An Elementary Treatise on Trigonometry; With Its Different Applications

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1825 Excerpt: ...with these equations, cot iA+cot4 Acot A--....arfi«/: = cosec.-fcosec. 2A....adinf. From the above we can demonstrate a very elegant Geometrical Theorem, which we shall have occasion for in the sequel. If in a circle, a regular polygon of an odd number of sides be inscribed, and if lines be drawn from any point in the circumference to the several angles of the polygon, then denoting these lines in their order by the characters c', c", c'", c,v, &c, it may be proved that c/+c'//+cv+&c. = c"+c,v+cv', &c. Let A be half the arc subtended by the side c', and B half that subtended by a side of the polygon, let the number of the sides of the polygon be 2ra+l, then B = g J.: c"'=2. sin (A + 2B) = i 1 2. sin f A + 7,--r. ) Hence half the sum of the odd chords which we shall call S, is represented by the series sin A + m A + 27+1;+ sm U +27+I A + X--T7)-sin Similarly we have S', (half the sum of the even chords) In (Fig. 16.) let o be the point whence the chords are drawn; mvs=2B; v the middle point of mvs; andors=2; then the arc ov--2 A--J--S.'. the chord ovzi 2. sin (A----)andrv= tan--, whence the difference of the two systems of chords is a 4th proportional to Cv, vr, and vo. Several of these series are to be found in the excellent collection of examples on the calculus of finite differences, by Mr. Herchell, and several of them are due to Mr. Wallace, who communicated them to the Royal Society of Edinburgh, in the year 1808. We have them here deduced, perhaps for the first time, by Trigonometrical Artifice. Wallace used some of them as formulae of approximation to the arc of a circle, (when continued to infinity,) to which purpose their rapid convergence, even in the most unfavourable cases, well adapts them.--Fide Art. 6. of ...