(44.) Lengths of arcs in terms of radius, and their logarithms. 1° =0,017453292520; log10 1° +10=8,2418773675. (Enc. Met. V. I, p. 672.) (45.) Series for the construction of tables, m m sin n n + + + n9 m13 m15 mi na 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 + x 0,000025202042373 + x 0,000000471087478 n10 m14 n12 714 X0,000000006386603 + n16 m 20 720 m18 n18 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 m? x 0,000197580071520 n7 x 0,000021697737325 + 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 n m cot n 2 m m x 0,6866197723675813 x 0,3183098861837907 - .'. 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 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 m 226 mlo 9210 m 12 2212 m18 m 20 log 10 cos = log10 (n- m) + log10 (n+m)- 2 log10 n m4 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 {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.) + + (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 2 x14 36172016 + &c. 24.37.54.72.11.13.17 2010 &c.} + 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 929569x1 + &c. 62x 12728 x= + + + (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 n COS 2 + cos (2- +00) + cos (4 - + x) + &c. + &c.] +cos (2 - X) + cos — «) + &c. ) T (4 -{ 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. x= dp tan 0.tan 1 o, or – šptan 0.cot 10; or x=Vq tan 10, or – Vq cot . The equations 202 + pæ+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. |