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(44.) Lengths of arcs in terms of radius, and their logarithms.

1° =0,017453292520; log10 1° +10=8,2418773675.
1'=0,000290888209; log10 1' +10=6,4637261171.
1"=0,000004848137; logio 1"+10=4,6855748667.
18 =0,0157079632679; log10 16 +10=8,1961198769.
log10 T=0,4971498726.

(Enc. Met. V. I, p. 672.)

(45.) Series for the construction of tables,

m

m

sin

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n

n

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+ + +

n9

m13

m15

mi
X 1,570796326794827 x0,645964097506246

na
ms

m? x0,079692626246167

7 x 0,004681754135319 mo x 0,000160441184787

mli X0,000003598843235 X0,000000056921729

215 X0,000000000668804 x 0,0000000000006067

215 X 0,000000000000044. ma 1 x 1,233700550136170 n2

x0,253669507901048 m6

m8 X 0,020863480763353 + x 0,000919260274839 n6

713 m 17 n17

m 19

+

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x 0,000025202042373 +

x 0,000000471087478

n10

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m14

n12 714 X0,000000006386603 +

n16 m 20 720

m18 n18

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X0,000000000000529 +

X 0,000000000000003.

A few terms of these series will generally be sufficient, as it will never be necessary to take -> ..

m

n

The sines of the lesser divisions, as minutes, are usually found by the method of differences, which will be explained hereafter.

Series for calculating the tangent and cotangent, independently of the sine and cosine.

2 mn

m tan

n 2

m

n

n5

mil

na-ma

x0,6366197723675813 x 0,297556782059734 +

m3

x0,018688650277330

n3
m5
+ X0,001842475203510 +

m?

x 0,000197580071520

n7
mo
+

x 0,000021697737325 +
no

n11

x0,000002401136991 + 213 * 0,000000266413303 + x 0,000000029586468 m17 X0,000000003286788 +

n19

X0,000000000365175 + mir *0,000000000040754 +

7223

X 0,000000000004508

m27 + 2 *0,000000000000501 + x 0,000000000000056.

n27

m13

m15 715 m 19

+

n17

m21

m 23

m 25

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n

m cot

n

2

m

m

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x 0,6866197723675813
4mn

x 0,3183098861837907 - .'.
4n2m2
*0,205288889414508

m3
m3

x 0,006551074788218 ms

m? x 0,00034529255397

X0,000020279106052

m7 mo

m11 * 0,000001236652718

nii X0,000000076495882

ml 513 X0,000000004759738 775 X 0,000000000296905 717 *0,000000000018541 x 0,000000000001158

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log10 sin

= log10m+log10 (2n-m) + log10 (2n+m)-3 log 10 m n .2

+9,594059885702190 m2

m4 x 0,070022826605902 x 0,001117266441662 na

m8 x 0,000039229146454 x 0,000001729270798

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m

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226 mlo 9210

m 12

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2212

m18

m 20

log 10 cos = log10 (n- m) + log10 (n+m)- 2 log10 n
n 2

m4
X0,101494859341893
n2

x 0,003187294065451

n4 mo

m8 x 0,000209485800017

x 0,000016848348598

n8 x 0,000001480193987

X0,000000136502272 m14

m16 214 X0,000000012981715

2016 *0,000000001261471 218 0,000000000124567

720

x 0,000000000012456 722 *0,000000000001258 X0,000000000000128. If m is small, log (1

m.) may be expanded into the series

ma - 2m

+}

(2*)*+&c.

2n? - m? which will render the calculation independent of logarithmic tables.

m22

m24
n21

{21

If we find the logarithmic cosines of arcs < 45°, and the logarithmic sines of arcs between 45°, and 90°, the rest may be found from the formula log., sin a=log, sin 2a - logo cos a +9,698970004336019.

(Enc. Met. Trig. . 10.)

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(47.) Logarithmic series for the sine, cosine, and tangent. r 02

206 log, sin x=log, a

+ 12.3 22.3.5

34.5.7 23.33.59.7
691 212

2 x14
+
34.52.7.11 + 2.39.53.72.11.13 * 39.52.72.11.13

36172016
+

+ &c. 24.37.54.72.11.13.17

2010

&c.}

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+ 2

14

16

+

206 1708 31 x 10 log, cos x =

+ +

+ 22.3 32.5 23.3.5.7

23.3.5.7 34.52.7
691 r12 10922 r?

929569x1
+
+

+ &c.
2.39.52.7.11 39.52.7.11.13 24.36.53.79.11.13

62x

12728 x=

+

+
13 2.3.5 * 34.5.7 * 2.3.52.7
146x10 1414477212 32764.224
+
+

+
39.52.11 * 37.53.79.11.13* 36.52.72.11.13
16931177 2:26

(C. 405_-7.)

+

2.3”.64.7.11.13.17+ &c.}

(48.) To find the logarithmic sines and tangents of small arcs.

logo Sin x=log.on +4.6855749 - } (10 – logio Cosw);

log.. Tan x=logion + 4.6855749 + (10 – logi. Cos x); n being the value of w in seconds and decimal parts of seconds.

(Taylor, Log. Introd. p. 17.)

(49.) Formulæ for the verification of tables. sin x + sin (36o — «) + sin (72° + x)=sin (36° +w) + sin (72o — «).

(Euler, Anal. Inf. V. 1. p. 201.) sin (90° — x)=sin (54° + 2) + sin (54° —- )

- sin (18° + w) - sin (18° — ). (Leg. 40.) cos (36° + x) + cos (360—~)=COS X + cos (72° + x)+cos (72°—2): this is only a particular case of the more general theorem

T

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n

COS 2 + cos (2- +00) + cos (4 - + x) + &c.

+ &c.] +cos (2 - X) + cos — «) + &c. )

T

(4

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-{

n

n

+) + cos (3 - +®) + &c.

" + c. + cos - X) + cos (3

-~) + &c. in which each series is to be continued until the angle attains its greatest value next below 90°. (Enc. Met. V. 1. p. 695.)

TRIGONOMETRICAL SOLUTION OF EQUATIONS. (50.) Quadratic equations. [1] x? — px +q=0. Assume (sin e)' = 49; then

x=p (sin ( 0)?, or p (cos 10)”;

or x=vq. tan 10, or Vq.cot 10. [2] 2? +px-q=0.

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x= dp tan 0.tan 1 o, or – šptan 0.cot 10; or x=Vq tan 10, or – Vq cot .

The equations 202 + +q, and w-pa-q=0, have respectively the same roots as the above forms, but with contrary signs.

(C. 810_-23; L. 426; Hind, Trig. 285.) (51.) Cubic equations. [1] 03 + qFr=0.

2 Assume tan =

9

p=10

+2({9)".cot 20. [2] 203 — qx Fr=0, and 4 q* < 27 pl.

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