[3] Assume tan a = cos A. tan b; -log Tan a = log Cos A + log Tan b — log r : log Cos Blog Cos a + log Cos a-log Cos b: then caß, under the same conditions as above. Sixth Case: given A, B, a. To determine b: log Sin blog Sin a + log Sin B — log Sin A. The above remarks on sin a sin b sin A may here be applied to sin B C, and c may be determined as above; or thus, log Sinẞ=log Sin a + log Cos A – log Cos B : then Caẞ, as above. Assume tan a tan a. cos B, log Sin Blog Sina + log Cos A - log Cos B ; then caẞ, as above. (W. Ch. xi; C. Ch. xviii; L. 221-62: Legendre, Trig. 84-91; Delambre, Astron. 138-83.) (28.) Series expressing the inverse circular functions. 1 x3 1.3 25 1.3.5 x7 + 4813 33.5.7 33.52.7 35.5.7.11 15 36.59.7.11.13 3617x 37.5.7.11.13.17 &c. B (C. 266—302.) + + &c: 2 3 2.4 5 2.4.6 7 (29.) Series for determining the value of π. T=3,1415926535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 0938446, &c. (Lacroix, Introd. Diff. Calc. 43-5.) (30.) Formulæ involving impossible quantities. sin nx= 1 (cosa±√1 sin x)TM {(cos +/-1 sin x)" — (cos æ—√/—1 sin x)"} · ́If (a + b√ = 1)8+n√ Cos c +/-1 Sin c, b = and - = tana, then c=gx+h.log (a2 + b2), a log, 1=2(m −1) π-1; m being any integer. Of these values only one is possible, which is obtained by putting m=1. (Lacr. Introd. 81.) (31.) Formula for the sums of arcs, and multiple arcs. Let the sum of the continued products of each combination of the sines of m of the arcs, a1, α, &c. a,, into the cosines of the remaining n-m arcs, be represented by Cn-mSm; then sin (a1 + a2+ &c. + an) = Cn− C n − 2 S2 + Cn−4S4- &c. cos (a1 + a2+ &c. + an) = Cn-1 S1 − Cn - 3S3 + &c. Let T be the sum of the continued products of each combination of m tangents, to (n + 1) terms, if n is odd, and to in terms, if n is even. to (n + 1) terms, if n is odd, and to n+1 terms, if n is even. |