", and

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4-V2-W6,&c.

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