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

(45.) Series for the construction of tables,

A few terms of these series will generally be sufficient, as it

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.

m13

7213

+ + + +

IL Y LY LULU

+

m21

m25

m3

+

× 0,018688650277330

× 0,001842475203510 + × 0,000197580071520

× 0,000021697737325 +1 × 0,000002401136991

* 0,000000266413303 + × 0,000000029586468

x0,000000003286788 +

m15

n15 m19

m23

0,000000000365175

×0,000000000040754+ × 0,000000000004508

X0,000000000000501 + X 0,000000000000056.

n

-=

m

m3

m13

m

7211

m15

× 0,000000076495882

n

13×0,000000004759788

715
m19

× 0,000000000296905

X 0,000000000001158

= log10m+log 10 (2 n − m)+log10(2n+m) — 3 log10m

which will render the calculation independent of logarithmic tables.

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

log10 sin a = log10 sin 2a - log1, cos a +9,698970004336019.

10

(Enc. Met. Trig. §. 10.)

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

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

log1 Sin ~=log1n + 4.6855749 — (10 — log1 Cos x);

10

log1 Tan x=log1n + 4.6855749 + (10 – log1 Cos x);

10

10

n being the value of x in seconds and decimal parts of seconds.

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

(49.) Formula for the verification of tables.

sin x + sin (36° - x) + sin (72° + x) = sin (36° + x) + sin (72° — ∞).

(Euler, Anal. Inf. V. 1. p. 201.)

(Leg. 40.)

sin (90° — x) = sin (54° + ∞) + sin (54° — x)

- sin (18+) - sin (18°-x).

cos (36° + x) + cos (36°—x) = cos x + cos (72° + x) + cos (72°—x):

this is only a particular case of the more general theorem

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.

or x=√q tan 10, or -q cot 10.

The equations x2+px+q, and x2-px-q=0, have respectively the same roots as the above forms, but with contrary (C. 810-23; L. 426; Hind, Trig. 285.)

signs.

x = ± 2 (§ q)13. cot 24.

[2] x3−qxFr=0, and 4 q3 < 27 r2.

Assume sin 0:

=

2

(3)',

x= ±

and tan = (tan), then
2(q)*
sin 24